From 8a2bd8ccb42775eb611405df15106d64464cbd71 Mon Sep 17 00:00:00 2001
From: Aleks Kissinger <aleks0@gmail.com>
Date: Sun, 22 Dec 2024 15:59:49 +0000
Subject: [PATCH] updated demo notebook

---
 demos/PauliWebs.ipynb | 4461 ++++++++++++++++++++++++++++++++++++++---
 pyzx/pauliweb.py      |   56 +-
 2 files changed, 4174 insertions(+), 343 deletions(-)

diff --git a/demos/PauliWebs.ipynb b/demos/PauliWebs.ipynb
index 4a71e074..af93c289 100644
--- a/demos/PauliWebs.ipynb
+++ b/demos/PauliWebs.ipynb
@@ -33,7 +33,7 @@
     "4. output vertices have two webs `zweb[v]` and `xweb[v]` corresponding to both colors at `v`\n",
     "5. for non-Pauli spiders `v, w`, if `v` anti-commutes with the Pauli web of `w` then `order[v] < order[w]`\n",
     "\n",
-    "Under the hood, this is using PyZX's gflow-finding algorithm to compute a focussed gflow and translate this data into Pauli webs."
+    "Under the hood, this is using PyZX's gflow-finding algorithm to compute a focussed Pauli flow and translate this data into Pauli webs."
    ]
   },
   {
@@ -51,7 +51,7 @@
     {
      "data": {
       "text/html": [
-       "<div style=\"overflow:auto; background-color: white\" id=\"graph-output-khqP2ICQ\"></div>\n",
+       "<div style=\"overflow:auto; background-color: white\" id=\"graph-output-RzjAwSWd\"></div>\n",
        "<script type=\"module\">\n",
        "var d3;\n",
        "if (d3 == null) { d3 = await import(\"https://cdn.skypack.dev/d3@5\"); }\n",
@@ -414,7 +414,7 @@
        "}\n",
        "\n",
        "\n",
-       "showGraph('#graph-output-khqP2ICQ',\n",
+       "showGraph('#graph-output-RzjAwSWd',\n",
        "  JSON.parse('{\"nodes\": [{\"name\": \"0\", \"x\": 50, \"y\": 100, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"1\", \"x\": 50, \"y\": 150, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"2\", \"x\": 100, \"y\": 150, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"3\", \"x\": 100, \"y\": 100, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"4\", \"x\": 150, \"y\": 100, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"5\", \"x\": 150, \"y\": 150, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}], \"links\": [{\"source\": \"0\", \"target\": \"3\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"1\", \"target\": \"2\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"2\", \"target\": \"3\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"2\", \"target\": \"5\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"3\", \"target\": \"4\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}], \"pauli_web\": []}'), 200, 200, 50,\n",
        "  10.0, false, true, '');\n",
        "</script>"
@@ -447,7 +447,7 @@
     {
      "data": {
       "text/html": [
-       "<div style=\"overflow:auto; background-color: white\" id=\"graph-output-Fvju4MEV\"></div>\n",
+       "<div style=\"overflow:auto; background-color: white\" id=\"graph-output-nQKbr5VU\"></div>\n",
        "<script type=\"module\">\n",
        "var d3;\n",
        "if (d3 == null) { d3 = await import(\"https://cdn.skypack.dev/d3@5\"); }\n",
@@ -810,8 +810,8 @@
        "}\n",
        "\n",
        "\n",
-       "showGraph('#graph-output-Fvju4MEV',\n",
-       "  JSON.parse('{\"nodes\": [{\"name\": \"0\", \"x\": 50, \"y\": 100, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"1\", \"x\": 50, \"y\": 150, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"2\", \"x\": 100, \"y\": 150, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"3\", \"x\": 100, \"y\": 100, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"4\", \"x\": 150, \"y\": 100, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"5\", \"x\": 150, \"y\": 150, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}], \"links\": [{\"source\": \"0\", \"target\": \"3\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"1\", \"target\": \"2\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"2\", \"target\": \"3\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"2\", \"target\": \"5\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"3\", \"target\": \"4\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}], \"pauli_web\": [{\"source\": 3, \"target\": 0, \"t\": \"X\"}, {\"source\": 0, \"target\": 3, \"t\": \"X\"}, {\"source\": 3, \"target\": 2, \"t\": \"X\"}, {\"source\": 2, \"target\": 3, \"t\": \"X\"}, {\"source\": 3, \"target\": 4, \"t\": \"X\"}, {\"source\": 4, \"target\": 3, \"t\": \"X\"}, {\"source\": 2, \"target\": 1, \"t\": \"X\"}, {\"source\": 1, \"target\": 2, \"t\": \"X\"}]}'), 200, 200, 50,\n",
+       "showGraph('#graph-output-nQKbr5VU',\n",
+       "  JSON.parse('{\"nodes\": [{\"name\": \"0\", \"x\": 50, \"y\": 100, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"1\", \"x\": 50, \"y\": 150, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"2\", \"x\": 100, \"y\": 150, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"3\", \"x\": 100, \"y\": 100, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"4\", \"x\": 150, \"y\": 100, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"5\", \"x\": 150, \"y\": 150, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}], \"links\": [{\"source\": \"0\", \"target\": \"3\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"1\", \"target\": \"2\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"2\", \"target\": \"3\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"2\", \"target\": \"5\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"3\", \"target\": \"4\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}], \"pauli_web\": [{\"source\": 1, \"target\": 2, \"t\": \"X\"}, {\"source\": 2, \"target\": 1, \"t\": \"X\"}, {\"source\": 3, \"target\": 0, \"t\": \"X\"}, {\"source\": 0, \"target\": 3, \"t\": \"X\"}, {\"source\": 3, \"target\": 2, \"t\": \"X\"}, {\"source\": 2, \"target\": 3, \"t\": \"X\"}, {\"source\": 3, \"target\": 4, \"t\": \"X\"}, {\"source\": 4, \"target\": 3, \"t\": \"X\"}]}'), 200, 200, 50,\n",
        "  10.0, false, true, '');\n",
        "</script>"
       ],
@@ -825,7 +825,7 @@
     {
      "data": {
       "text/html": [
-       "<div style=\"overflow:auto; background-color: white\" id=\"graph-output-NLJZKxsn\"></div>\n",
+       "<div style=\"overflow:auto; background-color: white\" id=\"graph-output-lVO5zvXU\"></div>\n",
        "<script type=\"module\">\n",
        "var d3;\n",
        "if (d3 == null) { d3 = await import(\"https://cdn.skypack.dev/d3@5\"); }\n",
@@ -1188,8 +1188,8 @@
        "}\n",
        "\n",
        "\n",
-       "showGraph('#graph-output-NLJZKxsn',\n",
-       "  JSON.parse('{\"nodes\": [{\"name\": \"0\", \"x\": 50, \"y\": 100, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"1\", \"x\": 50, \"y\": 150, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"2\", \"x\": 100, \"y\": 150, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"3\", \"x\": 100, \"y\": 100, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"4\", \"x\": 150, \"y\": 100, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"5\", \"x\": 150, \"y\": 150, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}], \"links\": [{\"source\": \"0\", \"target\": \"3\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"1\", \"target\": \"2\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"2\", \"target\": \"3\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"2\", \"target\": \"5\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"3\", \"target\": \"4\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}], \"pauli_web\": [{\"source\": 5, \"target\": 2, \"t\": \"X\"}, {\"source\": 2, \"target\": 5, \"t\": \"X\"}, {\"source\": 2, \"target\": 1, \"t\": \"X\"}, {\"source\": 1, \"target\": 2, \"t\": \"X\"}]}'), 200, 200, 50,\n",
+       "showGraph('#graph-output-lVO5zvXU',\n",
+       "  JSON.parse('{\"nodes\": [{\"name\": \"0\", \"x\": 50, \"y\": 100, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"1\", \"x\": 50, \"y\": 150, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"2\", \"x\": 100, \"y\": 150, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"3\", \"x\": 100, \"y\": 100, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"4\", \"x\": 150, \"y\": 100, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"5\", \"x\": 150, \"y\": 150, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}], \"links\": [{\"source\": \"0\", \"target\": \"3\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"1\", \"target\": \"2\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"2\", \"target\": \"3\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"2\", \"target\": \"5\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"3\", \"target\": \"4\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}], \"pauli_web\": [{\"source\": 1, \"target\": 2, \"t\": \"X\"}, {\"source\": 2, \"target\": 1, \"t\": \"X\"}, {\"source\": 5, \"target\": 2, \"t\": \"X\"}, {\"source\": 2, \"target\": 5, \"t\": \"X\"}]}'), 200, 200, 50,\n",
        "  10.0, false, true, '');\n",
        "</script>"
       ],
@@ -1203,7 +1203,7 @@
     {
      "data": {
       "text/html": [
-       "<div style=\"overflow:auto; background-color: white\" id=\"graph-output-MxsbpzGq\"></div>\n",
+       "<div style=\"overflow:auto; background-color: white\" id=\"graph-output-0SXRbHvE\"></div>\n",
        "<script type=\"module\">\n",
        "var d3;\n",
        "if (d3 == null) { d3 = await import(\"https://cdn.skypack.dev/d3@5\"); }\n",
@@ -1566,8 +1566,8 @@
        "}\n",
        "\n",
        "\n",
-       "showGraph('#graph-output-MxsbpzGq',\n",
-       "  JSON.parse('{\"nodes\": [{\"name\": \"0\", \"x\": 50, \"y\": 100, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"1\", \"x\": 50, \"y\": 150, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"2\", \"x\": 100, \"y\": 150, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"3\", \"x\": 100, \"y\": 100, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"4\", \"x\": 150, \"y\": 100, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"5\", \"x\": 150, \"y\": 150, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}], \"links\": [{\"source\": \"0\", \"target\": \"3\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"1\", \"target\": \"2\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"2\", \"target\": \"3\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"2\", \"target\": \"5\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"3\", \"target\": \"4\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}], \"pauli_web\": [{\"source\": 4, \"target\": 3, \"t\": \"Z\"}, {\"source\": 3, \"target\": 4, \"t\": \"Z\"}, {\"source\": 3, \"target\": 0, \"t\": \"Z\"}, {\"source\": 0, \"target\": 3, \"t\": \"Z\"}]}'), 200, 200, 50,\n",
+       "showGraph('#graph-output-0SXRbHvE',\n",
+       "  JSON.parse('{\"nodes\": [{\"name\": \"0\", \"x\": 50, \"y\": 100, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"1\", \"x\": 50, \"y\": 150, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"2\", \"x\": 100, \"y\": 150, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"3\", \"x\": 100, \"y\": 100, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"4\", \"x\": 150, \"y\": 100, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"5\", \"x\": 150, \"y\": 150, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}], \"links\": [{\"source\": \"0\", \"target\": \"3\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"1\", \"target\": \"2\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"2\", \"target\": \"3\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"2\", \"target\": \"5\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"3\", \"target\": \"4\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}], \"pauli_web\": [{\"source\": 4, \"target\": 3, \"t\": \"Z\"}, {\"source\": 3, \"target\": 4, \"t\": \"Z\"}, {\"source\": 0, \"target\": 3, \"t\": \"Z\"}, {\"source\": 3, \"target\": 0, \"t\": \"Z\"}]}'), 200, 200, 50,\n",
        "  10.0, false, true, '');\n",
        "</script>"
       ],
@@ -1581,7 +1581,7 @@
     {
      "data": {
       "text/html": [
-       "<div style=\"overflow:auto; background-color: white\" id=\"graph-output-AgXbcai7\"></div>\n",
+       "<div style=\"overflow:auto; background-color: white\" id=\"graph-output-1hwCBFeb\"></div>\n",
        "<script type=\"module\">\n",
        "var d3;\n",
        "if (d3 == null) { d3 = await import(\"https://cdn.skypack.dev/d3@5\"); }\n",
@@ -1944,8 +1944,8 @@
        "}\n",
        "\n",
        "\n",
-       "showGraph('#graph-output-AgXbcai7',\n",
-       "  JSON.parse('{\"nodes\": [{\"name\": \"0\", \"x\": 50, \"y\": 100, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"1\", \"x\": 50, \"y\": 150, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"2\", \"x\": 100, \"y\": 150, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"3\", \"x\": 100, \"y\": 100, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"4\", \"x\": 150, \"y\": 100, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"5\", \"x\": 150, \"y\": 150, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}], \"links\": [{\"source\": \"0\", \"target\": \"3\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"1\", \"target\": \"2\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"2\", \"target\": \"3\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"2\", \"target\": \"5\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"3\", \"target\": \"4\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}], \"pauli_web\": [{\"source\": 2, \"target\": 1, \"t\": \"Z\"}, {\"source\": 1, \"target\": 2, \"t\": \"Z\"}, {\"source\": 2, \"target\": 3, \"t\": \"Z\"}, {\"source\": 3, \"target\": 2, \"t\": \"Z\"}, {\"source\": 2, \"target\": 5, \"t\": \"Z\"}, {\"source\": 5, \"target\": 2, \"t\": \"Z\"}, {\"source\": 3, \"target\": 0, \"t\": \"Z\"}, {\"source\": 0, \"target\": 3, \"t\": \"Z\"}]}'), 200, 200, 50,\n",
+       "showGraph('#graph-output-1hwCBFeb',\n",
+       "  JSON.parse('{\"nodes\": [{\"name\": \"0\", \"x\": 50, \"y\": 100, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"1\", \"x\": 50, \"y\": 150, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"2\", \"x\": 100, \"y\": 150, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"3\", \"x\": 100, \"y\": 100, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"4\", \"x\": 150, \"y\": 100, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"5\", \"x\": 150, \"y\": 150, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}], \"links\": [{\"source\": \"0\", \"target\": \"3\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"1\", \"target\": \"2\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"2\", \"target\": \"3\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"2\", \"target\": \"5\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"3\", \"target\": \"4\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}], \"pauli_web\": [{\"source\": 2, \"target\": 1, \"t\": \"Z\"}, {\"source\": 1, \"target\": 2, \"t\": \"Z\"}, {\"source\": 2, \"target\": 3, \"t\": \"Z\"}, {\"source\": 3, \"target\": 2, \"t\": \"Z\"}, {\"source\": 2, \"target\": 5, \"t\": \"Z\"}, {\"source\": 5, \"target\": 2, \"t\": \"Z\"}, {\"source\": 0, \"target\": 3, \"t\": \"Z\"}, {\"source\": 3, \"target\": 0, \"t\": \"Z\"}]}'), 200, 200, 50,\n",
        "  10.0, false, true, '');\n",
        "</script>"
       ],
@@ -1958,7 +1958,7 @@
     }
    ],
    "source": [
-    "# There are 4 interesting Pauli webs corresponding to the two outputs\n",
+    "# There are 4 Pauli webs corresponding to the two outputs\n",
     "zx.draw(g, labels=True, pauli_web=xwebs[4])\n",
     "zx.draw(g, labels=True, pauli_web=xwebs[5])\n",
     "zx.draw(g, labels=True, pauli_web=zwebs[4])\n",
@@ -1973,7 +1973,7 @@
     {
      "data": {
       "text/html": [
-       "<div style=\"overflow:auto; background-color: white\" id=\"graph-output-1Hx4Khea\"></div>\n",
+       "<div style=\"overflow:auto; background-color: white\" id=\"graph-output-tyWMaC9E\"></div>\n",
        "<script type=\"module\">\n",
        "var d3;\n",
        "if (d3 == null) { d3 = await import(\"https://cdn.skypack.dev/d3@5\"); }\n",
@@ -2336,8 +2336,8 @@
        "}\n",
        "\n",
        "\n",
-       "showGraph('#graph-output-1Hx4Khea',\n",
-       "  JSON.parse('{\"nodes\": [{\"name\": \"0\", \"x\": 50, \"y\": 100, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"1\", \"x\": 50, \"y\": 150, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"2\", \"x\": 100, \"y\": 150, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"3\", \"x\": 100, \"y\": 100, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"4\", \"x\": 150, \"y\": 100, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"5\", \"x\": 150, \"y\": 150, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}], \"links\": [{\"source\": \"0\", \"target\": \"3\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"1\", \"target\": \"2\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"2\", \"target\": \"3\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"2\", \"target\": \"5\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"3\", \"target\": \"4\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}], \"pauli_web\": [{\"source\": 3, \"target\": 0, \"t\": \"Z\"}, {\"source\": 0, \"target\": 3, \"t\": \"Z\"}]}'), 200, 200, 50,\n",
+       "showGraph('#graph-output-tyWMaC9E',\n",
+       "  JSON.parse('{\"nodes\": [{\"name\": \"0\", \"x\": 50, \"y\": 100, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"1\", \"x\": 50, \"y\": 150, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"2\", \"x\": 100, \"y\": 150, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"3\", \"x\": 100, \"y\": 100, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"4\", \"x\": 150, \"y\": 100, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"5\", \"x\": 150, \"y\": 150, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}], \"links\": [{\"source\": \"0\", \"target\": \"3\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"1\", \"target\": \"2\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"2\", \"target\": \"3\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"2\", \"target\": \"5\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"3\", \"target\": \"4\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}], \"pauli_web\": [{\"source\": 3, \"target\": 0, \"t\": \"X\"}, {\"source\": 0, \"target\": 3, \"t\": \"X\"}, {\"source\": 3, \"target\": 2, \"t\": \"X\"}, {\"source\": 2, \"target\": 3, \"t\": \"X\"}, {\"source\": 3, \"target\": 4, \"t\": \"X\"}, {\"source\": 4, \"target\": 3, \"t\": \"X\"}, {\"source\": 5, \"target\": 2, \"t\": \"X\"}, {\"source\": 2, \"target\": 5, \"t\": \"X\"}]}'), 200, 200, 50,\n",
        "  10.0, false, true, '');\n",
        "</script>"
       ],
@@ -2351,7 +2351,7 @@
     {
      "data": {
       "text/html": [
-       "<div style=\"overflow:auto; background-color: white\" id=\"graph-output-JQ1pt4qy\"></div>\n",
+       "<div style=\"overflow:auto; background-color: white\" id=\"graph-output-tBTcFFGN\"></div>\n",
        "<script type=\"module\">\n",
        "var d3;\n",
        "if (d3 == null) { d3 = await import(\"https://cdn.skypack.dev/d3@5\"); }\n",
@@ -2714,8 +2714,3845 @@
        "}\n",
        "\n",
        "\n",
-       "showGraph('#graph-output-JQ1pt4qy',\n",
-       "  JSON.parse('{\"nodes\": [{\"name\": \"0\", \"x\": 50, \"y\": 100, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"1\", \"x\": 50, \"y\": 150, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"2\", \"x\": 100, \"y\": 150, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"3\", \"x\": 100, \"y\": 100, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"4\", \"x\": 150, \"y\": 100, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"5\", \"x\": 150, \"y\": 150, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}], \"links\": [{\"source\": \"0\", \"target\": \"3\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"1\", \"target\": \"2\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"2\", \"target\": \"3\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"2\", \"target\": \"5\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"3\", \"target\": \"4\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}], \"pauli_web\": [{\"source\": 2, \"target\": 1, \"t\": \"X\"}, {\"source\": 1, \"target\": 2, \"t\": \"X\"}]}'), 200, 200, 50,\n",
+       "showGraph('#graph-output-tBTcFFGN',\n",
+       "  JSON.parse('{\"nodes\": [{\"name\": \"0\", \"x\": 50, \"y\": 100, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"1\", \"x\": 50, \"y\": 150, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"2\", \"x\": 100, \"y\": 150, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"3\", \"x\": 100, \"y\": 100, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"4\", \"x\": 150, \"y\": 100, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"5\", \"x\": 150, \"y\": 150, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}], \"links\": [{\"source\": \"0\", \"target\": \"3\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"1\", \"target\": \"2\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"2\", \"target\": \"3\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"2\", \"target\": \"5\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"3\", \"target\": \"4\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}], \"pauli_web\": [{\"source\": 1, \"target\": 2, \"t\": \"X\"}, {\"source\": 2, \"target\": 1, \"t\": \"X\"}, {\"source\": 5, \"target\": 2, \"t\": \"X\"}, {\"source\": 2, \"target\": 5, \"t\": \"X\"}]}'), 200, 200, 50,\n",
+       "  10.0, false, true, '');\n",
+       "</script>"
+      ],
+      "text/plain": [
+       "<IPython.core.display.HTML object>"
+      ]
+     },
+     "metadata": {},
+     "output_type": "display_data"
+    },
+    {
+     "data": {
+      "text/html": [
+       "<div style=\"overflow:auto; background-color: white\" id=\"graph-output-lvtxDfgY\"></div>\n",
+       "<script type=\"module\">\n",
+       "var d3;\n",
+       "if (d3 == null) { d3 = await import(\"https://cdn.skypack.dev/d3@5\"); }\n",
+       "// PyZX - Python library for quantum circuit rewriting \n",
+       "//        and optimisation using the ZX-calculus\n",
+       "// Copyright (C) 2018 - Aleks Kissinger and John van de Wetering\n",
+       "\n",
+       "// Licensed under the Apache License, Version 2.0 (the \"License\");\n",
+       "// you may not use this file except in compliance with the License.\n",
+       "// You may obtain a copy of the License at\n",
+       "\n",
+       "//    http://www.apache.org/licenses/LICENSE-2.0\n",
+       "\n",
+       "// Unless required by applicable law or agreed to in writing, software\n",
+       "// distributed under the License is distributed on an \"AS IS\" BASIS,\n",
+       "// WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.\n",
+       "// See the License for the specific language governing permissions and\n",
+       "// limitations under the License.\n",
+       "\n",
+       "// styling functions\n",
+       "function nodeColor(t) {\n",
+       "    if (t == 0) return \"black\";\n",
+       "    else if (t == 1) return \"#ccffcc\";\n",
+       "    else if (t == 2) return \"#ff8888\";\n",
+       "    else if (t == 3) return \"yellow\";\n",
+       "    else if (t == 4) return \"black\";\n",
+       "    else if (t == 5) return \"black\";\n",
+       "    else if (t == 6) return \"#ccffcc\";\n",
+       "}\n",
+       "\n",
+       "function edgeColor(t) {\n",
+       "    if (t == 1) return \"black\";\n",
+       "    else if (t == 2) return \"#08f\";\n",
+       "    else if (t == 3) return \"gray\";\n",
+       "}\n",
+       "\n",
+       "function webColor(t) {\n",
+       "    if (t == 'X') return \"#ff8888\";\n",
+       "    else if (t == 'Y') return \"#eebb00\";\n",
+       "    else if (t == 'Z') return \"#66cc66\";\n",
+       "    else if (t == 'I') return \"#eee\";\n",
+       "}\n",
+       "\n",
+       "function nodeStyle(selected) {\n",
+       "    return selected ? \"stroke-width: 2px; stroke: #00f\" : \"stroke-width: 1.5px\";\n",
+       "}\n",
+       "\n",
+       "var symbolGround = {\n",
+       "    draw: function(context, size){\n",
+       "        let s = size/2;\n",
+       "\n",
+       "        context.moveTo(0,-s);\n",
+       "        context.lineTo(0,0);\n",
+       "\n",
+       "        context.moveTo(-s,0);\n",
+       "        context.lineTo(s,0);\n",
+       "\n",
+       "        context.moveTo(-2*s/3,s/3);\n",
+       "        context.lineTo(2*s/3,s/3);\n",
+       "\n",
+       "        context.moveTo(-s/3,2*s/3);\n",
+       "        context.lineTo(s/3,2*s/3);\n",
+       "    }\n",
+       "}\n",
+       "\n",
+       "function showGraph(tag, graph, width, height, scale, node_size, auto_hbox, show_labels, scalar_str) {\n",
+       "    var ntab = {};\n",
+       "\n",
+       "    var groundOffset = 2.5 * node_size;\n",
+       "\n",
+       "    graph.nodes.forEach(function(d) {\n",
+       "        ntab[d.name] = d;\n",
+       "        d.selected = false;\n",
+       "        d.previouslySelected = false;\n",
+       "        d.nhd = [];\n",
+       "    });\n",
+       "\n",
+       "    var spiders_and_boundaries = graph.nodes.filter(function(d) {\n",
+       "        return d.t != 3;\n",
+       "    });\n",
+       "\n",
+       "    graph.links.forEach(function(d) {\n",
+       "        var s = ntab[d.source];\n",
+       "        var t = ntab[d.target];\n",
+       "        d.source = s;\n",
+       "        d.target = t;\n",
+       "        s.nhd.push(t);\n",
+       "        t.nhd.push(s);\n",
+       "    });\n",
+       "\n",
+       "    graph.pauli_web.forEach(function(d) {\n",
+       "        var s = ntab[d.source];\n",
+       "        var t = ntab[d.target];\n",
+       "        d.source = s;\n",
+       "        d.target = t;\n",
+       "    });\n",
+       "\n",
+       "    var shiftKey;\n",
+       "\n",
+       "    // SETUP SVG ITEMS\n",
+       "\n",
+       "    var svg = d3.select(tag)\n",
+       "    //.attr(\"tabindex\", 1)\n",
+       "        .on(\"keydown.brush\", function() {shiftKey = d3.event.shiftKey || d3.event.metaKey;})\n",
+       "        .on(\"keyup.brush\", function() {shiftKey = d3.event.shiftKey || d3.event.metaKey;})\n",
+       "    //.each(function() { this.focus(); })\n",
+       "        .append(\"svg\")\n",
+       "        .attr(\"style\", \"max-width: none; max-height: none\")\n",
+       "        .attr(\"width\", width)\n",
+       "        .attr(\"height\", height);\n",
+       "\n",
+       "    var web = svg.append(\"g\")\n",
+       "        .attr(\"class\", \"web\")\n",
+       "        .selectAll(\"line\")\n",
+       "        .data(graph.pauli_web)\n",
+       "        .enter().append(\"path\")\n",
+       "        .attr(\"stroke\", function(d) { return webColor(d.t); })\n",
+       "        .attr(\"fill\", \"transparent\")\n",
+       "        .attr(\"style\", \"stroke-width: 7px\");\n",
+       "\n",
+       "    var link = svg.append(\"g\")\n",
+       "        .attr(\"class\", \"link\")\n",
+       "        .selectAll(\"line\")\n",
+       "        .data(graph.links)\n",
+       "        .enter().append(\"path\")\n",
+       "        .attr(\"stroke\", function(d) { return edgeColor(d.t); })\n",
+       "        .attr(\"fill\", \"transparent\")\n",
+       "        .attr(\"style\", \"stroke-width: 1.5px\");\n",
+       "\n",
+       "    var brush = svg.append(\"g\")\n",
+       "        .attr(\"class\", \"brush\");\n",
+       "\n",
+       "    var node = svg.append(\"g\")\n",
+       "        .attr(\"class\", \"node\")\n",
+       "        .selectAll(\"g\")\n",
+       "        .data(graph.nodes)\n",
+       "        .enter().append(\"g\")\n",
+       "        .attr(\"transform\", function(d) {\n",
+       "            return \"translate(\" + d.x + \",\" + d.y +\")\";\n",
+       "        });\n",
+       "\n",
+       "    // Draw a ground symbol connected to the node.\n",
+       "    node.filter(function(d) { return d.ground; })\n",
+       "        .append(\"path\")\n",
+       "        .attr(\"stroke\", \"black\")\n",
+       "        .attr(\"style\", \"stroke-width: 1.5px\")\n",
+       "        .attr(\"fill\", \"none\")\n",
+       "        .attr(\"d\", \"M 0 0 L 0 \"+(groundOffset))\n",
+       "        .attr(\"class\", \"selectable\");\n",
+       "    node.filter(function(d) { return d.ground; })\n",
+       "        .append(\"path\")\n",
+       "        .attr(\"stroke\", \"black\")\n",
+       "        .attr(\"style\", \"stroke-width: 1.5px\")\n",
+       "        .attr(\"fill\", \"none\")\n",
+       "        .attr(\"d\", d3.symbol().type(symbolGround).size(node_size*1.5))\n",
+       "        .attr(\"transform\", \"translate(0,\"+groundOffset+\")\")\n",
+       "        .attr(\"class\", \"selectable\");\n",
+       "\n",
+       "    node.filter(function(d) { return d.t != 3 && d.t != 5 && d.t != 6; })\n",
+       "        .append(\"circle\")\n",
+       "        .attr(\"r\", function(d) {\n",
+       "            if (d.t == 0) return 0.5 * node_size;\n",
+       "            else if (d.t == 4) return 0.25 * node_size;\n",
+       "            else return node_size;\n",
+       "        })\n",
+       "        .attr(\"fill\", function(d) { return nodeColor(d.t); })\n",
+       "        .attr(\"stroke\", \"black\")\n",
+       "        .attr(\"class\", \"selectable\");\n",
+       "\n",
+       "    var hbox = node.filter(function(d) { return d.t == 3; });\n",
+       "\n",
+       "    hbox.append(\"rect\")\n",
+       "        .attr(\"x\", -0.75 * node_size).attr(\"y\", -0.75 * node_size)\n",
+       "        .attr(\"width\", node_size * 1.5).attr(\"height\", node_size * 1.5)\n",
+       "        .attr(\"fill\", function(d) { return nodeColor(d.t); })\n",
+       "        .attr(\"stroke\", \"black\")\n",
+       "        .attr(\"class\", \"selectable\");\n",
+       "\n",
+       "    // draw a triangle for d.t == 5\n",
+       "    node.filter(function(d) { return d.t == 5; })\n",
+       "        .append(\"path\")\n",
+       "        .attr(\"d\", \"M 0 0 L \"+node_size+\" \"+node_size+\" L -\"+node_size+\" \"+node_size+\" Z\")\n",
+       "        .attr(\"fill\", function(d) { return nodeColor(d.t); })\n",
+       "        .attr(\"stroke\", \"black\")\n",
+       "        .attr(\"class\", \"selectable\")\n",
+       "        .attr(\"transform\", \"translate(\" + (-node_size/2) + \", 0) rotate(-90)\");\n",
+       "\n",
+       "    // draw a square for Z box: d.t == 6\n",
+       "    node.filter(function(d) { return d.t == 6; })\n",
+       "        .append(\"rect\")\n",
+       "        .attr(\"x\", -0.75 * node_size).attr(\"y\", -0.75 * node_size)\n",
+       "        .attr(\"width\", node_size * 1.5).attr(\"height\", node_size * 1.5)\n",
+       "        .attr(\"fill\", function(d) { return nodeColor(d.t); })\n",
+       "        .attr(\"stroke\", \"black\")\n",
+       "        .attr(\"class\", \"selectable\");\n",
+       "\n",
+       "    node.filter(function(d) { return d.phase != ''; })\n",
+       "        .append(\"text\")\n",
+       "        .attr(\"y\", 0.7 * node_size + 14)\n",
+       "        .text(function (d) { return d.phase })\n",
+       "        .attr(\"text-anchor\", \"middle\")\n",
+       "        .attr(\"font-size\", \"12px\")\n",
+       "        .attr(\"font-family\", \"monospace\")\n",
+       "        .attr(\"fill\", \"#00d\")\n",
+       "        .attr('style', 'pointer-events: none; user-select: none;');\n",
+       "\n",
+       "    if (show_labels) {\n",
+       "        node.append(\"text\")\n",
+       "            .attr(\"y\", -0.7 * node_size - 8)\n",
+       "            .text(function (d) { return d.name; })\n",
+       "            .attr(\"text-anchor\", \"middle\")\n",
+       "            .attr(\"font-size\", \"10px\")\n",
+       "            .attr(\"font-family\", \"monospace\")\n",
+       "            .attr(\"fill\", \"#999\")\n",
+       "            .attr('style', 'pointer-events: none; user-select: none;');\n",
+       "    }\n",
+       "\n",
+       "    // Display the chosen data fields over the node.\n",
+       "    node.filter(d => d.vdata.length > 0)\n",
+       "        .append(\"text\")\n",
+       "        .attr(\"y\", d => -0.7 * node_size - 14 - 10 * d.vdata.length)\n",
+       "        .attr(\"text-anchor\", \"middle\")\n",
+       "        .attr(\"font-size\", \"8px\")\n",
+       "        .attr(\"font-family\", \"monospace\")\n",
+       "        .attr(\"fill\", \"#c66\")\n",
+       "        .attr('style', 'pointer-events: none; user-select: none;')\n",
+       "        .selectAll(\"tspan\")\n",
+       "        .data(d => d.vdata)\n",
+       "        .enter()\n",
+       "        .append(\"tspan\")\n",
+       "        .attr(\"x\", \"0\")\n",
+       "        .attr(\"dy\", \"1.2em\")\n",
+       "        .text(x => x.join(\": \"));\n",
+       "\n",
+       "    if (scalar_str != \"\") {\n",
+       "        svg.append(\"text\")\n",
+       "            .text(scalar_str)\n",
+       "            .attr(\"x\", 60).attr(\"y\", 40)\n",
+       "            .attr(\"text-anchor\", \"middle\")\n",
+       "    }\n",
+       "\n",
+       "    function update_hboxes() {\n",
+       "        if (auto_hbox) {\n",
+       "            var pos = {};\n",
+       "            hbox.attr(\"transform\", function(d) {\n",
+       "                // calculate barycenter of non-hbox neighbours, then nudge a bit\n",
+       "                // to the NE.\n",
+       "                var x=0,y=0,sz=0;\n",
+       "                for (var i = 0; i < d.nhd.length; ++i) {\n",
+       "                    if (d.nhd[i].t != 3) {\n",
+       "                        sz++;\n",
+       "                        x += d.nhd[i].x;\n",
+       "                        y += d.nhd[i].y;\n",
+       "                    }\n",
+       "                }\n",
+       "\n",
+       "                offset = 0.25 * scale;\n",
+       "\n",
+       "                if (sz != 0) {\n",
+       "                    x = (x/sz) + offset;\n",
+       "                    y = (y/sz) - offset;\n",
+       "\n",
+       "                    while (pos[[x,y]]) {\n",
+       "                        x += offset;\n",
+       "                    }\n",
+       "                    d.x = x;\n",
+       "                    d.y = y;\n",
+       "                    pos[[x,y]] = true;\n",
+       "                }\n",
+       "\n",
+       "                return \"translate(\"+d.x+\",\"+d.y+\")\";\n",
+       "            });\n",
+       "        }\n",
+       "    }\n",
+       "\n",
+       "    update_hboxes();\n",
+       "\n",
+       "    var link_curve = function(d) {\n",
+       "        var x1 = d.source.x, x2 = d.target.x, y1 = d.source.y, y2 = d.target.y;\n",
+       "        if (d.num_parallel == 1) {\n",
+       "            return `M ${x1} ${y1} L ${x2} ${y2}`;\n",
+       "        } else {\n",
+       "            var dx = x2 - x1, dy = y2 - y1;\n",
+       "            var midx = 0.5 * (x1 + x2), midy = 0.5 * (y1 + y2);\n",
+       "            var pos = (d.index / (d.num_parallel-1)) - 0.5;\n",
+       "            var cx = midx - pos * dy;\n",
+       "            var cy = midy + pos * dx;\n",
+       "            return `M ${x1} ${y1} Q ${cx} ${cy}, ${x2} ${y2}`;\n",
+       "            // return `M ${x1} ${y1} L ${x2} ${y2}`;\n",
+       "        }\n",
+       "    };\n",
+       "    link.attr(\"d\", link_curve);\n",
+       "\n",
+       "\n",
+       "    var web_curve = function(d) {\n",
+       "        var x1 = d.source.x, x2 = (x1 + d.target.x)/2, y1 = d.source.y, y2 = (y1 + d.target.y)/2;\n",
+       "        return `M ${x1} ${y1} L ${x2} ${y2}`;\n",
+       "    }\n",
+       "    web.attr(\"d\", web_curve);\n",
+       "\n",
+       "    // EVENTS FOR DRAGGING AND SELECTION\n",
+       "\n",
+       "    node.on(\"mousedown\", function(d) {\n",
+       "        if (shiftKey) {\n",
+       "            d3.select(this).selectAll(\".selectable\").attr(\"style\", nodeStyle(d.selected = !d.selected));\n",
+       "            d3.event.stopImmediatePropagation();\n",
+       "        } else if (!d.selected) {\n",
+       "            node.selectAll(\".selectable\").attr(\"style\", function(p) { return nodeStyle(p.selected = d === p); });\n",
+       "        }\n",
+       "    })\n",
+       "        .call(d3.drag().on(\"drag\", function(d) {\n",
+       "            var dx = d3.event.dx;\n",
+       "            var dy = d3.event.dy;\n",
+       "            // node.filter(function(d) { return d.selected; })\n",
+       "            //     .attr(\"cx\", function(d) { return d.x += dx; })\n",
+       "            //     .attr(\"cy\", function(d) { return d.y += dy; });\n",
+       "            node.filter(function(d) { return d.selected; })\n",
+       "                .attr(\"transform\", function(d) {\n",
+       "                    d.x += dx;\n",
+       "                    d.y += dy;\n",
+       "                    return \"translate(\" + d.x + \",\" + d.y +\")\";\n",
+       "                });\n",
+       "\n",
+       "            update_hboxes();\n",
+       "\n",
+       "            link.filter(function(d) { return d.source.selected || d.target.selected ||\n",
+       "                    (auto_hbox && d.source.t == 3); })\n",
+       "                .attr(\"d\", link_curve);\n",
+       "            web.filter(function(d) { return d.source.selected || d.target.selected; })\n",
+       "                .attr(\"d\", web_curve);\n",
+       "        }));\n",
+       "\n",
+       "    brush.call(d3.brush().keyModifiers(false)\n",
+       "        .extent([[0, 0], [width, height]])\n",
+       "        .on(\"start\", function() {\n",
+       "            if (d3.event.sourceEvent.type !== \"end\") {\n",
+       "                node.selectAll(\".selectable\").attr(\"style\", function(d) {\n",
+       "                    return nodeStyle(\n",
+       "                        d.selected = d.previouslySelected = shiftKey &&\n",
+       "                        d.selected);\n",
+       "                });\n",
+       "            }\n",
+       "        })\n",
+       "        .on(\"brush\", function() {\n",
+       "            if (d3.event.sourceEvent.type !== \"end\") {\n",
+       "                var selection = d3.event.selection;\n",
+       "                node.selectAll(\".selectable\").attr(\"style\", function(d) {\n",
+       "                    return nodeStyle(d.selected = d.previouslySelected ^\n",
+       "                        (selection != null\n",
+       "                            && selection[0][0] <= d.x && d.x < selection[1][0]\n",
+       "                            && selection[0][1] <= d.y && d.y < selection[1][1]));\n",
+       "                });\n",
+       "            }\n",
+       "        })\n",
+       "        .on(\"end\", function() {\n",
+       "            if (d3.event.selection != null) {\n",
+       "                d3.select(this).call(d3.event.target.move, null);\n",
+       "            }\n",
+       "        }));\n",
+       "}\n",
+       "\n",
+       "\n",
+       "showGraph('#graph-output-lvtxDfgY',\n",
+       "  JSON.parse('{\"nodes\": [{\"name\": \"0\", \"x\": 50, \"y\": 100, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"1\", \"x\": 50, \"y\": 150, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"2\", \"x\": 100, \"y\": 150, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"3\", \"x\": 100, \"y\": 100, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"4\", \"x\": 150, \"y\": 100, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"5\", \"x\": 150, \"y\": 150, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}], \"links\": [{\"source\": \"0\", \"target\": \"3\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"1\", \"target\": \"2\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"2\", \"target\": \"3\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"2\", \"target\": \"5\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"3\", \"target\": \"4\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}], \"pauli_web\": [{\"source\": 4, \"target\": 3, \"t\": \"Z\"}, {\"source\": 3, \"target\": 4, \"t\": \"Z\"}, {\"source\": 0, \"target\": 3, \"t\": \"Z\"}, {\"source\": 3, \"target\": 0, \"t\": \"Z\"}]}'), 200, 200, 50,\n",
+       "  10.0, false, true, '');\n",
+       "</script>"
+      ],
+      "text/plain": [
+       "<IPython.core.display.HTML object>"
+      ]
+     },
+     "metadata": {},
+     "output_type": "display_data"
+    },
+    {
+     "data": {
+      "text/html": [
+       "<div style=\"overflow:auto; background-color: white\" id=\"graph-output-komyousH\"></div>\n",
+       "<script type=\"module\">\n",
+       "var d3;\n",
+       "if (d3 == null) { d3 = await import(\"https://cdn.skypack.dev/d3@5\"); }\n",
+       "// PyZX - Python library for quantum circuit rewriting \n",
+       "//        and optimisation using the ZX-calculus\n",
+       "// Copyright (C) 2018 - Aleks Kissinger and John van de Wetering\n",
+       "\n",
+       "// Licensed under the Apache License, Version 2.0 (the \"License\");\n",
+       "// you may not use this file except in compliance with the License.\n",
+       "// You may obtain a copy of the License at\n",
+       "\n",
+       "//    http://www.apache.org/licenses/LICENSE-2.0\n",
+       "\n",
+       "// Unless required by applicable law or agreed to in writing, software\n",
+       "// distributed under the License is distributed on an \"AS IS\" BASIS,\n",
+       "// WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.\n",
+       "// See the License for the specific language governing permissions and\n",
+       "// limitations under the License.\n",
+       "\n",
+       "// styling functions\n",
+       "function nodeColor(t) {\n",
+       "    if (t == 0) return \"black\";\n",
+       "    else if (t == 1) return \"#ccffcc\";\n",
+       "    else if (t == 2) return \"#ff8888\";\n",
+       "    else if (t == 3) return \"yellow\";\n",
+       "    else if (t == 4) return \"black\";\n",
+       "    else if (t == 5) return \"black\";\n",
+       "    else if (t == 6) return \"#ccffcc\";\n",
+       "}\n",
+       "\n",
+       "function edgeColor(t) {\n",
+       "    if (t == 1) return \"black\";\n",
+       "    else if (t == 2) return \"#08f\";\n",
+       "    else if (t == 3) return \"gray\";\n",
+       "}\n",
+       "\n",
+       "function webColor(t) {\n",
+       "    if (t == 'X') return \"#ff8888\";\n",
+       "    else if (t == 'Y') return \"#eebb00\";\n",
+       "    else if (t == 'Z') return \"#66cc66\";\n",
+       "    else if (t == 'I') return \"#eee\";\n",
+       "}\n",
+       "\n",
+       "function nodeStyle(selected) {\n",
+       "    return selected ? \"stroke-width: 2px; stroke: #00f\" : \"stroke-width: 1.5px\";\n",
+       "}\n",
+       "\n",
+       "var symbolGround = {\n",
+       "    draw: function(context, size){\n",
+       "        let s = size/2;\n",
+       "\n",
+       "        context.moveTo(0,-s);\n",
+       "        context.lineTo(0,0);\n",
+       "\n",
+       "        context.moveTo(-s,0);\n",
+       "        context.lineTo(s,0);\n",
+       "\n",
+       "        context.moveTo(-2*s/3,s/3);\n",
+       "        context.lineTo(2*s/3,s/3);\n",
+       "\n",
+       "        context.moveTo(-s/3,2*s/3);\n",
+       "        context.lineTo(s/3,2*s/3);\n",
+       "    }\n",
+       "}\n",
+       "\n",
+       "function showGraph(tag, graph, width, height, scale, node_size, auto_hbox, show_labels, scalar_str) {\n",
+       "    var ntab = {};\n",
+       "\n",
+       "    var groundOffset = 2.5 * node_size;\n",
+       "\n",
+       "    graph.nodes.forEach(function(d) {\n",
+       "        ntab[d.name] = d;\n",
+       "        d.selected = false;\n",
+       "        d.previouslySelected = false;\n",
+       "        d.nhd = [];\n",
+       "    });\n",
+       "\n",
+       "    var spiders_and_boundaries = graph.nodes.filter(function(d) {\n",
+       "        return d.t != 3;\n",
+       "    });\n",
+       "\n",
+       "    graph.links.forEach(function(d) {\n",
+       "        var s = ntab[d.source];\n",
+       "        var t = ntab[d.target];\n",
+       "        d.source = s;\n",
+       "        d.target = t;\n",
+       "        s.nhd.push(t);\n",
+       "        t.nhd.push(s);\n",
+       "    });\n",
+       "\n",
+       "    graph.pauli_web.forEach(function(d) {\n",
+       "        var s = ntab[d.source];\n",
+       "        var t = ntab[d.target];\n",
+       "        d.source = s;\n",
+       "        d.target = t;\n",
+       "    });\n",
+       "\n",
+       "    var shiftKey;\n",
+       "\n",
+       "    // SETUP SVG ITEMS\n",
+       "\n",
+       "    var svg = d3.select(tag)\n",
+       "    //.attr(\"tabindex\", 1)\n",
+       "        .on(\"keydown.brush\", function() {shiftKey = d3.event.shiftKey || d3.event.metaKey;})\n",
+       "        .on(\"keyup.brush\", function() {shiftKey = d3.event.shiftKey || d3.event.metaKey;})\n",
+       "    //.each(function() { this.focus(); })\n",
+       "        .append(\"svg\")\n",
+       "        .attr(\"style\", \"max-width: none; max-height: none\")\n",
+       "        .attr(\"width\", width)\n",
+       "        .attr(\"height\", height);\n",
+       "\n",
+       "    var web = svg.append(\"g\")\n",
+       "        .attr(\"class\", \"web\")\n",
+       "        .selectAll(\"line\")\n",
+       "        .data(graph.pauli_web)\n",
+       "        .enter().append(\"path\")\n",
+       "        .attr(\"stroke\", function(d) { return webColor(d.t); })\n",
+       "        .attr(\"fill\", \"transparent\")\n",
+       "        .attr(\"style\", \"stroke-width: 7px\");\n",
+       "\n",
+       "    var link = svg.append(\"g\")\n",
+       "        .attr(\"class\", \"link\")\n",
+       "        .selectAll(\"line\")\n",
+       "        .data(graph.links)\n",
+       "        .enter().append(\"path\")\n",
+       "        .attr(\"stroke\", function(d) { return edgeColor(d.t); })\n",
+       "        .attr(\"fill\", \"transparent\")\n",
+       "        .attr(\"style\", \"stroke-width: 1.5px\");\n",
+       "\n",
+       "    var brush = svg.append(\"g\")\n",
+       "        .attr(\"class\", \"brush\");\n",
+       "\n",
+       "    var node = svg.append(\"g\")\n",
+       "        .attr(\"class\", \"node\")\n",
+       "        .selectAll(\"g\")\n",
+       "        .data(graph.nodes)\n",
+       "        .enter().append(\"g\")\n",
+       "        .attr(\"transform\", function(d) {\n",
+       "            return \"translate(\" + d.x + \",\" + d.y +\")\";\n",
+       "        });\n",
+       "\n",
+       "    // Draw a ground symbol connected to the node.\n",
+       "    node.filter(function(d) { return d.ground; })\n",
+       "        .append(\"path\")\n",
+       "        .attr(\"stroke\", \"black\")\n",
+       "        .attr(\"style\", \"stroke-width: 1.5px\")\n",
+       "        .attr(\"fill\", \"none\")\n",
+       "        .attr(\"d\", \"M 0 0 L 0 \"+(groundOffset))\n",
+       "        .attr(\"class\", \"selectable\");\n",
+       "    node.filter(function(d) { return d.ground; })\n",
+       "        .append(\"path\")\n",
+       "        .attr(\"stroke\", \"black\")\n",
+       "        .attr(\"style\", \"stroke-width: 1.5px\")\n",
+       "        .attr(\"fill\", \"none\")\n",
+       "        .attr(\"d\", d3.symbol().type(symbolGround).size(node_size*1.5))\n",
+       "        .attr(\"transform\", \"translate(0,\"+groundOffset+\")\")\n",
+       "        .attr(\"class\", \"selectable\");\n",
+       "\n",
+       "    node.filter(function(d) { return d.t != 3 && d.t != 5 && d.t != 6; })\n",
+       "        .append(\"circle\")\n",
+       "        .attr(\"r\", function(d) {\n",
+       "            if (d.t == 0) return 0.5 * node_size;\n",
+       "            else if (d.t == 4) return 0.25 * node_size;\n",
+       "            else return node_size;\n",
+       "        })\n",
+       "        .attr(\"fill\", function(d) { return nodeColor(d.t); })\n",
+       "        .attr(\"stroke\", \"black\")\n",
+       "        .attr(\"class\", \"selectable\");\n",
+       "\n",
+       "    var hbox = node.filter(function(d) { return d.t == 3; });\n",
+       "\n",
+       "    hbox.append(\"rect\")\n",
+       "        .attr(\"x\", -0.75 * node_size).attr(\"y\", -0.75 * node_size)\n",
+       "        .attr(\"width\", node_size * 1.5).attr(\"height\", node_size * 1.5)\n",
+       "        .attr(\"fill\", function(d) { return nodeColor(d.t); })\n",
+       "        .attr(\"stroke\", \"black\")\n",
+       "        .attr(\"class\", \"selectable\");\n",
+       "\n",
+       "    // draw a triangle for d.t == 5\n",
+       "    node.filter(function(d) { return d.t == 5; })\n",
+       "        .append(\"path\")\n",
+       "        .attr(\"d\", \"M 0 0 L \"+node_size+\" \"+node_size+\" L -\"+node_size+\" \"+node_size+\" Z\")\n",
+       "        .attr(\"fill\", function(d) { return nodeColor(d.t); })\n",
+       "        .attr(\"stroke\", \"black\")\n",
+       "        .attr(\"class\", \"selectable\")\n",
+       "        .attr(\"transform\", \"translate(\" + (-node_size/2) + \", 0) rotate(-90)\");\n",
+       "\n",
+       "    // draw a square for Z box: d.t == 6\n",
+       "    node.filter(function(d) { return d.t == 6; })\n",
+       "        .append(\"rect\")\n",
+       "        .attr(\"x\", -0.75 * node_size).attr(\"y\", -0.75 * node_size)\n",
+       "        .attr(\"width\", node_size * 1.5).attr(\"height\", node_size * 1.5)\n",
+       "        .attr(\"fill\", function(d) { return nodeColor(d.t); })\n",
+       "        .attr(\"stroke\", \"black\")\n",
+       "        .attr(\"class\", \"selectable\");\n",
+       "\n",
+       "    node.filter(function(d) { return d.phase != ''; })\n",
+       "        .append(\"text\")\n",
+       "        .attr(\"y\", 0.7 * node_size + 14)\n",
+       "        .text(function (d) { return d.phase })\n",
+       "        .attr(\"text-anchor\", \"middle\")\n",
+       "        .attr(\"font-size\", \"12px\")\n",
+       "        .attr(\"font-family\", \"monospace\")\n",
+       "        .attr(\"fill\", \"#00d\")\n",
+       "        .attr('style', 'pointer-events: none; user-select: none;');\n",
+       "\n",
+       "    if (show_labels) {\n",
+       "        node.append(\"text\")\n",
+       "            .attr(\"y\", -0.7 * node_size - 8)\n",
+       "            .text(function (d) { return d.name; })\n",
+       "            .attr(\"text-anchor\", \"middle\")\n",
+       "            .attr(\"font-size\", \"10px\")\n",
+       "            .attr(\"font-family\", \"monospace\")\n",
+       "            .attr(\"fill\", \"#999\")\n",
+       "            .attr('style', 'pointer-events: none; user-select: none;');\n",
+       "    }\n",
+       "\n",
+       "    // Display the chosen data fields over the node.\n",
+       "    node.filter(d => d.vdata.length > 0)\n",
+       "        .append(\"text\")\n",
+       "        .attr(\"y\", d => -0.7 * node_size - 14 - 10 * d.vdata.length)\n",
+       "        .attr(\"text-anchor\", \"middle\")\n",
+       "        .attr(\"font-size\", \"8px\")\n",
+       "        .attr(\"font-family\", \"monospace\")\n",
+       "        .attr(\"fill\", \"#c66\")\n",
+       "        .attr('style', 'pointer-events: none; user-select: none;')\n",
+       "        .selectAll(\"tspan\")\n",
+       "        .data(d => d.vdata)\n",
+       "        .enter()\n",
+       "        .append(\"tspan\")\n",
+       "        .attr(\"x\", \"0\")\n",
+       "        .attr(\"dy\", \"1.2em\")\n",
+       "        .text(x => x.join(\": \"));\n",
+       "\n",
+       "    if (scalar_str != \"\") {\n",
+       "        svg.append(\"text\")\n",
+       "            .text(scalar_str)\n",
+       "            .attr(\"x\", 60).attr(\"y\", 40)\n",
+       "            .attr(\"text-anchor\", \"middle\")\n",
+       "    }\n",
+       "\n",
+       "    function update_hboxes() {\n",
+       "        if (auto_hbox) {\n",
+       "            var pos = {};\n",
+       "            hbox.attr(\"transform\", function(d) {\n",
+       "                // calculate barycenter of non-hbox neighbours, then nudge a bit\n",
+       "                // to the NE.\n",
+       "                var x=0,y=0,sz=0;\n",
+       "                for (var i = 0; i < d.nhd.length; ++i) {\n",
+       "                    if (d.nhd[i].t != 3) {\n",
+       "                        sz++;\n",
+       "                        x += d.nhd[i].x;\n",
+       "                        y += d.nhd[i].y;\n",
+       "                    }\n",
+       "                }\n",
+       "\n",
+       "                offset = 0.25 * scale;\n",
+       "\n",
+       "                if (sz != 0) {\n",
+       "                    x = (x/sz) + offset;\n",
+       "                    y = (y/sz) - offset;\n",
+       "\n",
+       "                    while (pos[[x,y]]) {\n",
+       "                        x += offset;\n",
+       "                    }\n",
+       "                    d.x = x;\n",
+       "                    d.y = y;\n",
+       "                    pos[[x,y]] = true;\n",
+       "                }\n",
+       "\n",
+       "                return \"translate(\"+d.x+\",\"+d.y+\")\";\n",
+       "            });\n",
+       "        }\n",
+       "    }\n",
+       "\n",
+       "    update_hboxes();\n",
+       "\n",
+       "    var link_curve = function(d) {\n",
+       "        var x1 = d.source.x, x2 = d.target.x, y1 = d.source.y, y2 = d.target.y;\n",
+       "        if (d.num_parallel == 1) {\n",
+       "            return `M ${x1} ${y1} L ${x2} ${y2}`;\n",
+       "        } else {\n",
+       "            var dx = x2 - x1, dy = y2 - y1;\n",
+       "            var midx = 0.5 * (x1 + x2), midy = 0.5 * (y1 + y2);\n",
+       "            var pos = (d.index / (d.num_parallel-1)) - 0.5;\n",
+       "            var cx = midx - pos * dy;\n",
+       "            var cy = midy + pos * dx;\n",
+       "            return `M ${x1} ${y1} Q ${cx} ${cy}, ${x2} ${y2}`;\n",
+       "            // return `M ${x1} ${y1} L ${x2} ${y2}`;\n",
+       "        }\n",
+       "    };\n",
+       "    link.attr(\"d\", link_curve);\n",
+       "\n",
+       "\n",
+       "    var web_curve = function(d) {\n",
+       "        var x1 = d.source.x, x2 = (x1 + d.target.x)/2, y1 = d.source.y, y2 = (y1 + d.target.y)/2;\n",
+       "        return `M ${x1} ${y1} L ${x2} ${y2}`;\n",
+       "    }\n",
+       "    web.attr(\"d\", web_curve);\n",
+       "\n",
+       "    // EVENTS FOR DRAGGING AND SELECTION\n",
+       "\n",
+       "    node.on(\"mousedown\", function(d) {\n",
+       "        if (shiftKey) {\n",
+       "            d3.select(this).selectAll(\".selectable\").attr(\"style\", nodeStyle(d.selected = !d.selected));\n",
+       "            d3.event.stopImmediatePropagation();\n",
+       "        } else if (!d.selected) {\n",
+       "            node.selectAll(\".selectable\").attr(\"style\", function(p) { return nodeStyle(p.selected = d === p); });\n",
+       "        }\n",
+       "    })\n",
+       "        .call(d3.drag().on(\"drag\", function(d) {\n",
+       "            var dx = d3.event.dx;\n",
+       "            var dy = d3.event.dy;\n",
+       "            // node.filter(function(d) { return d.selected; })\n",
+       "            //     .attr(\"cx\", function(d) { return d.x += dx; })\n",
+       "            //     .attr(\"cy\", function(d) { return d.y += dy; });\n",
+       "            node.filter(function(d) { return d.selected; })\n",
+       "                .attr(\"transform\", function(d) {\n",
+       "                    d.x += dx;\n",
+       "                    d.y += dy;\n",
+       "                    return \"translate(\" + d.x + \",\" + d.y +\")\";\n",
+       "                });\n",
+       "\n",
+       "            update_hboxes();\n",
+       "\n",
+       "            link.filter(function(d) { return d.source.selected || d.target.selected ||\n",
+       "                    (auto_hbox && d.source.t == 3); })\n",
+       "                .attr(\"d\", link_curve);\n",
+       "            web.filter(function(d) { return d.source.selected || d.target.selected; })\n",
+       "                .attr(\"d\", web_curve);\n",
+       "        }));\n",
+       "\n",
+       "    brush.call(d3.brush().keyModifiers(false)\n",
+       "        .extent([[0, 0], [width, height]])\n",
+       "        .on(\"start\", function() {\n",
+       "            if (d3.event.sourceEvent.type !== \"end\") {\n",
+       "                node.selectAll(\".selectable\").attr(\"style\", function(d) {\n",
+       "                    return nodeStyle(\n",
+       "                        d.selected = d.previouslySelected = shiftKey &&\n",
+       "                        d.selected);\n",
+       "                });\n",
+       "            }\n",
+       "        })\n",
+       "        .on(\"brush\", function() {\n",
+       "            if (d3.event.sourceEvent.type !== \"end\") {\n",
+       "                var selection = d3.event.selection;\n",
+       "                node.selectAll(\".selectable\").attr(\"style\", function(d) {\n",
+       "                    return nodeStyle(d.selected = d.previouslySelected ^\n",
+       "                        (selection != null\n",
+       "                            && selection[0][0] <= d.x && d.x < selection[1][0]\n",
+       "                            && selection[0][1] <= d.y && d.y < selection[1][1]));\n",
+       "                });\n",
+       "            }\n",
+       "        })\n",
+       "        .on(\"end\", function() {\n",
+       "            if (d3.event.selection != null) {\n",
+       "                d3.select(this).call(d3.event.target.move, null);\n",
+       "            }\n",
+       "        }));\n",
+       "}\n",
+       "\n",
+       "\n",
+       "showGraph('#graph-output-komyousH',\n",
+       "  JSON.parse('{\"nodes\": [{\"name\": \"0\", \"x\": 50, \"y\": 100, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"1\", \"x\": 50, \"y\": 150, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"2\", \"x\": 100, \"y\": 150, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"3\", \"x\": 100, \"y\": 100, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"4\", \"x\": 150, \"y\": 100, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"5\", \"x\": 150, \"y\": 150, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}], \"links\": [{\"source\": \"0\", \"target\": \"3\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"1\", \"target\": \"2\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"2\", \"target\": \"3\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"2\", \"target\": \"5\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"3\", \"target\": \"4\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}], \"pauli_web\": [{\"source\": 2, \"target\": 1, \"t\": \"Z\"}, {\"source\": 1, \"target\": 2, \"t\": \"Z\"}, {\"source\": 2, \"target\": 3, \"t\": \"Z\"}, {\"source\": 3, \"target\": 2, \"t\": \"Z\"}, {\"source\": 2, \"target\": 5, \"t\": \"Z\"}, {\"source\": 5, \"target\": 2, \"t\": \"Z\"}, {\"source\": 4, \"target\": 3, \"t\": \"Z\"}, {\"source\": 3, \"target\": 4, \"t\": \"Z\"}]}'), 200, 200, 50,\n",
+       "  10.0, false, true, '');\n",
+       "</script>"
+      ],
+      "text/plain": [
+       "<IPython.core.display.HTML object>"
+      ]
+     },
+     "metadata": {},
+     "output_type": "display_data"
+    }
+   ],
+   "source": [
+    "# Pauli webs are computed in the backwards (i.e. Heisenberg-style) by default,\n",
+    "# propegating Paulis from outputs to inputs. Paulis can be propegated forward\n",
+    "# instead by passing backwards=False.\n",
+    "\n",
+    "order, zwebsf, xwebsf = compute_pauli_webs(g, backwards=False)\n",
+    "zx.draw(g, labels=True, pauli_web=xwebsf[0])\n",
+    "zx.draw(g, labels=True, pauli_web=xwebsf[1])\n",
+    "zx.draw(g, labels=True, pauli_web=zwebsf[0])\n",
+    "zx.draw(g, labels=True, pauli_web=zwebsf[1])"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 5,
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "text/html": [
+       "<div style=\"overflow:auto; background-color: white\" id=\"graph-output-FNfQuO89\"></div>\n",
+       "<script type=\"module\">\n",
+       "var d3;\n",
+       "if (d3 == null) { d3 = await import(\"https://cdn.skypack.dev/d3@5\"); }\n",
+       "// PyZX - Python library for quantum circuit rewriting \n",
+       "//        and optimisation using the ZX-calculus\n",
+       "// Copyright (C) 2018 - Aleks Kissinger and John van de Wetering\n",
+       "\n",
+       "// Licensed under the Apache License, Version 2.0 (the \"License\");\n",
+       "// you may not use this file except in compliance with the License.\n",
+       "// You may obtain a copy of the License at\n",
+       "\n",
+       "//    http://www.apache.org/licenses/LICENSE-2.0\n",
+       "\n",
+       "// Unless required by applicable law or agreed to in writing, software\n",
+       "// distributed under the License is distributed on an \"AS IS\" BASIS,\n",
+       "// WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.\n",
+       "// See the License for the specific language governing permissions and\n",
+       "// limitations under the License.\n",
+       "\n",
+       "// styling functions\n",
+       "function nodeColor(t) {\n",
+       "    if (t == 0) return \"black\";\n",
+       "    else if (t == 1) return \"#ccffcc\";\n",
+       "    else if (t == 2) return \"#ff8888\";\n",
+       "    else if (t == 3) return \"yellow\";\n",
+       "    else if (t == 4) return \"black\";\n",
+       "    else if (t == 5) return \"black\";\n",
+       "    else if (t == 6) return \"#ccffcc\";\n",
+       "}\n",
+       "\n",
+       "function edgeColor(t) {\n",
+       "    if (t == 1) return \"black\";\n",
+       "    else if (t == 2) return \"#08f\";\n",
+       "    else if (t == 3) return \"gray\";\n",
+       "}\n",
+       "\n",
+       "function webColor(t) {\n",
+       "    if (t == 'X') return \"#ff8888\";\n",
+       "    else if (t == 'Y') return \"#eebb00\";\n",
+       "    else if (t == 'Z') return \"#66cc66\";\n",
+       "    else if (t == 'I') return \"#eee\";\n",
+       "}\n",
+       "\n",
+       "function nodeStyle(selected) {\n",
+       "    return selected ? \"stroke-width: 2px; stroke: #00f\" : \"stroke-width: 1.5px\";\n",
+       "}\n",
+       "\n",
+       "var symbolGround = {\n",
+       "    draw: function(context, size){\n",
+       "        let s = size/2;\n",
+       "\n",
+       "        context.moveTo(0,-s);\n",
+       "        context.lineTo(0,0);\n",
+       "\n",
+       "        context.moveTo(-s,0);\n",
+       "        context.lineTo(s,0);\n",
+       "\n",
+       "        context.moveTo(-2*s/3,s/3);\n",
+       "        context.lineTo(2*s/3,s/3);\n",
+       "\n",
+       "        context.moveTo(-s/3,2*s/3);\n",
+       "        context.lineTo(s/3,2*s/3);\n",
+       "    }\n",
+       "}\n",
+       "\n",
+       "function showGraph(tag, graph, width, height, scale, node_size, auto_hbox, show_labels, scalar_str) {\n",
+       "    var ntab = {};\n",
+       "\n",
+       "    var groundOffset = 2.5 * node_size;\n",
+       "\n",
+       "    graph.nodes.forEach(function(d) {\n",
+       "        ntab[d.name] = d;\n",
+       "        d.selected = false;\n",
+       "        d.previouslySelected = false;\n",
+       "        d.nhd = [];\n",
+       "    });\n",
+       "\n",
+       "    var spiders_and_boundaries = graph.nodes.filter(function(d) {\n",
+       "        return d.t != 3;\n",
+       "    });\n",
+       "\n",
+       "    graph.links.forEach(function(d) {\n",
+       "        var s = ntab[d.source];\n",
+       "        var t = ntab[d.target];\n",
+       "        d.source = s;\n",
+       "        d.target = t;\n",
+       "        s.nhd.push(t);\n",
+       "        t.nhd.push(s);\n",
+       "    });\n",
+       "\n",
+       "    graph.pauli_web.forEach(function(d) {\n",
+       "        var s = ntab[d.source];\n",
+       "        var t = ntab[d.target];\n",
+       "        d.source = s;\n",
+       "        d.target = t;\n",
+       "    });\n",
+       "\n",
+       "    var shiftKey;\n",
+       "\n",
+       "    // SETUP SVG ITEMS\n",
+       "\n",
+       "    var svg = d3.select(tag)\n",
+       "    //.attr(\"tabindex\", 1)\n",
+       "        .on(\"keydown.brush\", function() {shiftKey = d3.event.shiftKey || d3.event.metaKey;})\n",
+       "        .on(\"keyup.brush\", function() {shiftKey = d3.event.shiftKey || d3.event.metaKey;})\n",
+       "    //.each(function() { this.focus(); })\n",
+       "        .append(\"svg\")\n",
+       "        .attr(\"style\", \"max-width: none; max-height: none\")\n",
+       "        .attr(\"width\", width)\n",
+       "        .attr(\"height\", height);\n",
+       "\n",
+       "    var web = svg.append(\"g\")\n",
+       "        .attr(\"class\", \"web\")\n",
+       "        .selectAll(\"line\")\n",
+       "        .data(graph.pauli_web)\n",
+       "        .enter().append(\"path\")\n",
+       "        .attr(\"stroke\", function(d) { return webColor(d.t); })\n",
+       "        .attr(\"fill\", \"transparent\")\n",
+       "        .attr(\"style\", \"stroke-width: 7px\");\n",
+       "\n",
+       "    var link = svg.append(\"g\")\n",
+       "        .attr(\"class\", \"link\")\n",
+       "        .selectAll(\"line\")\n",
+       "        .data(graph.links)\n",
+       "        .enter().append(\"path\")\n",
+       "        .attr(\"stroke\", function(d) { return edgeColor(d.t); })\n",
+       "        .attr(\"fill\", \"transparent\")\n",
+       "        .attr(\"style\", \"stroke-width: 1.5px\");\n",
+       "\n",
+       "    var brush = svg.append(\"g\")\n",
+       "        .attr(\"class\", \"brush\");\n",
+       "\n",
+       "    var node = svg.append(\"g\")\n",
+       "        .attr(\"class\", \"node\")\n",
+       "        .selectAll(\"g\")\n",
+       "        .data(graph.nodes)\n",
+       "        .enter().append(\"g\")\n",
+       "        .attr(\"transform\", function(d) {\n",
+       "            return \"translate(\" + d.x + \",\" + d.y +\")\";\n",
+       "        });\n",
+       "\n",
+       "    // Draw a ground symbol connected to the node.\n",
+       "    node.filter(function(d) { return d.ground; })\n",
+       "        .append(\"path\")\n",
+       "        .attr(\"stroke\", \"black\")\n",
+       "        .attr(\"style\", \"stroke-width: 1.5px\")\n",
+       "        .attr(\"fill\", \"none\")\n",
+       "        .attr(\"d\", \"M 0 0 L 0 \"+(groundOffset))\n",
+       "        .attr(\"class\", \"selectable\");\n",
+       "    node.filter(function(d) { return d.ground; })\n",
+       "        .append(\"path\")\n",
+       "        .attr(\"stroke\", \"black\")\n",
+       "        .attr(\"style\", \"stroke-width: 1.5px\")\n",
+       "        .attr(\"fill\", \"none\")\n",
+       "        .attr(\"d\", d3.symbol().type(symbolGround).size(node_size*1.5))\n",
+       "        .attr(\"transform\", \"translate(0,\"+groundOffset+\")\")\n",
+       "        .attr(\"class\", \"selectable\");\n",
+       "\n",
+       "    node.filter(function(d) { return d.t != 3 && d.t != 5 && d.t != 6; })\n",
+       "        .append(\"circle\")\n",
+       "        .attr(\"r\", function(d) {\n",
+       "            if (d.t == 0) return 0.5 * node_size;\n",
+       "            else if (d.t == 4) return 0.25 * node_size;\n",
+       "            else return node_size;\n",
+       "        })\n",
+       "        .attr(\"fill\", function(d) { return nodeColor(d.t); })\n",
+       "        .attr(\"stroke\", \"black\")\n",
+       "        .attr(\"class\", \"selectable\");\n",
+       "\n",
+       "    var hbox = node.filter(function(d) { return d.t == 3; });\n",
+       "\n",
+       "    hbox.append(\"rect\")\n",
+       "        .attr(\"x\", -0.75 * node_size).attr(\"y\", -0.75 * node_size)\n",
+       "        .attr(\"width\", node_size * 1.5).attr(\"height\", node_size * 1.5)\n",
+       "        .attr(\"fill\", function(d) { return nodeColor(d.t); })\n",
+       "        .attr(\"stroke\", \"black\")\n",
+       "        .attr(\"class\", \"selectable\");\n",
+       "\n",
+       "    // draw a triangle for d.t == 5\n",
+       "    node.filter(function(d) { return d.t == 5; })\n",
+       "        .append(\"path\")\n",
+       "        .attr(\"d\", \"M 0 0 L \"+node_size+\" \"+node_size+\" L -\"+node_size+\" \"+node_size+\" Z\")\n",
+       "        .attr(\"fill\", function(d) { return nodeColor(d.t); })\n",
+       "        .attr(\"stroke\", \"black\")\n",
+       "        .attr(\"class\", \"selectable\")\n",
+       "        .attr(\"transform\", \"translate(\" + (-node_size/2) + \", 0) rotate(-90)\");\n",
+       "\n",
+       "    // draw a square for Z box: d.t == 6\n",
+       "    node.filter(function(d) { return d.t == 6; })\n",
+       "        .append(\"rect\")\n",
+       "        .attr(\"x\", -0.75 * node_size).attr(\"y\", -0.75 * node_size)\n",
+       "        .attr(\"width\", node_size * 1.5).attr(\"height\", node_size * 1.5)\n",
+       "        .attr(\"fill\", function(d) { return nodeColor(d.t); })\n",
+       "        .attr(\"stroke\", \"black\")\n",
+       "        .attr(\"class\", \"selectable\");\n",
+       "\n",
+       "    node.filter(function(d) { return d.phase != ''; })\n",
+       "        .append(\"text\")\n",
+       "        .attr(\"y\", 0.7 * node_size + 14)\n",
+       "        .text(function (d) { return d.phase })\n",
+       "        .attr(\"text-anchor\", \"middle\")\n",
+       "        .attr(\"font-size\", \"12px\")\n",
+       "        .attr(\"font-family\", \"monospace\")\n",
+       "        .attr(\"fill\", \"#00d\")\n",
+       "        .attr('style', 'pointer-events: none; user-select: none;');\n",
+       "\n",
+       "    if (show_labels) {\n",
+       "        node.append(\"text\")\n",
+       "            .attr(\"y\", -0.7 * node_size - 8)\n",
+       "            .text(function (d) { return d.name; })\n",
+       "            .attr(\"text-anchor\", \"middle\")\n",
+       "            .attr(\"font-size\", \"10px\")\n",
+       "            .attr(\"font-family\", \"monospace\")\n",
+       "            .attr(\"fill\", \"#999\")\n",
+       "            .attr('style', 'pointer-events: none; user-select: none;');\n",
+       "    }\n",
+       "\n",
+       "    // Display the chosen data fields over the node.\n",
+       "    node.filter(d => d.vdata.length > 0)\n",
+       "        .append(\"text\")\n",
+       "        .attr(\"y\", d => -0.7 * node_size - 14 - 10 * d.vdata.length)\n",
+       "        .attr(\"text-anchor\", \"middle\")\n",
+       "        .attr(\"font-size\", \"8px\")\n",
+       "        .attr(\"font-family\", \"monospace\")\n",
+       "        .attr(\"fill\", \"#c66\")\n",
+       "        .attr('style', 'pointer-events: none; user-select: none;')\n",
+       "        .selectAll(\"tspan\")\n",
+       "        .data(d => d.vdata)\n",
+       "        .enter()\n",
+       "        .append(\"tspan\")\n",
+       "        .attr(\"x\", \"0\")\n",
+       "        .attr(\"dy\", \"1.2em\")\n",
+       "        .text(x => x.join(\": \"));\n",
+       "\n",
+       "    if (scalar_str != \"\") {\n",
+       "        svg.append(\"text\")\n",
+       "            .text(scalar_str)\n",
+       "            .attr(\"x\", 60).attr(\"y\", 40)\n",
+       "            .attr(\"text-anchor\", \"middle\")\n",
+       "    }\n",
+       "\n",
+       "    function update_hboxes() {\n",
+       "        if (auto_hbox) {\n",
+       "            var pos = {};\n",
+       "            hbox.attr(\"transform\", function(d) {\n",
+       "                // calculate barycenter of non-hbox neighbours, then nudge a bit\n",
+       "                // to the NE.\n",
+       "                var x=0,y=0,sz=0;\n",
+       "                for (var i = 0; i < d.nhd.length; ++i) {\n",
+       "                    if (d.nhd[i].t != 3) {\n",
+       "                        sz++;\n",
+       "                        x += d.nhd[i].x;\n",
+       "                        y += d.nhd[i].y;\n",
+       "                    }\n",
+       "                }\n",
+       "\n",
+       "                offset = 0.25 * scale;\n",
+       "\n",
+       "                if (sz != 0) {\n",
+       "                    x = (x/sz) + offset;\n",
+       "                    y = (y/sz) - offset;\n",
+       "\n",
+       "                    while (pos[[x,y]]) {\n",
+       "                        x += offset;\n",
+       "                    }\n",
+       "                    d.x = x;\n",
+       "                    d.y = y;\n",
+       "                    pos[[x,y]] = true;\n",
+       "                }\n",
+       "\n",
+       "                return \"translate(\"+d.x+\",\"+d.y+\")\";\n",
+       "            });\n",
+       "        }\n",
+       "    }\n",
+       "\n",
+       "    update_hboxes();\n",
+       "\n",
+       "    var link_curve = function(d) {\n",
+       "        var x1 = d.source.x, x2 = d.target.x, y1 = d.source.y, y2 = d.target.y;\n",
+       "        if (d.num_parallel == 1) {\n",
+       "            return `M ${x1} ${y1} L ${x2} ${y2}`;\n",
+       "        } else {\n",
+       "            var dx = x2 - x1, dy = y2 - y1;\n",
+       "            var midx = 0.5 * (x1 + x2), midy = 0.5 * (y1 + y2);\n",
+       "            var pos = (d.index / (d.num_parallel-1)) - 0.5;\n",
+       "            var cx = midx - pos * dy;\n",
+       "            var cy = midy + pos * dx;\n",
+       "            return `M ${x1} ${y1} Q ${cx} ${cy}, ${x2} ${y2}`;\n",
+       "            // return `M ${x1} ${y1} L ${x2} ${y2}`;\n",
+       "        }\n",
+       "    };\n",
+       "    link.attr(\"d\", link_curve);\n",
+       "\n",
+       "\n",
+       "    var web_curve = function(d) {\n",
+       "        var x1 = d.source.x, x2 = (x1 + d.target.x)/2, y1 = d.source.y, y2 = (y1 + d.target.y)/2;\n",
+       "        return `M ${x1} ${y1} L ${x2} ${y2}`;\n",
+       "    }\n",
+       "    web.attr(\"d\", web_curve);\n",
+       "\n",
+       "    // EVENTS FOR DRAGGING AND SELECTION\n",
+       "\n",
+       "    node.on(\"mousedown\", function(d) {\n",
+       "        if (shiftKey) {\n",
+       "            d3.select(this).selectAll(\".selectable\").attr(\"style\", nodeStyle(d.selected = !d.selected));\n",
+       "            d3.event.stopImmediatePropagation();\n",
+       "        } else if (!d.selected) {\n",
+       "            node.selectAll(\".selectable\").attr(\"style\", function(p) { return nodeStyle(p.selected = d === p); });\n",
+       "        }\n",
+       "    })\n",
+       "        .call(d3.drag().on(\"drag\", function(d) {\n",
+       "            var dx = d3.event.dx;\n",
+       "            var dy = d3.event.dy;\n",
+       "            // node.filter(function(d) { return d.selected; })\n",
+       "            //     .attr(\"cx\", function(d) { return d.x += dx; })\n",
+       "            //     .attr(\"cy\", function(d) { return d.y += dy; });\n",
+       "            node.filter(function(d) { return d.selected; })\n",
+       "                .attr(\"transform\", function(d) {\n",
+       "                    d.x += dx;\n",
+       "                    d.y += dy;\n",
+       "                    return \"translate(\" + d.x + \",\" + d.y +\")\";\n",
+       "                });\n",
+       "\n",
+       "            update_hboxes();\n",
+       "\n",
+       "            link.filter(function(d) { return d.source.selected || d.target.selected ||\n",
+       "                    (auto_hbox && d.source.t == 3); })\n",
+       "                .attr(\"d\", link_curve);\n",
+       "            web.filter(function(d) { return d.source.selected || d.target.selected; })\n",
+       "                .attr(\"d\", web_curve);\n",
+       "        }));\n",
+       "\n",
+       "    brush.call(d3.brush().keyModifiers(false)\n",
+       "        .extent([[0, 0], [width, height]])\n",
+       "        .on(\"start\", function() {\n",
+       "            if (d3.event.sourceEvent.type !== \"end\") {\n",
+       "                node.selectAll(\".selectable\").attr(\"style\", function(d) {\n",
+       "                    return nodeStyle(\n",
+       "                        d.selected = d.previouslySelected = shiftKey &&\n",
+       "                        d.selected);\n",
+       "                });\n",
+       "            }\n",
+       "        })\n",
+       "        .on(\"brush\", function() {\n",
+       "            if (d3.event.sourceEvent.type !== \"end\") {\n",
+       "                var selection = d3.event.selection;\n",
+       "                node.selectAll(\".selectable\").attr(\"style\", function(d) {\n",
+       "                    return nodeStyle(d.selected = d.previouslySelected ^\n",
+       "                        (selection != null\n",
+       "                            && selection[0][0] <= d.x && d.x < selection[1][0]\n",
+       "                            && selection[0][1] <= d.y && d.y < selection[1][1]));\n",
+       "                });\n",
+       "            }\n",
+       "        })\n",
+       "        .on(\"end\", function() {\n",
+       "            if (d3.event.selection != null) {\n",
+       "                d3.select(this).call(d3.event.target.move, null);\n",
+       "            }\n",
+       "        }));\n",
+       "}\n",
+       "\n",
+       "\n",
+       "showGraph('#graph-output-FNfQuO89',\n",
+       "  JSON.parse('{\"nodes\": [{\"name\": \"0\", \"x\": 50, \"y\": 100, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"1\", \"x\": 50, \"y\": 150, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"2\", \"x\": 50, \"y\": 200, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"3\", \"x\": 100, \"y\": 150, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"4\", \"x\": 100, \"y\": 100, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"5\", \"x\": 150, \"y\": 200, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"6\", \"x\": 150, \"y\": 150, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"7\", \"x\": 200, \"y\": 200, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"8\", \"x\": 200, \"y\": 100, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"9\", \"x\": 250, \"y\": 150, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"10\", \"x\": 250, \"y\": 200, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"11\", \"x\": 300, \"y\": 150, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"12\", \"x\": 300, \"y\": 100, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"13\", \"x\": 350, \"y\": 100, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"14\", \"x\": 350, \"y\": 200, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"15\", \"x\": 400, \"y\": 100, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"16\", \"x\": 400, \"y\": 150, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"17\", \"x\": 400, \"y\": 200, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}], \"links\": [{\"source\": \"0\", \"target\": \"4\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"1\", \"target\": \"3\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"2\", \"target\": \"5\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"3\", \"target\": \"4\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"3\", \"target\": \"6\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"4\", \"target\": \"8\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"5\", \"target\": \"6\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"5\", \"target\": \"7\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"6\", \"target\": \"9\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"7\", \"target\": \"8\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"7\", \"target\": \"10\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"8\", \"target\": \"12\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"9\", \"target\": \"10\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"9\", \"target\": \"11\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"10\", \"target\": \"14\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"11\", \"target\": \"12\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"11\", \"target\": \"16\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"12\", \"target\": \"13\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"13\", \"target\": \"14\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"13\", \"target\": \"15\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"14\", \"target\": \"17\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}], \"pauli_web\": []}'), 450, 250, 50,\n",
+       "  10.0, false, true, '');\n",
+       "</script>"
+      ],
+      "text/plain": [
+       "<IPython.core.display.HTML object>"
+      ]
+     },
+     "metadata": {},
+     "output_type": "display_data"
+    }
+   ],
+   "source": [
+    "# Here's an example with more CNOT gates\n",
+    "\n",
+    "c = zx.qasm(\"\"\"\n",
+    "qreg q[3];\n",
+    "cx q[0], q[1];\n",
+    "cx q[1], q[2];\n",
+    "cx q[0], q[2];\n",
+    "cx q[2], q[1];\n",
+    "cx q[0], q[1];\n",
+    "cx q[2], q[0];\n",
+    "\"\"\")\n",
+    "g = c.to_graph()\n",
+    "\n",
+    "zx.draw(g, labels=True)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 6,
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "text/html": [
+       "<div style=\"overflow:auto; background-color: white\" id=\"graph-output-JsV1plp5\"></div>\n",
+       "<script type=\"module\">\n",
+       "var d3;\n",
+       "if (d3 == null) { d3 = await import(\"https://cdn.skypack.dev/d3@5\"); }\n",
+       "// PyZX - Python library for quantum circuit rewriting \n",
+       "//        and optimisation using the ZX-calculus\n",
+       "// Copyright (C) 2018 - Aleks Kissinger and John van de Wetering\n",
+       "\n",
+       "// Licensed under the Apache License, Version 2.0 (the \"License\");\n",
+       "// you may not use this file except in compliance with the License.\n",
+       "// You may obtain a copy of the License at\n",
+       "\n",
+       "//    http://www.apache.org/licenses/LICENSE-2.0\n",
+       "\n",
+       "// Unless required by applicable law or agreed to in writing, software\n",
+       "// distributed under the License is distributed on an \"AS IS\" BASIS,\n",
+       "// WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.\n",
+       "// See the License for the specific language governing permissions and\n",
+       "// limitations under the License.\n",
+       "\n",
+       "// styling functions\n",
+       "function nodeColor(t) {\n",
+       "    if (t == 0) return \"black\";\n",
+       "    else if (t == 1) return \"#ccffcc\";\n",
+       "    else if (t == 2) return \"#ff8888\";\n",
+       "    else if (t == 3) return \"yellow\";\n",
+       "    else if (t == 4) return \"black\";\n",
+       "    else if (t == 5) return \"black\";\n",
+       "    else if (t == 6) return \"#ccffcc\";\n",
+       "}\n",
+       "\n",
+       "function edgeColor(t) {\n",
+       "    if (t == 1) return \"black\";\n",
+       "    else if (t == 2) return \"#08f\";\n",
+       "    else if (t == 3) return \"gray\";\n",
+       "}\n",
+       "\n",
+       "function webColor(t) {\n",
+       "    if (t == 'X') return \"#ff8888\";\n",
+       "    else if (t == 'Y') return \"#eebb00\";\n",
+       "    else if (t == 'Z') return \"#66cc66\";\n",
+       "    else if (t == 'I') return \"#eee\";\n",
+       "}\n",
+       "\n",
+       "function nodeStyle(selected) {\n",
+       "    return selected ? \"stroke-width: 2px; stroke: #00f\" : \"stroke-width: 1.5px\";\n",
+       "}\n",
+       "\n",
+       "var symbolGround = {\n",
+       "    draw: function(context, size){\n",
+       "        let s = size/2;\n",
+       "\n",
+       "        context.moveTo(0,-s);\n",
+       "        context.lineTo(0,0);\n",
+       "\n",
+       "        context.moveTo(-s,0);\n",
+       "        context.lineTo(s,0);\n",
+       "\n",
+       "        context.moveTo(-2*s/3,s/3);\n",
+       "        context.lineTo(2*s/3,s/3);\n",
+       "\n",
+       "        context.moveTo(-s/3,2*s/3);\n",
+       "        context.lineTo(s/3,2*s/3);\n",
+       "    }\n",
+       "}\n",
+       "\n",
+       "function showGraph(tag, graph, width, height, scale, node_size, auto_hbox, show_labels, scalar_str) {\n",
+       "    var ntab = {};\n",
+       "\n",
+       "    var groundOffset = 2.5 * node_size;\n",
+       "\n",
+       "    graph.nodes.forEach(function(d) {\n",
+       "        ntab[d.name] = d;\n",
+       "        d.selected = false;\n",
+       "        d.previouslySelected = false;\n",
+       "        d.nhd = [];\n",
+       "    });\n",
+       "\n",
+       "    var spiders_and_boundaries = graph.nodes.filter(function(d) {\n",
+       "        return d.t != 3;\n",
+       "    });\n",
+       "\n",
+       "    graph.links.forEach(function(d) {\n",
+       "        var s = ntab[d.source];\n",
+       "        var t = ntab[d.target];\n",
+       "        d.source = s;\n",
+       "        d.target = t;\n",
+       "        s.nhd.push(t);\n",
+       "        t.nhd.push(s);\n",
+       "    });\n",
+       "\n",
+       "    graph.pauli_web.forEach(function(d) {\n",
+       "        var s = ntab[d.source];\n",
+       "        var t = ntab[d.target];\n",
+       "        d.source = s;\n",
+       "        d.target = t;\n",
+       "    });\n",
+       "\n",
+       "    var shiftKey;\n",
+       "\n",
+       "    // SETUP SVG ITEMS\n",
+       "\n",
+       "    var svg = d3.select(tag)\n",
+       "    //.attr(\"tabindex\", 1)\n",
+       "        .on(\"keydown.brush\", function() {shiftKey = d3.event.shiftKey || d3.event.metaKey;})\n",
+       "        .on(\"keyup.brush\", function() {shiftKey = d3.event.shiftKey || d3.event.metaKey;})\n",
+       "    //.each(function() { this.focus(); })\n",
+       "        .append(\"svg\")\n",
+       "        .attr(\"style\", \"max-width: none; max-height: none\")\n",
+       "        .attr(\"width\", width)\n",
+       "        .attr(\"height\", height);\n",
+       "\n",
+       "    var web = svg.append(\"g\")\n",
+       "        .attr(\"class\", \"web\")\n",
+       "        .selectAll(\"line\")\n",
+       "        .data(graph.pauli_web)\n",
+       "        .enter().append(\"path\")\n",
+       "        .attr(\"stroke\", function(d) { return webColor(d.t); })\n",
+       "        .attr(\"fill\", \"transparent\")\n",
+       "        .attr(\"style\", \"stroke-width: 7px\");\n",
+       "\n",
+       "    var link = svg.append(\"g\")\n",
+       "        .attr(\"class\", \"link\")\n",
+       "        .selectAll(\"line\")\n",
+       "        .data(graph.links)\n",
+       "        .enter().append(\"path\")\n",
+       "        .attr(\"stroke\", function(d) { return edgeColor(d.t); })\n",
+       "        .attr(\"fill\", \"transparent\")\n",
+       "        .attr(\"style\", \"stroke-width: 1.5px\");\n",
+       "\n",
+       "    var brush = svg.append(\"g\")\n",
+       "        .attr(\"class\", \"brush\");\n",
+       "\n",
+       "    var node = svg.append(\"g\")\n",
+       "        .attr(\"class\", \"node\")\n",
+       "        .selectAll(\"g\")\n",
+       "        .data(graph.nodes)\n",
+       "        .enter().append(\"g\")\n",
+       "        .attr(\"transform\", function(d) {\n",
+       "            return \"translate(\" + d.x + \",\" + d.y +\")\";\n",
+       "        });\n",
+       "\n",
+       "    // Draw a ground symbol connected to the node.\n",
+       "    node.filter(function(d) { return d.ground; })\n",
+       "        .append(\"path\")\n",
+       "        .attr(\"stroke\", \"black\")\n",
+       "        .attr(\"style\", \"stroke-width: 1.5px\")\n",
+       "        .attr(\"fill\", \"none\")\n",
+       "        .attr(\"d\", \"M 0 0 L 0 \"+(groundOffset))\n",
+       "        .attr(\"class\", \"selectable\");\n",
+       "    node.filter(function(d) { return d.ground; })\n",
+       "        .append(\"path\")\n",
+       "        .attr(\"stroke\", \"black\")\n",
+       "        .attr(\"style\", \"stroke-width: 1.5px\")\n",
+       "        .attr(\"fill\", \"none\")\n",
+       "        .attr(\"d\", d3.symbol().type(symbolGround).size(node_size*1.5))\n",
+       "        .attr(\"transform\", \"translate(0,\"+groundOffset+\")\")\n",
+       "        .attr(\"class\", \"selectable\");\n",
+       "\n",
+       "    node.filter(function(d) { return d.t != 3 && d.t != 5 && d.t != 6; })\n",
+       "        .append(\"circle\")\n",
+       "        .attr(\"r\", function(d) {\n",
+       "            if (d.t == 0) return 0.5 * node_size;\n",
+       "            else if (d.t == 4) return 0.25 * node_size;\n",
+       "            else return node_size;\n",
+       "        })\n",
+       "        .attr(\"fill\", function(d) { return nodeColor(d.t); })\n",
+       "        .attr(\"stroke\", \"black\")\n",
+       "        .attr(\"class\", \"selectable\");\n",
+       "\n",
+       "    var hbox = node.filter(function(d) { return d.t == 3; });\n",
+       "\n",
+       "    hbox.append(\"rect\")\n",
+       "        .attr(\"x\", -0.75 * node_size).attr(\"y\", -0.75 * node_size)\n",
+       "        .attr(\"width\", node_size * 1.5).attr(\"height\", node_size * 1.5)\n",
+       "        .attr(\"fill\", function(d) { return nodeColor(d.t); })\n",
+       "        .attr(\"stroke\", \"black\")\n",
+       "        .attr(\"class\", \"selectable\");\n",
+       "\n",
+       "    // draw a triangle for d.t == 5\n",
+       "    node.filter(function(d) { return d.t == 5; })\n",
+       "        .append(\"path\")\n",
+       "        .attr(\"d\", \"M 0 0 L \"+node_size+\" \"+node_size+\" L -\"+node_size+\" \"+node_size+\" Z\")\n",
+       "        .attr(\"fill\", function(d) { return nodeColor(d.t); })\n",
+       "        .attr(\"stroke\", \"black\")\n",
+       "        .attr(\"class\", \"selectable\")\n",
+       "        .attr(\"transform\", \"translate(\" + (-node_size/2) + \", 0) rotate(-90)\");\n",
+       "\n",
+       "    // draw a square for Z box: d.t == 6\n",
+       "    node.filter(function(d) { return d.t == 6; })\n",
+       "        .append(\"rect\")\n",
+       "        .attr(\"x\", -0.75 * node_size).attr(\"y\", -0.75 * node_size)\n",
+       "        .attr(\"width\", node_size * 1.5).attr(\"height\", node_size * 1.5)\n",
+       "        .attr(\"fill\", function(d) { return nodeColor(d.t); })\n",
+       "        .attr(\"stroke\", \"black\")\n",
+       "        .attr(\"class\", \"selectable\");\n",
+       "\n",
+       "    node.filter(function(d) { return d.phase != ''; })\n",
+       "        .append(\"text\")\n",
+       "        .attr(\"y\", 0.7 * node_size + 14)\n",
+       "        .text(function (d) { return d.phase })\n",
+       "        .attr(\"text-anchor\", \"middle\")\n",
+       "        .attr(\"font-size\", \"12px\")\n",
+       "        .attr(\"font-family\", \"monospace\")\n",
+       "        .attr(\"fill\", \"#00d\")\n",
+       "        .attr('style', 'pointer-events: none; user-select: none;');\n",
+       "\n",
+       "    if (show_labels) {\n",
+       "        node.append(\"text\")\n",
+       "            .attr(\"y\", -0.7 * node_size - 8)\n",
+       "            .text(function (d) { return d.name; })\n",
+       "            .attr(\"text-anchor\", \"middle\")\n",
+       "            .attr(\"font-size\", \"10px\")\n",
+       "            .attr(\"font-family\", \"monospace\")\n",
+       "            .attr(\"fill\", \"#999\")\n",
+       "            .attr('style', 'pointer-events: none; user-select: none;');\n",
+       "    }\n",
+       "\n",
+       "    // Display the chosen data fields over the node.\n",
+       "    node.filter(d => d.vdata.length > 0)\n",
+       "        .append(\"text\")\n",
+       "        .attr(\"y\", d => -0.7 * node_size - 14 - 10 * d.vdata.length)\n",
+       "        .attr(\"text-anchor\", \"middle\")\n",
+       "        .attr(\"font-size\", \"8px\")\n",
+       "        .attr(\"font-family\", \"monospace\")\n",
+       "        .attr(\"fill\", \"#c66\")\n",
+       "        .attr('style', 'pointer-events: none; user-select: none;')\n",
+       "        .selectAll(\"tspan\")\n",
+       "        .data(d => d.vdata)\n",
+       "        .enter()\n",
+       "        .append(\"tspan\")\n",
+       "        .attr(\"x\", \"0\")\n",
+       "        .attr(\"dy\", \"1.2em\")\n",
+       "        .text(x => x.join(\": \"));\n",
+       "\n",
+       "    if (scalar_str != \"\") {\n",
+       "        svg.append(\"text\")\n",
+       "            .text(scalar_str)\n",
+       "            .attr(\"x\", 60).attr(\"y\", 40)\n",
+       "            .attr(\"text-anchor\", \"middle\")\n",
+       "    }\n",
+       "\n",
+       "    function update_hboxes() {\n",
+       "        if (auto_hbox) {\n",
+       "            var pos = {};\n",
+       "            hbox.attr(\"transform\", function(d) {\n",
+       "                // calculate barycenter of non-hbox neighbours, then nudge a bit\n",
+       "                // to the NE.\n",
+       "                var x=0,y=0,sz=0;\n",
+       "                for (var i = 0; i < d.nhd.length; ++i) {\n",
+       "                    if (d.nhd[i].t != 3) {\n",
+       "                        sz++;\n",
+       "                        x += d.nhd[i].x;\n",
+       "                        y += d.nhd[i].y;\n",
+       "                    }\n",
+       "                }\n",
+       "\n",
+       "                offset = 0.25 * scale;\n",
+       "\n",
+       "                if (sz != 0) {\n",
+       "                    x = (x/sz) + offset;\n",
+       "                    y = (y/sz) - offset;\n",
+       "\n",
+       "                    while (pos[[x,y]]) {\n",
+       "                        x += offset;\n",
+       "                    }\n",
+       "                    d.x = x;\n",
+       "                    d.y = y;\n",
+       "                    pos[[x,y]] = true;\n",
+       "                }\n",
+       "\n",
+       "                return \"translate(\"+d.x+\",\"+d.y+\")\";\n",
+       "            });\n",
+       "        }\n",
+       "    }\n",
+       "\n",
+       "    update_hboxes();\n",
+       "\n",
+       "    var link_curve = function(d) {\n",
+       "        var x1 = d.source.x, x2 = d.target.x, y1 = d.source.y, y2 = d.target.y;\n",
+       "        if (d.num_parallel == 1) {\n",
+       "            return `M ${x1} ${y1} L ${x2} ${y2}`;\n",
+       "        } else {\n",
+       "            var dx = x2 - x1, dy = y2 - y1;\n",
+       "            var midx = 0.5 * (x1 + x2), midy = 0.5 * (y1 + y2);\n",
+       "            var pos = (d.index / (d.num_parallel-1)) - 0.5;\n",
+       "            var cx = midx - pos * dy;\n",
+       "            var cy = midy + pos * dx;\n",
+       "            return `M ${x1} ${y1} Q ${cx} ${cy}, ${x2} ${y2}`;\n",
+       "            // return `M ${x1} ${y1} L ${x2} ${y2}`;\n",
+       "        }\n",
+       "    };\n",
+       "    link.attr(\"d\", link_curve);\n",
+       "\n",
+       "\n",
+       "    var web_curve = function(d) {\n",
+       "        var x1 = d.source.x, x2 = (x1 + d.target.x)/2, y1 = d.source.y, y2 = (y1 + d.target.y)/2;\n",
+       "        return `M ${x1} ${y1} L ${x2} ${y2}`;\n",
+       "    }\n",
+       "    web.attr(\"d\", web_curve);\n",
+       "\n",
+       "    // EVENTS FOR DRAGGING AND SELECTION\n",
+       "\n",
+       "    node.on(\"mousedown\", function(d) {\n",
+       "        if (shiftKey) {\n",
+       "            d3.select(this).selectAll(\".selectable\").attr(\"style\", nodeStyle(d.selected = !d.selected));\n",
+       "            d3.event.stopImmediatePropagation();\n",
+       "        } else if (!d.selected) {\n",
+       "            node.selectAll(\".selectable\").attr(\"style\", function(p) { return nodeStyle(p.selected = d === p); });\n",
+       "        }\n",
+       "    })\n",
+       "        .call(d3.drag().on(\"drag\", function(d) {\n",
+       "            var dx = d3.event.dx;\n",
+       "            var dy = d3.event.dy;\n",
+       "            // node.filter(function(d) { return d.selected; })\n",
+       "            //     .attr(\"cx\", function(d) { return d.x += dx; })\n",
+       "            //     .attr(\"cy\", function(d) { return d.y += dy; });\n",
+       "            node.filter(function(d) { return d.selected; })\n",
+       "                .attr(\"transform\", function(d) {\n",
+       "                    d.x += dx;\n",
+       "                    d.y += dy;\n",
+       "                    return \"translate(\" + d.x + \",\" + d.y +\")\";\n",
+       "                });\n",
+       "\n",
+       "            update_hboxes();\n",
+       "\n",
+       "            link.filter(function(d) { return d.source.selected || d.target.selected ||\n",
+       "                    (auto_hbox && d.source.t == 3); })\n",
+       "                .attr(\"d\", link_curve);\n",
+       "            web.filter(function(d) { return d.source.selected || d.target.selected; })\n",
+       "                .attr(\"d\", web_curve);\n",
+       "        }));\n",
+       "\n",
+       "    brush.call(d3.brush().keyModifiers(false)\n",
+       "        .extent([[0, 0], [width, height]])\n",
+       "        .on(\"start\", function() {\n",
+       "            if (d3.event.sourceEvent.type !== \"end\") {\n",
+       "                node.selectAll(\".selectable\").attr(\"style\", function(d) {\n",
+       "                    return nodeStyle(\n",
+       "                        d.selected = d.previouslySelected = shiftKey &&\n",
+       "                        d.selected);\n",
+       "                });\n",
+       "            }\n",
+       "        })\n",
+       "        .on(\"brush\", function() {\n",
+       "            if (d3.event.sourceEvent.type !== \"end\") {\n",
+       "                var selection = d3.event.selection;\n",
+       "                node.selectAll(\".selectable\").attr(\"style\", function(d) {\n",
+       "                    return nodeStyle(d.selected = d.previouslySelected ^\n",
+       "                        (selection != null\n",
+       "                            && selection[0][0] <= d.x && d.x < selection[1][0]\n",
+       "                            && selection[0][1] <= d.y && d.y < selection[1][1]));\n",
+       "                });\n",
+       "            }\n",
+       "        })\n",
+       "        .on(\"end\", function() {\n",
+       "            if (d3.event.selection != null) {\n",
+       "                d3.select(this).call(d3.event.target.move, null);\n",
+       "            }\n",
+       "        }));\n",
+       "}\n",
+       "\n",
+       "\n",
+       "showGraph('#graph-output-JsV1plp5',\n",
+       "  JSON.parse('{\"nodes\": [{\"name\": \"0\", \"x\": 50, \"y\": 100, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"1\", \"x\": 50, \"y\": 150, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"2\", \"x\": 50, \"y\": 200, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"3\", \"x\": 100, \"y\": 150, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"4\", \"x\": 100, \"y\": 100, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"5\", \"x\": 150, \"y\": 200, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"6\", \"x\": 150, \"y\": 150, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"7\", \"x\": 200, \"y\": 200, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"8\", \"x\": 200, \"y\": 100, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"9\", \"x\": 250, \"y\": 150, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"10\", \"x\": 250, \"y\": 200, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"11\", \"x\": 300, \"y\": 150, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"12\", \"x\": 300, \"y\": 100, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"13\", \"x\": 350, \"y\": 100, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"14\", \"x\": 350, \"y\": 200, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"15\", \"x\": 400, \"y\": 100, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"16\", \"x\": 400, \"y\": 150, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"17\", \"x\": 400, \"y\": 200, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}], \"links\": [{\"source\": \"0\", \"target\": \"4\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"1\", \"target\": \"3\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"2\", \"target\": \"5\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"3\", \"target\": \"4\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"3\", \"target\": \"6\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"4\", \"target\": \"8\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"5\", \"target\": \"6\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"5\", \"target\": \"7\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"6\", \"target\": \"9\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"7\", \"target\": \"8\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"7\", \"target\": \"10\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"8\", \"target\": \"12\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"9\", \"target\": \"10\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"9\", \"target\": \"11\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"10\", \"target\": \"14\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"11\", \"target\": \"12\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"11\", \"target\": \"16\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"12\", \"target\": \"13\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"13\", \"target\": \"14\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"13\", \"target\": \"15\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"14\", \"target\": \"17\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}], \"pauli_web\": [{\"source\": 4, \"target\": 0, \"t\": \"X\"}, {\"source\": 0, \"target\": 4, \"t\": \"X\"}, {\"source\": 4, \"target\": 3, \"t\": \"X\"}, {\"source\": 3, \"target\": 4, \"t\": \"X\"}, {\"source\": 4, \"target\": 8, \"t\": \"X\"}, {\"source\": 6, \"target\": 3, \"t\": \"X\"}, {\"source\": 3, \"target\": 6, \"t\": \"X\"}, {\"source\": 6, \"target\": 5, \"t\": \"X\"}, {\"source\": 5, \"target\": 6, \"t\": \"X\"}, {\"source\": 6, \"target\": 9, \"t\": \"X\"}, {\"source\": 9, \"target\": 6, \"t\": \"X\"}, {\"source\": 8, \"target\": 4, \"t\": \"X\"}, {\"source\": 8, \"target\": 7, \"t\": \"X\"}, {\"source\": 7, \"target\": 8, \"t\": \"X\"}, {\"source\": 8, \"target\": 12, \"t\": \"X\"}, {\"source\": 12, \"target\": 8, \"t\": \"X\"}, {\"source\": 12, \"target\": 11, \"t\": \"X\"}, {\"source\": 11, \"target\": 12, \"t\": \"X\"}, {\"source\": 12, \"target\": 13, \"t\": \"X\"}, {\"source\": 13, \"target\": 12, \"t\": \"X\"}, {\"source\": 5, \"target\": 7, \"t\": \"X\"}, {\"source\": 7, \"target\": 5, \"t\": \"X\"}, {\"source\": 9, \"target\": 11, \"t\": \"X\"}, {\"source\": 11, \"target\": 9, \"t\": \"X\"}, {\"source\": 15, \"target\": 13, \"t\": \"X\"}, {\"source\": 13, \"target\": 15, \"t\": \"X\"}]}'), 450, 250, 50,\n",
+       "  10.0, false, true, '');\n",
+       "</script>"
+      ],
+      "text/plain": [
+       "<IPython.core.display.HTML object>"
+      ]
+     },
+     "metadata": {},
+     "output_type": "display_data"
+    },
+    {
+     "data": {
+      "text/html": [
+       "<div style=\"overflow:auto; background-color: white\" id=\"graph-output-1Vwd2jRV\"></div>\n",
+       "<script type=\"module\">\n",
+       "var d3;\n",
+       "if (d3 == null) { d3 = await import(\"https://cdn.skypack.dev/d3@5\"); }\n",
+       "// PyZX - Python library for quantum circuit rewriting \n",
+       "//        and optimisation using the ZX-calculus\n",
+       "// Copyright (C) 2018 - Aleks Kissinger and John van de Wetering\n",
+       "\n",
+       "// Licensed under the Apache License, Version 2.0 (the \"License\");\n",
+       "// you may not use this file except in compliance with the License.\n",
+       "// You may obtain a copy of the License at\n",
+       "\n",
+       "//    http://www.apache.org/licenses/LICENSE-2.0\n",
+       "\n",
+       "// Unless required by applicable law or agreed to in writing, software\n",
+       "// distributed under the License is distributed on an \"AS IS\" BASIS,\n",
+       "// WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.\n",
+       "// See the License for the specific language governing permissions and\n",
+       "// limitations under the License.\n",
+       "\n",
+       "// styling functions\n",
+       "function nodeColor(t) {\n",
+       "    if (t == 0) return \"black\";\n",
+       "    else if (t == 1) return \"#ccffcc\";\n",
+       "    else if (t == 2) return \"#ff8888\";\n",
+       "    else if (t == 3) return \"yellow\";\n",
+       "    else if (t == 4) return \"black\";\n",
+       "    else if (t == 5) return \"black\";\n",
+       "    else if (t == 6) return \"#ccffcc\";\n",
+       "}\n",
+       "\n",
+       "function edgeColor(t) {\n",
+       "    if (t == 1) return \"black\";\n",
+       "    else if (t == 2) return \"#08f\";\n",
+       "    else if (t == 3) return \"gray\";\n",
+       "}\n",
+       "\n",
+       "function webColor(t) {\n",
+       "    if (t == 'X') return \"#ff8888\";\n",
+       "    else if (t == 'Y') return \"#eebb00\";\n",
+       "    else if (t == 'Z') return \"#66cc66\";\n",
+       "    else if (t == 'I') return \"#eee\";\n",
+       "}\n",
+       "\n",
+       "function nodeStyle(selected) {\n",
+       "    return selected ? \"stroke-width: 2px; stroke: #00f\" : \"stroke-width: 1.5px\";\n",
+       "}\n",
+       "\n",
+       "var symbolGround = {\n",
+       "    draw: function(context, size){\n",
+       "        let s = size/2;\n",
+       "\n",
+       "        context.moveTo(0,-s);\n",
+       "        context.lineTo(0,0);\n",
+       "\n",
+       "        context.moveTo(-s,0);\n",
+       "        context.lineTo(s,0);\n",
+       "\n",
+       "        context.moveTo(-2*s/3,s/3);\n",
+       "        context.lineTo(2*s/3,s/3);\n",
+       "\n",
+       "        context.moveTo(-s/3,2*s/3);\n",
+       "        context.lineTo(s/3,2*s/3);\n",
+       "    }\n",
+       "}\n",
+       "\n",
+       "function showGraph(tag, graph, width, height, scale, node_size, auto_hbox, show_labels, scalar_str) {\n",
+       "    var ntab = {};\n",
+       "\n",
+       "    var groundOffset = 2.5 * node_size;\n",
+       "\n",
+       "    graph.nodes.forEach(function(d) {\n",
+       "        ntab[d.name] = d;\n",
+       "        d.selected = false;\n",
+       "        d.previouslySelected = false;\n",
+       "        d.nhd = [];\n",
+       "    });\n",
+       "\n",
+       "    var spiders_and_boundaries = graph.nodes.filter(function(d) {\n",
+       "        return d.t != 3;\n",
+       "    });\n",
+       "\n",
+       "    graph.links.forEach(function(d) {\n",
+       "        var s = ntab[d.source];\n",
+       "        var t = ntab[d.target];\n",
+       "        d.source = s;\n",
+       "        d.target = t;\n",
+       "        s.nhd.push(t);\n",
+       "        t.nhd.push(s);\n",
+       "    });\n",
+       "\n",
+       "    graph.pauli_web.forEach(function(d) {\n",
+       "        var s = ntab[d.source];\n",
+       "        var t = ntab[d.target];\n",
+       "        d.source = s;\n",
+       "        d.target = t;\n",
+       "    });\n",
+       "\n",
+       "    var shiftKey;\n",
+       "\n",
+       "    // SETUP SVG ITEMS\n",
+       "\n",
+       "    var svg = d3.select(tag)\n",
+       "    //.attr(\"tabindex\", 1)\n",
+       "        .on(\"keydown.brush\", function() {shiftKey = d3.event.shiftKey || d3.event.metaKey;})\n",
+       "        .on(\"keyup.brush\", function() {shiftKey = d3.event.shiftKey || d3.event.metaKey;})\n",
+       "    //.each(function() { this.focus(); })\n",
+       "        .append(\"svg\")\n",
+       "        .attr(\"style\", \"max-width: none; max-height: none\")\n",
+       "        .attr(\"width\", width)\n",
+       "        .attr(\"height\", height);\n",
+       "\n",
+       "    var web = svg.append(\"g\")\n",
+       "        .attr(\"class\", \"web\")\n",
+       "        .selectAll(\"line\")\n",
+       "        .data(graph.pauli_web)\n",
+       "        .enter().append(\"path\")\n",
+       "        .attr(\"stroke\", function(d) { return webColor(d.t); })\n",
+       "        .attr(\"fill\", \"transparent\")\n",
+       "        .attr(\"style\", \"stroke-width: 7px\");\n",
+       "\n",
+       "    var link = svg.append(\"g\")\n",
+       "        .attr(\"class\", \"link\")\n",
+       "        .selectAll(\"line\")\n",
+       "        .data(graph.links)\n",
+       "        .enter().append(\"path\")\n",
+       "        .attr(\"stroke\", function(d) { return edgeColor(d.t); })\n",
+       "        .attr(\"fill\", \"transparent\")\n",
+       "        .attr(\"style\", \"stroke-width: 1.5px\");\n",
+       "\n",
+       "    var brush = svg.append(\"g\")\n",
+       "        .attr(\"class\", \"brush\");\n",
+       "\n",
+       "    var node = svg.append(\"g\")\n",
+       "        .attr(\"class\", \"node\")\n",
+       "        .selectAll(\"g\")\n",
+       "        .data(graph.nodes)\n",
+       "        .enter().append(\"g\")\n",
+       "        .attr(\"transform\", function(d) {\n",
+       "            return \"translate(\" + d.x + \",\" + d.y +\")\";\n",
+       "        });\n",
+       "\n",
+       "    // Draw a ground symbol connected to the node.\n",
+       "    node.filter(function(d) { return d.ground; })\n",
+       "        .append(\"path\")\n",
+       "        .attr(\"stroke\", \"black\")\n",
+       "        .attr(\"style\", \"stroke-width: 1.5px\")\n",
+       "        .attr(\"fill\", \"none\")\n",
+       "        .attr(\"d\", \"M 0 0 L 0 \"+(groundOffset))\n",
+       "        .attr(\"class\", \"selectable\");\n",
+       "    node.filter(function(d) { return d.ground; })\n",
+       "        .append(\"path\")\n",
+       "        .attr(\"stroke\", \"black\")\n",
+       "        .attr(\"style\", \"stroke-width: 1.5px\")\n",
+       "        .attr(\"fill\", \"none\")\n",
+       "        .attr(\"d\", d3.symbol().type(symbolGround).size(node_size*1.5))\n",
+       "        .attr(\"transform\", \"translate(0,\"+groundOffset+\")\")\n",
+       "        .attr(\"class\", \"selectable\");\n",
+       "\n",
+       "    node.filter(function(d) { return d.t != 3 && d.t != 5 && d.t != 6; })\n",
+       "        .append(\"circle\")\n",
+       "        .attr(\"r\", function(d) {\n",
+       "            if (d.t == 0) return 0.5 * node_size;\n",
+       "            else if (d.t == 4) return 0.25 * node_size;\n",
+       "            else return node_size;\n",
+       "        })\n",
+       "        .attr(\"fill\", function(d) { return nodeColor(d.t); })\n",
+       "        .attr(\"stroke\", \"black\")\n",
+       "        .attr(\"class\", \"selectable\");\n",
+       "\n",
+       "    var hbox = node.filter(function(d) { return d.t == 3; });\n",
+       "\n",
+       "    hbox.append(\"rect\")\n",
+       "        .attr(\"x\", -0.75 * node_size).attr(\"y\", -0.75 * node_size)\n",
+       "        .attr(\"width\", node_size * 1.5).attr(\"height\", node_size * 1.5)\n",
+       "        .attr(\"fill\", function(d) { return nodeColor(d.t); })\n",
+       "        .attr(\"stroke\", \"black\")\n",
+       "        .attr(\"class\", \"selectable\");\n",
+       "\n",
+       "    // draw a triangle for d.t == 5\n",
+       "    node.filter(function(d) { return d.t == 5; })\n",
+       "        .append(\"path\")\n",
+       "        .attr(\"d\", \"M 0 0 L \"+node_size+\" \"+node_size+\" L -\"+node_size+\" \"+node_size+\" Z\")\n",
+       "        .attr(\"fill\", function(d) { return nodeColor(d.t); })\n",
+       "        .attr(\"stroke\", \"black\")\n",
+       "        .attr(\"class\", \"selectable\")\n",
+       "        .attr(\"transform\", \"translate(\" + (-node_size/2) + \", 0) rotate(-90)\");\n",
+       "\n",
+       "    // draw a square for Z box: d.t == 6\n",
+       "    node.filter(function(d) { return d.t == 6; })\n",
+       "        .append(\"rect\")\n",
+       "        .attr(\"x\", -0.75 * node_size).attr(\"y\", -0.75 * node_size)\n",
+       "        .attr(\"width\", node_size * 1.5).attr(\"height\", node_size * 1.5)\n",
+       "        .attr(\"fill\", function(d) { return nodeColor(d.t); })\n",
+       "        .attr(\"stroke\", \"black\")\n",
+       "        .attr(\"class\", \"selectable\");\n",
+       "\n",
+       "    node.filter(function(d) { return d.phase != ''; })\n",
+       "        .append(\"text\")\n",
+       "        .attr(\"y\", 0.7 * node_size + 14)\n",
+       "        .text(function (d) { return d.phase })\n",
+       "        .attr(\"text-anchor\", \"middle\")\n",
+       "        .attr(\"font-size\", \"12px\")\n",
+       "        .attr(\"font-family\", \"monospace\")\n",
+       "        .attr(\"fill\", \"#00d\")\n",
+       "        .attr('style', 'pointer-events: none; user-select: none;');\n",
+       "\n",
+       "    if (show_labels) {\n",
+       "        node.append(\"text\")\n",
+       "            .attr(\"y\", -0.7 * node_size - 8)\n",
+       "            .text(function (d) { return d.name; })\n",
+       "            .attr(\"text-anchor\", \"middle\")\n",
+       "            .attr(\"font-size\", \"10px\")\n",
+       "            .attr(\"font-family\", \"monospace\")\n",
+       "            .attr(\"fill\", \"#999\")\n",
+       "            .attr('style', 'pointer-events: none; user-select: none;');\n",
+       "    }\n",
+       "\n",
+       "    // Display the chosen data fields over the node.\n",
+       "    node.filter(d => d.vdata.length > 0)\n",
+       "        .append(\"text\")\n",
+       "        .attr(\"y\", d => -0.7 * node_size - 14 - 10 * d.vdata.length)\n",
+       "        .attr(\"text-anchor\", \"middle\")\n",
+       "        .attr(\"font-size\", \"8px\")\n",
+       "        .attr(\"font-family\", \"monospace\")\n",
+       "        .attr(\"fill\", \"#c66\")\n",
+       "        .attr('style', 'pointer-events: none; user-select: none;')\n",
+       "        .selectAll(\"tspan\")\n",
+       "        .data(d => d.vdata)\n",
+       "        .enter()\n",
+       "        .append(\"tspan\")\n",
+       "        .attr(\"x\", \"0\")\n",
+       "        .attr(\"dy\", \"1.2em\")\n",
+       "        .text(x => x.join(\": \"));\n",
+       "\n",
+       "    if (scalar_str != \"\") {\n",
+       "        svg.append(\"text\")\n",
+       "            .text(scalar_str)\n",
+       "            .attr(\"x\", 60).attr(\"y\", 40)\n",
+       "            .attr(\"text-anchor\", \"middle\")\n",
+       "    }\n",
+       "\n",
+       "    function update_hboxes() {\n",
+       "        if (auto_hbox) {\n",
+       "            var pos = {};\n",
+       "            hbox.attr(\"transform\", function(d) {\n",
+       "                // calculate barycenter of non-hbox neighbours, then nudge a bit\n",
+       "                // to the NE.\n",
+       "                var x=0,y=0,sz=0;\n",
+       "                for (var i = 0; i < d.nhd.length; ++i) {\n",
+       "                    if (d.nhd[i].t != 3) {\n",
+       "                        sz++;\n",
+       "                        x += d.nhd[i].x;\n",
+       "                        y += d.nhd[i].y;\n",
+       "                    }\n",
+       "                }\n",
+       "\n",
+       "                offset = 0.25 * scale;\n",
+       "\n",
+       "                if (sz != 0) {\n",
+       "                    x = (x/sz) + offset;\n",
+       "                    y = (y/sz) - offset;\n",
+       "\n",
+       "                    while (pos[[x,y]]) {\n",
+       "                        x += offset;\n",
+       "                    }\n",
+       "                    d.x = x;\n",
+       "                    d.y = y;\n",
+       "                    pos[[x,y]] = true;\n",
+       "                }\n",
+       "\n",
+       "                return \"translate(\"+d.x+\",\"+d.y+\")\";\n",
+       "            });\n",
+       "        }\n",
+       "    }\n",
+       "\n",
+       "    update_hboxes();\n",
+       "\n",
+       "    var link_curve = function(d) {\n",
+       "        var x1 = d.source.x, x2 = d.target.x, y1 = d.source.y, y2 = d.target.y;\n",
+       "        if (d.num_parallel == 1) {\n",
+       "            return `M ${x1} ${y1} L ${x2} ${y2}`;\n",
+       "        } else {\n",
+       "            var dx = x2 - x1, dy = y2 - y1;\n",
+       "            var midx = 0.5 * (x1 + x2), midy = 0.5 * (y1 + y2);\n",
+       "            var pos = (d.index / (d.num_parallel-1)) - 0.5;\n",
+       "            var cx = midx - pos * dy;\n",
+       "            var cy = midy + pos * dx;\n",
+       "            return `M ${x1} ${y1} Q ${cx} ${cy}, ${x2} ${y2}`;\n",
+       "            // return `M ${x1} ${y1} L ${x2} ${y2}`;\n",
+       "        }\n",
+       "    };\n",
+       "    link.attr(\"d\", link_curve);\n",
+       "\n",
+       "\n",
+       "    var web_curve = function(d) {\n",
+       "        var x1 = d.source.x, x2 = (x1 + d.target.x)/2, y1 = d.source.y, y2 = (y1 + d.target.y)/2;\n",
+       "        return `M ${x1} ${y1} L ${x2} ${y2}`;\n",
+       "    }\n",
+       "    web.attr(\"d\", web_curve);\n",
+       "\n",
+       "    // EVENTS FOR DRAGGING AND SELECTION\n",
+       "\n",
+       "    node.on(\"mousedown\", function(d) {\n",
+       "        if (shiftKey) {\n",
+       "            d3.select(this).selectAll(\".selectable\").attr(\"style\", nodeStyle(d.selected = !d.selected));\n",
+       "            d3.event.stopImmediatePropagation();\n",
+       "        } else if (!d.selected) {\n",
+       "            node.selectAll(\".selectable\").attr(\"style\", function(p) { return nodeStyle(p.selected = d === p); });\n",
+       "        }\n",
+       "    })\n",
+       "        .call(d3.drag().on(\"drag\", function(d) {\n",
+       "            var dx = d3.event.dx;\n",
+       "            var dy = d3.event.dy;\n",
+       "            // node.filter(function(d) { return d.selected; })\n",
+       "            //     .attr(\"cx\", function(d) { return d.x += dx; })\n",
+       "            //     .attr(\"cy\", function(d) { return d.y += dy; });\n",
+       "            node.filter(function(d) { return d.selected; })\n",
+       "                .attr(\"transform\", function(d) {\n",
+       "                    d.x += dx;\n",
+       "                    d.y += dy;\n",
+       "                    return \"translate(\" + d.x + \",\" + d.y +\")\";\n",
+       "                });\n",
+       "\n",
+       "            update_hboxes();\n",
+       "\n",
+       "            link.filter(function(d) { return d.source.selected || d.target.selected ||\n",
+       "                    (auto_hbox && d.source.t == 3); })\n",
+       "                .attr(\"d\", link_curve);\n",
+       "            web.filter(function(d) { return d.source.selected || d.target.selected; })\n",
+       "                .attr(\"d\", web_curve);\n",
+       "        }));\n",
+       "\n",
+       "    brush.call(d3.brush().keyModifiers(false)\n",
+       "        .extent([[0, 0], [width, height]])\n",
+       "        .on(\"start\", function() {\n",
+       "            if (d3.event.sourceEvent.type !== \"end\") {\n",
+       "                node.selectAll(\".selectable\").attr(\"style\", function(d) {\n",
+       "                    return nodeStyle(\n",
+       "                        d.selected = d.previouslySelected = shiftKey &&\n",
+       "                        d.selected);\n",
+       "                });\n",
+       "            }\n",
+       "        })\n",
+       "        .on(\"brush\", function() {\n",
+       "            if (d3.event.sourceEvent.type !== \"end\") {\n",
+       "                var selection = d3.event.selection;\n",
+       "                node.selectAll(\".selectable\").attr(\"style\", function(d) {\n",
+       "                    return nodeStyle(d.selected = d.previouslySelected ^\n",
+       "                        (selection != null\n",
+       "                            && selection[0][0] <= d.x && d.x < selection[1][0]\n",
+       "                            && selection[0][1] <= d.y && d.y < selection[1][1]));\n",
+       "                });\n",
+       "            }\n",
+       "        })\n",
+       "        .on(\"end\", function() {\n",
+       "            if (d3.event.selection != null) {\n",
+       "                d3.select(this).call(d3.event.target.move, null);\n",
+       "            }\n",
+       "        }));\n",
+       "}\n",
+       "\n",
+       "\n",
+       "showGraph('#graph-output-1Vwd2jRV',\n",
+       "  JSON.parse('{\"nodes\": [{\"name\": \"0\", \"x\": 50, \"y\": 100, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"1\", \"x\": 50, \"y\": 150, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"2\", \"x\": 50, \"y\": 200, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"3\", \"x\": 100, \"y\": 150, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"4\", \"x\": 100, \"y\": 100, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"5\", \"x\": 150, \"y\": 200, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"6\", \"x\": 150, \"y\": 150, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"7\", \"x\": 200, \"y\": 200, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"8\", \"x\": 200, \"y\": 100, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"9\", \"x\": 250, \"y\": 150, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"10\", \"x\": 250, \"y\": 200, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"11\", \"x\": 300, \"y\": 150, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"12\", \"x\": 300, \"y\": 100, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"13\", \"x\": 350, \"y\": 100, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"14\", \"x\": 350, \"y\": 200, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"15\", \"x\": 400, \"y\": 100, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"16\", \"x\": 400, \"y\": 150, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"17\", \"x\": 400, \"y\": 200, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}], \"links\": [{\"source\": \"0\", \"target\": \"4\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"1\", \"target\": \"3\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"2\", \"target\": \"5\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"3\", \"target\": \"4\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"3\", \"target\": \"6\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"4\", \"target\": \"8\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"5\", \"target\": \"6\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"5\", \"target\": \"7\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"6\", \"target\": \"9\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"7\", \"target\": \"8\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"7\", \"target\": \"10\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"8\", \"target\": \"12\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"9\", \"target\": \"10\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"9\", \"target\": \"11\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"10\", \"target\": \"14\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"11\", \"target\": \"12\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"11\", \"target\": \"16\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"12\", \"target\": \"13\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"13\", \"target\": \"14\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"13\", \"target\": \"15\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"14\", \"target\": \"17\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}], \"pauli_web\": [{\"source\": 3, \"target\": 1, \"t\": \"Z\"}, {\"source\": 1, \"target\": 3, \"t\": \"Z\"}, {\"source\": 3, \"target\": 4, \"t\": \"Z\"}, {\"source\": 4, \"target\": 3, \"t\": \"Z\"}, {\"source\": 3, \"target\": 6, \"t\": \"Z\"}, {\"source\": 6, \"target\": 3, \"t\": \"Z\"}, {\"source\": 5, \"target\": 2, \"t\": \"Z\"}, {\"source\": 2, \"target\": 5, \"t\": \"Z\"}, {\"source\": 5, \"target\": 6, \"t\": \"Z\"}, {\"source\": 6, \"target\": 5, \"t\": \"Z\"}, {\"source\": 5, \"target\": 7, \"t\": \"Z\"}, {\"source\": 7, \"target\": 5, \"t\": \"Z\"}, {\"source\": 7, \"target\": 8, \"t\": \"Z\"}, {\"source\": 8, \"target\": 7, \"t\": \"Z\"}, {\"source\": 7, \"target\": 10, \"t\": \"Z\"}, {\"source\": 10, \"target\": 7, \"t\": \"Z\"}, {\"source\": 13, \"target\": 12, \"t\": \"Z\"}, {\"source\": 12, \"target\": 13, \"t\": \"Z\"}, {\"source\": 13, \"target\": 14, \"t\": \"Z\"}, {\"source\": 14, \"target\": 13, \"t\": \"Z\"}, {\"source\": 13, \"target\": 15, \"t\": \"Z\"}, {\"source\": 15, \"target\": 13, \"t\": \"Z\"}, {\"source\": 8, \"target\": 12, \"t\": \"Z\"}, {\"source\": 12, \"target\": 8, \"t\": \"Z\"}, {\"source\": 10, \"target\": 14, \"t\": \"Z\"}, {\"source\": 14, \"target\": 10, \"t\": \"Z\"}, {\"source\": 0, \"target\": 4, \"t\": \"Z\"}, {\"source\": 4, \"target\": 0, \"t\": \"Z\"}]}'), 450, 250, 50,\n",
+       "  10.0, false, true, '');\n",
+       "</script>"
+      ],
+      "text/plain": [
+       "<IPython.core.display.HTML object>"
+      ]
+     },
+     "metadata": {},
+     "output_type": "display_data"
+    },
+    {
+     "data": {
+      "text/html": [
+       "<div style=\"overflow:auto; background-color: white\" id=\"graph-output-zNKtMoDU\"></div>\n",
+       "<script type=\"module\">\n",
+       "var d3;\n",
+       "if (d3 == null) { d3 = await import(\"https://cdn.skypack.dev/d3@5\"); }\n",
+       "// PyZX - Python library for quantum circuit rewriting \n",
+       "//        and optimisation using the ZX-calculus\n",
+       "// Copyright (C) 2018 - Aleks Kissinger and John van de Wetering\n",
+       "\n",
+       "// Licensed under the Apache License, Version 2.0 (the \"License\");\n",
+       "// you may not use this file except in compliance with the License.\n",
+       "// You may obtain a copy of the License at\n",
+       "\n",
+       "//    http://www.apache.org/licenses/LICENSE-2.0\n",
+       "\n",
+       "// Unless required by applicable law or agreed to in writing, software\n",
+       "// distributed under the License is distributed on an \"AS IS\" BASIS,\n",
+       "// WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.\n",
+       "// See the License for the specific language governing permissions and\n",
+       "// limitations under the License.\n",
+       "\n",
+       "// styling functions\n",
+       "function nodeColor(t) {\n",
+       "    if (t == 0) return \"black\";\n",
+       "    else if (t == 1) return \"#ccffcc\";\n",
+       "    else if (t == 2) return \"#ff8888\";\n",
+       "    else if (t == 3) return \"yellow\";\n",
+       "    else if (t == 4) return \"black\";\n",
+       "    else if (t == 5) return \"black\";\n",
+       "    else if (t == 6) return \"#ccffcc\";\n",
+       "}\n",
+       "\n",
+       "function edgeColor(t) {\n",
+       "    if (t == 1) return \"black\";\n",
+       "    else if (t == 2) return \"#08f\";\n",
+       "    else if (t == 3) return \"gray\";\n",
+       "}\n",
+       "\n",
+       "function webColor(t) {\n",
+       "    if (t == 'X') return \"#ff8888\";\n",
+       "    else if (t == 'Y') return \"#eebb00\";\n",
+       "    else if (t == 'Z') return \"#66cc66\";\n",
+       "    else if (t == 'I') return \"#eee\";\n",
+       "}\n",
+       "\n",
+       "function nodeStyle(selected) {\n",
+       "    return selected ? \"stroke-width: 2px; stroke: #00f\" : \"stroke-width: 1.5px\";\n",
+       "}\n",
+       "\n",
+       "var symbolGround = {\n",
+       "    draw: function(context, size){\n",
+       "        let s = size/2;\n",
+       "\n",
+       "        context.moveTo(0,-s);\n",
+       "        context.lineTo(0,0);\n",
+       "\n",
+       "        context.moveTo(-s,0);\n",
+       "        context.lineTo(s,0);\n",
+       "\n",
+       "        context.moveTo(-2*s/3,s/3);\n",
+       "        context.lineTo(2*s/3,s/3);\n",
+       "\n",
+       "        context.moveTo(-s/3,2*s/3);\n",
+       "        context.lineTo(s/3,2*s/3);\n",
+       "    }\n",
+       "}\n",
+       "\n",
+       "function showGraph(tag, graph, width, height, scale, node_size, auto_hbox, show_labels, scalar_str) {\n",
+       "    var ntab = {};\n",
+       "\n",
+       "    var groundOffset = 2.5 * node_size;\n",
+       "\n",
+       "    graph.nodes.forEach(function(d) {\n",
+       "        ntab[d.name] = d;\n",
+       "        d.selected = false;\n",
+       "        d.previouslySelected = false;\n",
+       "        d.nhd = [];\n",
+       "    });\n",
+       "\n",
+       "    var spiders_and_boundaries = graph.nodes.filter(function(d) {\n",
+       "        return d.t != 3;\n",
+       "    });\n",
+       "\n",
+       "    graph.links.forEach(function(d) {\n",
+       "        var s = ntab[d.source];\n",
+       "        var t = ntab[d.target];\n",
+       "        d.source = s;\n",
+       "        d.target = t;\n",
+       "        s.nhd.push(t);\n",
+       "        t.nhd.push(s);\n",
+       "    });\n",
+       "\n",
+       "    graph.pauli_web.forEach(function(d) {\n",
+       "        var s = ntab[d.source];\n",
+       "        var t = ntab[d.target];\n",
+       "        d.source = s;\n",
+       "        d.target = t;\n",
+       "    });\n",
+       "\n",
+       "    var shiftKey;\n",
+       "\n",
+       "    // SETUP SVG ITEMS\n",
+       "\n",
+       "    var svg = d3.select(tag)\n",
+       "    //.attr(\"tabindex\", 1)\n",
+       "        .on(\"keydown.brush\", function() {shiftKey = d3.event.shiftKey || d3.event.metaKey;})\n",
+       "        .on(\"keyup.brush\", function() {shiftKey = d3.event.shiftKey || d3.event.metaKey;})\n",
+       "    //.each(function() { this.focus(); })\n",
+       "        .append(\"svg\")\n",
+       "        .attr(\"style\", \"max-width: none; max-height: none\")\n",
+       "        .attr(\"width\", width)\n",
+       "        .attr(\"height\", height);\n",
+       "\n",
+       "    var web = svg.append(\"g\")\n",
+       "        .attr(\"class\", \"web\")\n",
+       "        .selectAll(\"line\")\n",
+       "        .data(graph.pauli_web)\n",
+       "        .enter().append(\"path\")\n",
+       "        .attr(\"stroke\", function(d) { return webColor(d.t); })\n",
+       "        .attr(\"fill\", \"transparent\")\n",
+       "        .attr(\"style\", \"stroke-width: 7px\");\n",
+       "\n",
+       "    var link = svg.append(\"g\")\n",
+       "        .attr(\"class\", \"link\")\n",
+       "        .selectAll(\"line\")\n",
+       "        .data(graph.links)\n",
+       "        .enter().append(\"path\")\n",
+       "        .attr(\"stroke\", function(d) { return edgeColor(d.t); })\n",
+       "        .attr(\"fill\", \"transparent\")\n",
+       "        .attr(\"style\", \"stroke-width: 1.5px\");\n",
+       "\n",
+       "    var brush = svg.append(\"g\")\n",
+       "        .attr(\"class\", \"brush\");\n",
+       "\n",
+       "    var node = svg.append(\"g\")\n",
+       "        .attr(\"class\", \"node\")\n",
+       "        .selectAll(\"g\")\n",
+       "        .data(graph.nodes)\n",
+       "        .enter().append(\"g\")\n",
+       "        .attr(\"transform\", function(d) {\n",
+       "            return \"translate(\" + d.x + \",\" + d.y +\")\";\n",
+       "        });\n",
+       "\n",
+       "    // Draw a ground symbol connected to the node.\n",
+       "    node.filter(function(d) { return d.ground; })\n",
+       "        .append(\"path\")\n",
+       "        .attr(\"stroke\", \"black\")\n",
+       "        .attr(\"style\", \"stroke-width: 1.5px\")\n",
+       "        .attr(\"fill\", \"none\")\n",
+       "        .attr(\"d\", \"M 0 0 L 0 \"+(groundOffset))\n",
+       "        .attr(\"class\", \"selectable\");\n",
+       "    node.filter(function(d) { return d.ground; })\n",
+       "        .append(\"path\")\n",
+       "        .attr(\"stroke\", \"black\")\n",
+       "        .attr(\"style\", \"stroke-width: 1.5px\")\n",
+       "        .attr(\"fill\", \"none\")\n",
+       "        .attr(\"d\", d3.symbol().type(symbolGround).size(node_size*1.5))\n",
+       "        .attr(\"transform\", \"translate(0,\"+groundOffset+\")\")\n",
+       "        .attr(\"class\", \"selectable\");\n",
+       "\n",
+       "    node.filter(function(d) { return d.t != 3 && d.t != 5 && d.t != 6; })\n",
+       "        .append(\"circle\")\n",
+       "        .attr(\"r\", function(d) {\n",
+       "            if (d.t == 0) return 0.5 * node_size;\n",
+       "            else if (d.t == 4) return 0.25 * node_size;\n",
+       "            else return node_size;\n",
+       "        })\n",
+       "        .attr(\"fill\", function(d) { return nodeColor(d.t); })\n",
+       "        .attr(\"stroke\", \"black\")\n",
+       "        .attr(\"class\", \"selectable\");\n",
+       "\n",
+       "    var hbox = node.filter(function(d) { return d.t == 3; });\n",
+       "\n",
+       "    hbox.append(\"rect\")\n",
+       "        .attr(\"x\", -0.75 * node_size).attr(\"y\", -0.75 * node_size)\n",
+       "        .attr(\"width\", node_size * 1.5).attr(\"height\", node_size * 1.5)\n",
+       "        .attr(\"fill\", function(d) { return nodeColor(d.t); })\n",
+       "        .attr(\"stroke\", \"black\")\n",
+       "        .attr(\"class\", \"selectable\");\n",
+       "\n",
+       "    // draw a triangle for d.t == 5\n",
+       "    node.filter(function(d) { return d.t == 5; })\n",
+       "        .append(\"path\")\n",
+       "        .attr(\"d\", \"M 0 0 L \"+node_size+\" \"+node_size+\" L -\"+node_size+\" \"+node_size+\" Z\")\n",
+       "        .attr(\"fill\", function(d) { return nodeColor(d.t); })\n",
+       "        .attr(\"stroke\", \"black\")\n",
+       "        .attr(\"class\", \"selectable\")\n",
+       "        .attr(\"transform\", \"translate(\" + (-node_size/2) + \", 0) rotate(-90)\");\n",
+       "\n",
+       "    // draw a square for Z box: d.t == 6\n",
+       "    node.filter(function(d) { return d.t == 6; })\n",
+       "        .append(\"rect\")\n",
+       "        .attr(\"x\", -0.75 * node_size).attr(\"y\", -0.75 * node_size)\n",
+       "        .attr(\"width\", node_size * 1.5).attr(\"height\", node_size * 1.5)\n",
+       "        .attr(\"fill\", function(d) { return nodeColor(d.t); })\n",
+       "        .attr(\"stroke\", \"black\")\n",
+       "        .attr(\"class\", \"selectable\");\n",
+       "\n",
+       "    node.filter(function(d) { return d.phase != ''; })\n",
+       "        .append(\"text\")\n",
+       "        .attr(\"y\", 0.7 * node_size + 14)\n",
+       "        .text(function (d) { return d.phase })\n",
+       "        .attr(\"text-anchor\", \"middle\")\n",
+       "        .attr(\"font-size\", \"12px\")\n",
+       "        .attr(\"font-family\", \"monospace\")\n",
+       "        .attr(\"fill\", \"#00d\")\n",
+       "        .attr('style', 'pointer-events: none; user-select: none;');\n",
+       "\n",
+       "    if (show_labels) {\n",
+       "        node.append(\"text\")\n",
+       "            .attr(\"y\", -0.7 * node_size - 8)\n",
+       "            .text(function (d) { return d.name; })\n",
+       "            .attr(\"text-anchor\", \"middle\")\n",
+       "            .attr(\"font-size\", \"10px\")\n",
+       "            .attr(\"font-family\", \"monospace\")\n",
+       "            .attr(\"fill\", \"#999\")\n",
+       "            .attr('style', 'pointer-events: none; user-select: none;');\n",
+       "    }\n",
+       "\n",
+       "    // Display the chosen data fields over the node.\n",
+       "    node.filter(d => d.vdata.length > 0)\n",
+       "        .append(\"text\")\n",
+       "        .attr(\"y\", d => -0.7 * node_size - 14 - 10 * d.vdata.length)\n",
+       "        .attr(\"text-anchor\", \"middle\")\n",
+       "        .attr(\"font-size\", \"8px\")\n",
+       "        .attr(\"font-family\", \"monospace\")\n",
+       "        .attr(\"fill\", \"#c66\")\n",
+       "        .attr('style', 'pointer-events: none; user-select: none;')\n",
+       "        .selectAll(\"tspan\")\n",
+       "        .data(d => d.vdata)\n",
+       "        .enter()\n",
+       "        .append(\"tspan\")\n",
+       "        .attr(\"x\", \"0\")\n",
+       "        .attr(\"dy\", \"1.2em\")\n",
+       "        .text(x => x.join(\": \"));\n",
+       "\n",
+       "    if (scalar_str != \"\") {\n",
+       "        svg.append(\"text\")\n",
+       "            .text(scalar_str)\n",
+       "            .attr(\"x\", 60).attr(\"y\", 40)\n",
+       "            .attr(\"text-anchor\", \"middle\")\n",
+       "    }\n",
+       "\n",
+       "    function update_hboxes() {\n",
+       "        if (auto_hbox) {\n",
+       "            var pos = {};\n",
+       "            hbox.attr(\"transform\", function(d) {\n",
+       "                // calculate barycenter of non-hbox neighbours, then nudge a bit\n",
+       "                // to the NE.\n",
+       "                var x=0,y=0,sz=0;\n",
+       "                for (var i = 0; i < d.nhd.length; ++i) {\n",
+       "                    if (d.nhd[i].t != 3) {\n",
+       "                        sz++;\n",
+       "                        x += d.nhd[i].x;\n",
+       "                        y += d.nhd[i].y;\n",
+       "                    }\n",
+       "                }\n",
+       "\n",
+       "                offset = 0.25 * scale;\n",
+       "\n",
+       "                if (sz != 0) {\n",
+       "                    x = (x/sz) + offset;\n",
+       "                    y = (y/sz) - offset;\n",
+       "\n",
+       "                    while (pos[[x,y]]) {\n",
+       "                        x += offset;\n",
+       "                    }\n",
+       "                    d.x = x;\n",
+       "                    d.y = y;\n",
+       "                    pos[[x,y]] = true;\n",
+       "                }\n",
+       "\n",
+       "                return \"translate(\"+d.x+\",\"+d.y+\")\";\n",
+       "            });\n",
+       "        }\n",
+       "    }\n",
+       "\n",
+       "    update_hboxes();\n",
+       "\n",
+       "    var link_curve = function(d) {\n",
+       "        var x1 = d.source.x, x2 = d.target.x, y1 = d.source.y, y2 = d.target.y;\n",
+       "        if (d.num_parallel == 1) {\n",
+       "            return `M ${x1} ${y1} L ${x2} ${y2}`;\n",
+       "        } else {\n",
+       "            var dx = x2 - x1, dy = y2 - y1;\n",
+       "            var midx = 0.5 * (x1 + x2), midy = 0.5 * (y1 + y2);\n",
+       "            var pos = (d.index / (d.num_parallel-1)) - 0.5;\n",
+       "            var cx = midx - pos * dy;\n",
+       "            var cy = midy + pos * dx;\n",
+       "            return `M ${x1} ${y1} Q ${cx} ${cy}, ${x2} ${y2}`;\n",
+       "            // return `M ${x1} ${y1} L ${x2} ${y2}`;\n",
+       "        }\n",
+       "    };\n",
+       "    link.attr(\"d\", link_curve);\n",
+       "\n",
+       "\n",
+       "    var web_curve = function(d) {\n",
+       "        var x1 = d.source.x, x2 = (x1 + d.target.x)/2, y1 = d.source.y, y2 = (y1 + d.target.y)/2;\n",
+       "        return `M ${x1} ${y1} L ${x2} ${y2}`;\n",
+       "    }\n",
+       "    web.attr(\"d\", web_curve);\n",
+       "\n",
+       "    // EVENTS FOR DRAGGING AND SELECTION\n",
+       "\n",
+       "    node.on(\"mousedown\", function(d) {\n",
+       "        if (shiftKey) {\n",
+       "            d3.select(this).selectAll(\".selectable\").attr(\"style\", nodeStyle(d.selected = !d.selected));\n",
+       "            d3.event.stopImmediatePropagation();\n",
+       "        } else if (!d.selected) {\n",
+       "            node.selectAll(\".selectable\").attr(\"style\", function(p) { return nodeStyle(p.selected = d === p); });\n",
+       "        }\n",
+       "    })\n",
+       "        .call(d3.drag().on(\"drag\", function(d) {\n",
+       "            var dx = d3.event.dx;\n",
+       "            var dy = d3.event.dy;\n",
+       "            // node.filter(function(d) { return d.selected; })\n",
+       "            //     .attr(\"cx\", function(d) { return d.x += dx; })\n",
+       "            //     .attr(\"cy\", function(d) { return d.y += dy; });\n",
+       "            node.filter(function(d) { return d.selected; })\n",
+       "                .attr(\"transform\", function(d) {\n",
+       "                    d.x += dx;\n",
+       "                    d.y += dy;\n",
+       "                    return \"translate(\" + d.x + \",\" + d.y +\")\";\n",
+       "                });\n",
+       "\n",
+       "            update_hboxes();\n",
+       "\n",
+       "            link.filter(function(d) { return d.source.selected || d.target.selected ||\n",
+       "                    (auto_hbox && d.source.t == 3); })\n",
+       "                .attr(\"d\", link_curve);\n",
+       "            web.filter(function(d) { return d.source.selected || d.target.selected; })\n",
+       "                .attr(\"d\", web_curve);\n",
+       "        }));\n",
+       "\n",
+       "    brush.call(d3.brush().keyModifiers(false)\n",
+       "        .extent([[0, 0], [width, height]])\n",
+       "        .on(\"start\", function() {\n",
+       "            if (d3.event.sourceEvent.type !== \"end\") {\n",
+       "                node.selectAll(\".selectable\").attr(\"style\", function(d) {\n",
+       "                    return nodeStyle(\n",
+       "                        d.selected = d.previouslySelected = shiftKey &&\n",
+       "                        d.selected);\n",
+       "                });\n",
+       "            }\n",
+       "        })\n",
+       "        .on(\"brush\", function() {\n",
+       "            if (d3.event.sourceEvent.type !== \"end\") {\n",
+       "                var selection = d3.event.selection;\n",
+       "                node.selectAll(\".selectable\").attr(\"style\", function(d) {\n",
+       "                    return nodeStyle(d.selected = d.previouslySelected ^\n",
+       "                        (selection != null\n",
+       "                            && selection[0][0] <= d.x && d.x < selection[1][0]\n",
+       "                            && selection[0][1] <= d.y && d.y < selection[1][1]));\n",
+       "                });\n",
+       "            }\n",
+       "        })\n",
+       "        .on(\"end\", function() {\n",
+       "            if (d3.event.selection != null) {\n",
+       "                d3.select(this).call(d3.event.target.move, null);\n",
+       "            }\n",
+       "        }));\n",
+       "}\n",
+       "\n",
+       "\n",
+       "showGraph('#graph-output-zNKtMoDU',\n",
+       "  JSON.parse('{\"nodes\": [{\"name\": \"0\", \"x\": 50, \"y\": 100, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"1\", \"x\": 50, \"y\": 150, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"2\", \"x\": 50, \"y\": 200, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"3\", \"x\": 100, \"y\": 150, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"4\", \"x\": 100, \"y\": 100, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"5\", \"x\": 150, \"y\": 200, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"6\", \"x\": 150, \"y\": 150, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"7\", \"x\": 200, \"y\": 200, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"8\", \"x\": 200, \"y\": 100, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"9\", \"x\": 250, \"y\": 150, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"10\", \"x\": 250, \"y\": 200, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"11\", \"x\": 300, \"y\": 150, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"12\", \"x\": 300, \"y\": 100, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"13\", \"x\": 350, \"y\": 100, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"14\", \"x\": 350, \"y\": 200, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"15\", \"x\": 400, \"y\": 100, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"16\", \"x\": 400, \"y\": 150, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"17\", \"x\": 400, \"y\": 200, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}], \"links\": [{\"source\": \"0\", \"target\": \"4\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"1\", \"target\": \"3\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"2\", \"target\": \"5\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"3\", \"target\": \"4\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"3\", \"target\": \"6\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"4\", \"target\": \"8\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"5\", \"target\": \"6\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"5\", \"target\": \"7\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"6\", \"target\": \"9\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"7\", \"target\": \"8\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"7\", \"target\": \"10\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"8\", \"target\": \"12\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"9\", \"target\": \"10\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"9\", \"target\": \"11\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"10\", \"target\": \"14\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"11\", \"target\": \"12\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"11\", \"target\": \"16\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"12\", \"target\": \"13\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"13\", \"target\": \"14\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"13\", \"target\": \"15\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"14\", \"target\": \"17\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}], \"pauli_web\": [{\"source\": 6, \"target\": 3, \"t\": \"X\"}, {\"source\": 3, \"target\": 6, \"t\": \"X\"}, {\"source\": 6, \"target\": 5, \"t\": \"X\"}, {\"source\": 5, \"target\": 6, \"t\": \"X\"}, {\"source\": 6, \"target\": 9, \"t\": \"X\"}, {\"source\": 9, \"target\": 6, \"t\": \"X\"}, {\"source\": 9, \"target\": 11, \"t\": \"X\"}, {\"source\": 11, \"target\": 9, \"t\": \"X\"}, {\"source\": 1, \"target\": 3, \"t\": \"X\"}, {\"source\": 3, \"target\": 1, \"t\": \"X\"}, {\"source\": 2, \"target\": 5, \"t\": \"X\"}, {\"source\": 5, \"target\": 2, \"t\": \"X\"}, {\"source\": 16, \"target\": 11, \"t\": \"X\"}, {\"source\": 11, \"target\": 16, \"t\": \"X\"}]}'), 450, 250, 50,\n",
+       "  10.0, false, true, '');\n",
+       "</script>"
+      ],
+      "text/plain": [
+       "<IPython.core.display.HTML object>"
+      ]
+     },
+     "metadata": {},
+     "output_type": "display_data"
+    },
+    {
+     "data": {
+      "text/html": [
+       "<div style=\"overflow:auto; background-color: white\" id=\"graph-output-Zqc7uEpe\"></div>\n",
+       "<script type=\"module\">\n",
+       "var d3;\n",
+       "if (d3 == null) { d3 = await import(\"https://cdn.skypack.dev/d3@5\"); }\n",
+       "// PyZX - Python library for quantum circuit rewriting \n",
+       "//        and optimisation using the ZX-calculus\n",
+       "// Copyright (C) 2018 - Aleks Kissinger and John van de Wetering\n",
+       "\n",
+       "// Licensed under the Apache License, Version 2.0 (the \"License\");\n",
+       "// you may not use this file except in compliance with the License.\n",
+       "// You may obtain a copy of the License at\n",
+       "\n",
+       "//    http://www.apache.org/licenses/LICENSE-2.0\n",
+       "\n",
+       "// Unless required by applicable law or agreed to in writing, software\n",
+       "// distributed under the License is distributed on an \"AS IS\" BASIS,\n",
+       "// WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.\n",
+       "// See the License for the specific language governing permissions and\n",
+       "// limitations under the License.\n",
+       "\n",
+       "// styling functions\n",
+       "function nodeColor(t) {\n",
+       "    if (t == 0) return \"black\";\n",
+       "    else if (t == 1) return \"#ccffcc\";\n",
+       "    else if (t == 2) return \"#ff8888\";\n",
+       "    else if (t == 3) return \"yellow\";\n",
+       "    else if (t == 4) return \"black\";\n",
+       "    else if (t == 5) return \"black\";\n",
+       "    else if (t == 6) return \"#ccffcc\";\n",
+       "}\n",
+       "\n",
+       "function edgeColor(t) {\n",
+       "    if (t == 1) return \"black\";\n",
+       "    else if (t == 2) return \"#08f\";\n",
+       "    else if (t == 3) return \"gray\";\n",
+       "}\n",
+       "\n",
+       "function webColor(t) {\n",
+       "    if (t == 'X') return \"#ff8888\";\n",
+       "    else if (t == 'Y') return \"#eebb00\";\n",
+       "    else if (t == 'Z') return \"#66cc66\";\n",
+       "    else if (t == 'I') return \"#eee\";\n",
+       "}\n",
+       "\n",
+       "function nodeStyle(selected) {\n",
+       "    return selected ? \"stroke-width: 2px; stroke: #00f\" : \"stroke-width: 1.5px\";\n",
+       "}\n",
+       "\n",
+       "var symbolGround = {\n",
+       "    draw: function(context, size){\n",
+       "        let s = size/2;\n",
+       "\n",
+       "        context.moveTo(0,-s);\n",
+       "        context.lineTo(0,0);\n",
+       "\n",
+       "        context.moveTo(-s,0);\n",
+       "        context.lineTo(s,0);\n",
+       "\n",
+       "        context.moveTo(-2*s/3,s/3);\n",
+       "        context.lineTo(2*s/3,s/3);\n",
+       "\n",
+       "        context.moveTo(-s/3,2*s/3);\n",
+       "        context.lineTo(s/3,2*s/3);\n",
+       "    }\n",
+       "}\n",
+       "\n",
+       "function showGraph(tag, graph, width, height, scale, node_size, auto_hbox, show_labels, scalar_str) {\n",
+       "    var ntab = {};\n",
+       "\n",
+       "    var groundOffset = 2.5 * node_size;\n",
+       "\n",
+       "    graph.nodes.forEach(function(d) {\n",
+       "        ntab[d.name] = d;\n",
+       "        d.selected = false;\n",
+       "        d.previouslySelected = false;\n",
+       "        d.nhd = [];\n",
+       "    });\n",
+       "\n",
+       "    var spiders_and_boundaries = graph.nodes.filter(function(d) {\n",
+       "        return d.t != 3;\n",
+       "    });\n",
+       "\n",
+       "    graph.links.forEach(function(d) {\n",
+       "        var s = ntab[d.source];\n",
+       "        var t = ntab[d.target];\n",
+       "        d.source = s;\n",
+       "        d.target = t;\n",
+       "        s.nhd.push(t);\n",
+       "        t.nhd.push(s);\n",
+       "    });\n",
+       "\n",
+       "    graph.pauli_web.forEach(function(d) {\n",
+       "        var s = ntab[d.source];\n",
+       "        var t = ntab[d.target];\n",
+       "        d.source = s;\n",
+       "        d.target = t;\n",
+       "    });\n",
+       "\n",
+       "    var shiftKey;\n",
+       "\n",
+       "    // SETUP SVG ITEMS\n",
+       "\n",
+       "    var svg = d3.select(tag)\n",
+       "    //.attr(\"tabindex\", 1)\n",
+       "        .on(\"keydown.brush\", function() {shiftKey = d3.event.shiftKey || d3.event.metaKey;})\n",
+       "        .on(\"keyup.brush\", function() {shiftKey = d3.event.shiftKey || d3.event.metaKey;})\n",
+       "    //.each(function() { this.focus(); })\n",
+       "        .append(\"svg\")\n",
+       "        .attr(\"style\", \"max-width: none; max-height: none\")\n",
+       "        .attr(\"width\", width)\n",
+       "        .attr(\"height\", height);\n",
+       "\n",
+       "    var web = svg.append(\"g\")\n",
+       "        .attr(\"class\", \"web\")\n",
+       "        .selectAll(\"line\")\n",
+       "        .data(graph.pauli_web)\n",
+       "        .enter().append(\"path\")\n",
+       "        .attr(\"stroke\", function(d) { return webColor(d.t); })\n",
+       "        .attr(\"fill\", \"transparent\")\n",
+       "        .attr(\"style\", \"stroke-width: 7px\");\n",
+       "\n",
+       "    var link = svg.append(\"g\")\n",
+       "        .attr(\"class\", \"link\")\n",
+       "        .selectAll(\"line\")\n",
+       "        .data(graph.links)\n",
+       "        .enter().append(\"path\")\n",
+       "        .attr(\"stroke\", function(d) { return edgeColor(d.t); })\n",
+       "        .attr(\"fill\", \"transparent\")\n",
+       "        .attr(\"style\", \"stroke-width: 1.5px\");\n",
+       "\n",
+       "    var brush = svg.append(\"g\")\n",
+       "        .attr(\"class\", \"brush\");\n",
+       "\n",
+       "    var node = svg.append(\"g\")\n",
+       "        .attr(\"class\", \"node\")\n",
+       "        .selectAll(\"g\")\n",
+       "        .data(graph.nodes)\n",
+       "        .enter().append(\"g\")\n",
+       "        .attr(\"transform\", function(d) {\n",
+       "            return \"translate(\" + d.x + \",\" + d.y +\")\";\n",
+       "        });\n",
+       "\n",
+       "    // Draw a ground symbol connected to the node.\n",
+       "    node.filter(function(d) { return d.ground; })\n",
+       "        .append(\"path\")\n",
+       "        .attr(\"stroke\", \"black\")\n",
+       "        .attr(\"style\", \"stroke-width: 1.5px\")\n",
+       "        .attr(\"fill\", \"none\")\n",
+       "        .attr(\"d\", \"M 0 0 L 0 \"+(groundOffset))\n",
+       "        .attr(\"class\", \"selectable\");\n",
+       "    node.filter(function(d) { return d.ground; })\n",
+       "        .append(\"path\")\n",
+       "        .attr(\"stroke\", \"black\")\n",
+       "        .attr(\"style\", \"stroke-width: 1.5px\")\n",
+       "        .attr(\"fill\", \"none\")\n",
+       "        .attr(\"d\", d3.symbol().type(symbolGround).size(node_size*1.5))\n",
+       "        .attr(\"transform\", \"translate(0,\"+groundOffset+\")\")\n",
+       "        .attr(\"class\", \"selectable\");\n",
+       "\n",
+       "    node.filter(function(d) { return d.t != 3 && d.t != 5 && d.t != 6; })\n",
+       "        .append(\"circle\")\n",
+       "        .attr(\"r\", function(d) {\n",
+       "            if (d.t == 0) return 0.5 * node_size;\n",
+       "            else if (d.t == 4) return 0.25 * node_size;\n",
+       "            else return node_size;\n",
+       "        })\n",
+       "        .attr(\"fill\", function(d) { return nodeColor(d.t); })\n",
+       "        .attr(\"stroke\", \"black\")\n",
+       "        .attr(\"class\", \"selectable\");\n",
+       "\n",
+       "    var hbox = node.filter(function(d) { return d.t == 3; });\n",
+       "\n",
+       "    hbox.append(\"rect\")\n",
+       "        .attr(\"x\", -0.75 * node_size).attr(\"y\", -0.75 * node_size)\n",
+       "        .attr(\"width\", node_size * 1.5).attr(\"height\", node_size * 1.5)\n",
+       "        .attr(\"fill\", function(d) { return nodeColor(d.t); })\n",
+       "        .attr(\"stroke\", \"black\")\n",
+       "        .attr(\"class\", \"selectable\");\n",
+       "\n",
+       "    // draw a triangle for d.t == 5\n",
+       "    node.filter(function(d) { return d.t == 5; })\n",
+       "        .append(\"path\")\n",
+       "        .attr(\"d\", \"M 0 0 L \"+node_size+\" \"+node_size+\" L -\"+node_size+\" \"+node_size+\" Z\")\n",
+       "        .attr(\"fill\", function(d) { return nodeColor(d.t); })\n",
+       "        .attr(\"stroke\", \"black\")\n",
+       "        .attr(\"class\", \"selectable\")\n",
+       "        .attr(\"transform\", \"translate(\" + (-node_size/2) + \", 0) rotate(-90)\");\n",
+       "\n",
+       "    // draw a square for Z box: d.t == 6\n",
+       "    node.filter(function(d) { return d.t == 6; })\n",
+       "        .append(\"rect\")\n",
+       "        .attr(\"x\", -0.75 * node_size).attr(\"y\", -0.75 * node_size)\n",
+       "        .attr(\"width\", node_size * 1.5).attr(\"height\", node_size * 1.5)\n",
+       "        .attr(\"fill\", function(d) { return nodeColor(d.t); })\n",
+       "        .attr(\"stroke\", \"black\")\n",
+       "        .attr(\"class\", \"selectable\");\n",
+       "\n",
+       "    node.filter(function(d) { return d.phase != ''; })\n",
+       "        .append(\"text\")\n",
+       "        .attr(\"y\", 0.7 * node_size + 14)\n",
+       "        .text(function (d) { return d.phase })\n",
+       "        .attr(\"text-anchor\", \"middle\")\n",
+       "        .attr(\"font-size\", \"12px\")\n",
+       "        .attr(\"font-family\", \"monospace\")\n",
+       "        .attr(\"fill\", \"#00d\")\n",
+       "        .attr('style', 'pointer-events: none; user-select: none;');\n",
+       "\n",
+       "    if (show_labels) {\n",
+       "        node.append(\"text\")\n",
+       "            .attr(\"y\", -0.7 * node_size - 8)\n",
+       "            .text(function (d) { return d.name; })\n",
+       "            .attr(\"text-anchor\", \"middle\")\n",
+       "            .attr(\"font-size\", \"10px\")\n",
+       "            .attr(\"font-family\", \"monospace\")\n",
+       "            .attr(\"fill\", \"#999\")\n",
+       "            .attr('style', 'pointer-events: none; user-select: none;');\n",
+       "    }\n",
+       "\n",
+       "    // Display the chosen data fields over the node.\n",
+       "    node.filter(d => d.vdata.length > 0)\n",
+       "        .append(\"text\")\n",
+       "        .attr(\"y\", d => -0.7 * node_size - 14 - 10 * d.vdata.length)\n",
+       "        .attr(\"text-anchor\", \"middle\")\n",
+       "        .attr(\"font-size\", \"8px\")\n",
+       "        .attr(\"font-family\", \"monospace\")\n",
+       "        .attr(\"fill\", \"#c66\")\n",
+       "        .attr('style', 'pointer-events: none; user-select: none;')\n",
+       "        .selectAll(\"tspan\")\n",
+       "        .data(d => d.vdata)\n",
+       "        .enter()\n",
+       "        .append(\"tspan\")\n",
+       "        .attr(\"x\", \"0\")\n",
+       "        .attr(\"dy\", \"1.2em\")\n",
+       "        .text(x => x.join(\": \"));\n",
+       "\n",
+       "    if (scalar_str != \"\") {\n",
+       "        svg.append(\"text\")\n",
+       "            .text(scalar_str)\n",
+       "            .attr(\"x\", 60).attr(\"y\", 40)\n",
+       "            .attr(\"text-anchor\", \"middle\")\n",
+       "    }\n",
+       "\n",
+       "    function update_hboxes() {\n",
+       "        if (auto_hbox) {\n",
+       "            var pos = {};\n",
+       "            hbox.attr(\"transform\", function(d) {\n",
+       "                // calculate barycenter of non-hbox neighbours, then nudge a bit\n",
+       "                // to the NE.\n",
+       "                var x=0,y=0,sz=0;\n",
+       "                for (var i = 0; i < d.nhd.length; ++i) {\n",
+       "                    if (d.nhd[i].t != 3) {\n",
+       "                        sz++;\n",
+       "                        x += d.nhd[i].x;\n",
+       "                        y += d.nhd[i].y;\n",
+       "                    }\n",
+       "                }\n",
+       "\n",
+       "                offset = 0.25 * scale;\n",
+       "\n",
+       "                if (sz != 0) {\n",
+       "                    x = (x/sz) + offset;\n",
+       "                    y = (y/sz) - offset;\n",
+       "\n",
+       "                    while (pos[[x,y]]) {\n",
+       "                        x += offset;\n",
+       "                    }\n",
+       "                    d.x = x;\n",
+       "                    d.y = y;\n",
+       "                    pos[[x,y]] = true;\n",
+       "                }\n",
+       "\n",
+       "                return \"translate(\"+d.x+\",\"+d.y+\")\";\n",
+       "            });\n",
+       "        }\n",
+       "    }\n",
+       "\n",
+       "    update_hboxes();\n",
+       "\n",
+       "    var link_curve = function(d) {\n",
+       "        var x1 = d.source.x, x2 = d.target.x, y1 = d.source.y, y2 = d.target.y;\n",
+       "        if (d.num_parallel == 1) {\n",
+       "            return `M ${x1} ${y1} L ${x2} ${y2}`;\n",
+       "        } else {\n",
+       "            var dx = x2 - x1, dy = y2 - y1;\n",
+       "            var midx = 0.5 * (x1 + x2), midy = 0.5 * (y1 + y2);\n",
+       "            var pos = (d.index / (d.num_parallel-1)) - 0.5;\n",
+       "            var cx = midx - pos * dy;\n",
+       "            var cy = midy + pos * dx;\n",
+       "            return `M ${x1} ${y1} Q ${cx} ${cy}, ${x2} ${y2}`;\n",
+       "            // return `M ${x1} ${y1} L ${x2} ${y2}`;\n",
+       "        }\n",
+       "    };\n",
+       "    link.attr(\"d\", link_curve);\n",
+       "\n",
+       "\n",
+       "    var web_curve = function(d) {\n",
+       "        var x1 = d.source.x, x2 = (x1 + d.target.x)/2, y1 = d.source.y, y2 = (y1 + d.target.y)/2;\n",
+       "        return `M ${x1} ${y1} L ${x2} ${y2}`;\n",
+       "    }\n",
+       "    web.attr(\"d\", web_curve);\n",
+       "\n",
+       "    // EVENTS FOR DRAGGING AND SELECTION\n",
+       "\n",
+       "    node.on(\"mousedown\", function(d) {\n",
+       "        if (shiftKey) {\n",
+       "            d3.select(this).selectAll(\".selectable\").attr(\"style\", nodeStyle(d.selected = !d.selected));\n",
+       "            d3.event.stopImmediatePropagation();\n",
+       "        } else if (!d.selected) {\n",
+       "            node.selectAll(\".selectable\").attr(\"style\", function(p) { return nodeStyle(p.selected = d === p); });\n",
+       "        }\n",
+       "    })\n",
+       "        .call(d3.drag().on(\"drag\", function(d) {\n",
+       "            var dx = d3.event.dx;\n",
+       "            var dy = d3.event.dy;\n",
+       "            // node.filter(function(d) { return d.selected; })\n",
+       "            //     .attr(\"cx\", function(d) { return d.x += dx; })\n",
+       "            //     .attr(\"cy\", function(d) { return d.y += dy; });\n",
+       "            node.filter(function(d) { return d.selected; })\n",
+       "                .attr(\"transform\", function(d) {\n",
+       "                    d.x += dx;\n",
+       "                    d.y += dy;\n",
+       "                    return \"translate(\" + d.x + \",\" + d.y +\")\";\n",
+       "                });\n",
+       "\n",
+       "            update_hboxes();\n",
+       "\n",
+       "            link.filter(function(d) { return d.source.selected || d.target.selected ||\n",
+       "                    (auto_hbox && d.source.t == 3); })\n",
+       "                .attr(\"d\", link_curve);\n",
+       "            web.filter(function(d) { return d.source.selected || d.target.selected; })\n",
+       "                .attr(\"d\", web_curve);\n",
+       "        }));\n",
+       "\n",
+       "    brush.call(d3.brush().keyModifiers(false)\n",
+       "        .extent([[0, 0], [width, height]])\n",
+       "        .on(\"start\", function() {\n",
+       "            if (d3.event.sourceEvent.type !== \"end\") {\n",
+       "                node.selectAll(\".selectable\").attr(\"style\", function(d) {\n",
+       "                    return nodeStyle(\n",
+       "                        d.selected = d.previouslySelected = shiftKey &&\n",
+       "                        d.selected);\n",
+       "                });\n",
+       "            }\n",
+       "        })\n",
+       "        .on(\"brush\", function() {\n",
+       "            if (d3.event.sourceEvent.type !== \"end\") {\n",
+       "                var selection = d3.event.selection;\n",
+       "                node.selectAll(\".selectable\").attr(\"style\", function(d) {\n",
+       "                    return nodeStyle(d.selected = d.previouslySelected ^\n",
+       "                        (selection != null\n",
+       "                            && selection[0][0] <= d.x && d.x < selection[1][0]\n",
+       "                            && selection[0][1] <= d.y && d.y < selection[1][1]));\n",
+       "                });\n",
+       "            }\n",
+       "        })\n",
+       "        .on(\"end\", function() {\n",
+       "            if (d3.event.selection != null) {\n",
+       "                d3.select(this).call(d3.event.target.move, null);\n",
+       "            }\n",
+       "        }));\n",
+       "}\n",
+       "\n",
+       "\n",
+       "showGraph('#graph-output-Zqc7uEpe',\n",
+       "  JSON.parse('{\"nodes\": [{\"name\": \"0\", \"x\": 50, \"y\": 100, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"1\", \"x\": 50, \"y\": 150, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"2\", \"x\": 50, \"y\": 200, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"3\", \"x\": 100, \"y\": 150, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"4\", \"x\": 100, \"y\": 100, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"5\", \"x\": 150, \"y\": 200, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"6\", \"x\": 150, \"y\": 150, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"7\", \"x\": 200, \"y\": 200, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"8\", \"x\": 200, \"y\": 100, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"9\", \"x\": 250, \"y\": 150, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"10\", \"x\": 250, \"y\": 200, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"11\", \"x\": 300, \"y\": 150, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"12\", \"x\": 300, \"y\": 100, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"13\", \"x\": 350, \"y\": 100, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"14\", \"x\": 350, \"y\": 200, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"15\", \"x\": 400, \"y\": 100, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"16\", \"x\": 400, \"y\": 150, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"17\", \"x\": 400, \"y\": 200, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}], \"links\": [{\"source\": \"0\", \"target\": \"4\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"1\", \"target\": \"3\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"2\", \"target\": \"5\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"3\", \"target\": \"4\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"3\", \"target\": \"6\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"4\", \"target\": \"8\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"5\", \"target\": \"6\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"5\", \"target\": \"7\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"6\", \"target\": \"9\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"7\", \"target\": \"8\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"7\", \"target\": \"10\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"8\", \"target\": \"12\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"9\", \"target\": \"10\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"9\", \"target\": \"11\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"10\", \"target\": \"14\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"11\", \"target\": \"12\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"11\", \"target\": \"16\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"12\", \"target\": \"13\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"13\", \"target\": \"14\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"13\", \"target\": \"15\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"14\", \"target\": \"17\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}], \"pauli_web\": [{\"source\": 5, \"target\": 2, \"t\": \"Z\"}, {\"source\": 2, \"target\": 5, \"t\": \"Z\"}, {\"source\": 5, \"target\": 6, \"t\": \"Z\"}, {\"source\": 6, \"target\": 5, \"t\": \"Z\"}, {\"source\": 5, \"target\": 7, \"t\": \"Z\"}, {\"source\": 7, \"target\": 5, \"t\": \"Z\"}, {\"source\": 7, \"target\": 8, \"t\": \"Z\"}, {\"source\": 8, \"target\": 7, \"t\": \"Z\"}, {\"source\": 7, \"target\": 10, \"t\": \"Z\"}, {\"source\": 10, \"target\": 7, \"t\": \"Z\"}, {\"source\": 9, \"target\": 6, \"t\": \"Z\"}, {\"source\": 6, \"target\": 9, \"t\": \"Z\"}, {\"source\": 9, \"target\": 10, \"t\": \"Z\"}, {\"source\": 10, \"target\": 9, \"t\": \"Z\"}, {\"source\": 9, \"target\": 11, \"t\": \"Z\"}, {\"source\": 11, \"target\": 9, \"t\": \"Z\"}, {\"source\": 11, \"target\": 12, \"t\": \"Z\"}, {\"source\": 12, \"target\": 11, \"t\": \"Z\"}, {\"source\": 11, \"target\": 16, \"t\": \"Z\"}, {\"source\": 16, \"target\": 11, \"t\": \"Z\"}, {\"source\": 8, \"target\": 12, \"t\": \"Z\"}, {\"source\": 12, \"target\": 8, \"t\": \"Z\"}]}'), 450, 250, 50,\n",
+       "  10.0, false, true, '');\n",
+       "</script>"
+      ],
+      "text/plain": [
+       "<IPython.core.display.HTML object>"
+      ]
+     },
+     "metadata": {},
+     "output_type": "display_data"
+    },
+    {
+     "data": {
+      "text/html": [
+       "<div style=\"overflow:auto; background-color: white\" id=\"graph-output-Q6uJmpId\"></div>\n",
+       "<script type=\"module\">\n",
+       "var d3;\n",
+       "if (d3 == null) { d3 = await import(\"https://cdn.skypack.dev/d3@5\"); }\n",
+       "// PyZX - Python library for quantum circuit rewriting \n",
+       "//        and optimisation using the ZX-calculus\n",
+       "// Copyright (C) 2018 - Aleks Kissinger and John van de Wetering\n",
+       "\n",
+       "// Licensed under the Apache License, Version 2.0 (the \"License\");\n",
+       "// you may not use this file except in compliance with the License.\n",
+       "// You may obtain a copy of the License at\n",
+       "\n",
+       "//    http://www.apache.org/licenses/LICENSE-2.0\n",
+       "\n",
+       "// Unless required by applicable law or agreed to in writing, software\n",
+       "// distributed under the License is distributed on an \"AS IS\" BASIS,\n",
+       "// WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.\n",
+       "// See the License for the specific language governing permissions and\n",
+       "// limitations under the License.\n",
+       "\n",
+       "// styling functions\n",
+       "function nodeColor(t) {\n",
+       "    if (t == 0) return \"black\";\n",
+       "    else if (t == 1) return \"#ccffcc\";\n",
+       "    else if (t == 2) return \"#ff8888\";\n",
+       "    else if (t == 3) return \"yellow\";\n",
+       "    else if (t == 4) return \"black\";\n",
+       "    else if (t == 5) return \"black\";\n",
+       "    else if (t == 6) return \"#ccffcc\";\n",
+       "}\n",
+       "\n",
+       "function edgeColor(t) {\n",
+       "    if (t == 1) return \"black\";\n",
+       "    else if (t == 2) return \"#08f\";\n",
+       "    else if (t == 3) return \"gray\";\n",
+       "}\n",
+       "\n",
+       "function webColor(t) {\n",
+       "    if (t == 'X') return \"#ff8888\";\n",
+       "    else if (t == 'Y') return \"#eebb00\";\n",
+       "    else if (t == 'Z') return \"#66cc66\";\n",
+       "    else if (t == 'I') return \"#eee\";\n",
+       "}\n",
+       "\n",
+       "function nodeStyle(selected) {\n",
+       "    return selected ? \"stroke-width: 2px; stroke: #00f\" : \"stroke-width: 1.5px\";\n",
+       "}\n",
+       "\n",
+       "var symbolGround = {\n",
+       "    draw: function(context, size){\n",
+       "        let s = size/2;\n",
+       "\n",
+       "        context.moveTo(0,-s);\n",
+       "        context.lineTo(0,0);\n",
+       "\n",
+       "        context.moveTo(-s,0);\n",
+       "        context.lineTo(s,0);\n",
+       "\n",
+       "        context.moveTo(-2*s/3,s/3);\n",
+       "        context.lineTo(2*s/3,s/3);\n",
+       "\n",
+       "        context.moveTo(-s/3,2*s/3);\n",
+       "        context.lineTo(s/3,2*s/3);\n",
+       "    }\n",
+       "}\n",
+       "\n",
+       "function showGraph(tag, graph, width, height, scale, node_size, auto_hbox, show_labels, scalar_str) {\n",
+       "    var ntab = {};\n",
+       "\n",
+       "    var groundOffset = 2.5 * node_size;\n",
+       "\n",
+       "    graph.nodes.forEach(function(d) {\n",
+       "        ntab[d.name] = d;\n",
+       "        d.selected = false;\n",
+       "        d.previouslySelected = false;\n",
+       "        d.nhd = [];\n",
+       "    });\n",
+       "\n",
+       "    var spiders_and_boundaries = graph.nodes.filter(function(d) {\n",
+       "        return d.t != 3;\n",
+       "    });\n",
+       "\n",
+       "    graph.links.forEach(function(d) {\n",
+       "        var s = ntab[d.source];\n",
+       "        var t = ntab[d.target];\n",
+       "        d.source = s;\n",
+       "        d.target = t;\n",
+       "        s.nhd.push(t);\n",
+       "        t.nhd.push(s);\n",
+       "    });\n",
+       "\n",
+       "    graph.pauli_web.forEach(function(d) {\n",
+       "        var s = ntab[d.source];\n",
+       "        var t = ntab[d.target];\n",
+       "        d.source = s;\n",
+       "        d.target = t;\n",
+       "    });\n",
+       "\n",
+       "    var shiftKey;\n",
+       "\n",
+       "    // SETUP SVG ITEMS\n",
+       "\n",
+       "    var svg = d3.select(tag)\n",
+       "    //.attr(\"tabindex\", 1)\n",
+       "        .on(\"keydown.brush\", function() {shiftKey = d3.event.shiftKey || d3.event.metaKey;})\n",
+       "        .on(\"keyup.brush\", function() {shiftKey = d3.event.shiftKey || d3.event.metaKey;})\n",
+       "    //.each(function() { this.focus(); })\n",
+       "        .append(\"svg\")\n",
+       "        .attr(\"style\", \"max-width: none; max-height: none\")\n",
+       "        .attr(\"width\", width)\n",
+       "        .attr(\"height\", height);\n",
+       "\n",
+       "    var web = svg.append(\"g\")\n",
+       "        .attr(\"class\", \"web\")\n",
+       "        .selectAll(\"line\")\n",
+       "        .data(graph.pauli_web)\n",
+       "        .enter().append(\"path\")\n",
+       "        .attr(\"stroke\", function(d) { return webColor(d.t); })\n",
+       "        .attr(\"fill\", \"transparent\")\n",
+       "        .attr(\"style\", \"stroke-width: 7px\");\n",
+       "\n",
+       "    var link = svg.append(\"g\")\n",
+       "        .attr(\"class\", \"link\")\n",
+       "        .selectAll(\"line\")\n",
+       "        .data(graph.links)\n",
+       "        .enter().append(\"path\")\n",
+       "        .attr(\"stroke\", function(d) { return edgeColor(d.t); })\n",
+       "        .attr(\"fill\", \"transparent\")\n",
+       "        .attr(\"style\", \"stroke-width: 1.5px\");\n",
+       "\n",
+       "    var brush = svg.append(\"g\")\n",
+       "        .attr(\"class\", \"brush\");\n",
+       "\n",
+       "    var node = svg.append(\"g\")\n",
+       "        .attr(\"class\", \"node\")\n",
+       "        .selectAll(\"g\")\n",
+       "        .data(graph.nodes)\n",
+       "        .enter().append(\"g\")\n",
+       "        .attr(\"transform\", function(d) {\n",
+       "            return \"translate(\" + d.x + \",\" + d.y +\")\";\n",
+       "        });\n",
+       "\n",
+       "    // Draw a ground symbol connected to the node.\n",
+       "    node.filter(function(d) { return d.ground; })\n",
+       "        .append(\"path\")\n",
+       "        .attr(\"stroke\", \"black\")\n",
+       "        .attr(\"style\", \"stroke-width: 1.5px\")\n",
+       "        .attr(\"fill\", \"none\")\n",
+       "        .attr(\"d\", \"M 0 0 L 0 \"+(groundOffset))\n",
+       "        .attr(\"class\", \"selectable\");\n",
+       "    node.filter(function(d) { return d.ground; })\n",
+       "        .append(\"path\")\n",
+       "        .attr(\"stroke\", \"black\")\n",
+       "        .attr(\"style\", \"stroke-width: 1.5px\")\n",
+       "        .attr(\"fill\", \"none\")\n",
+       "        .attr(\"d\", d3.symbol().type(symbolGround).size(node_size*1.5))\n",
+       "        .attr(\"transform\", \"translate(0,\"+groundOffset+\")\")\n",
+       "        .attr(\"class\", \"selectable\");\n",
+       "\n",
+       "    node.filter(function(d) { return d.t != 3 && d.t != 5 && d.t != 6; })\n",
+       "        .append(\"circle\")\n",
+       "        .attr(\"r\", function(d) {\n",
+       "            if (d.t == 0) return 0.5 * node_size;\n",
+       "            else if (d.t == 4) return 0.25 * node_size;\n",
+       "            else return node_size;\n",
+       "        })\n",
+       "        .attr(\"fill\", function(d) { return nodeColor(d.t); })\n",
+       "        .attr(\"stroke\", \"black\")\n",
+       "        .attr(\"class\", \"selectable\");\n",
+       "\n",
+       "    var hbox = node.filter(function(d) { return d.t == 3; });\n",
+       "\n",
+       "    hbox.append(\"rect\")\n",
+       "        .attr(\"x\", -0.75 * node_size).attr(\"y\", -0.75 * node_size)\n",
+       "        .attr(\"width\", node_size * 1.5).attr(\"height\", node_size * 1.5)\n",
+       "        .attr(\"fill\", function(d) { return nodeColor(d.t); })\n",
+       "        .attr(\"stroke\", \"black\")\n",
+       "        .attr(\"class\", \"selectable\");\n",
+       "\n",
+       "    // draw a triangle for d.t == 5\n",
+       "    node.filter(function(d) { return d.t == 5; })\n",
+       "        .append(\"path\")\n",
+       "        .attr(\"d\", \"M 0 0 L \"+node_size+\" \"+node_size+\" L -\"+node_size+\" \"+node_size+\" Z\")\n",
+       "        .attr(\"fill\", function(d) { return nodeColor(d.t); })\n",
+       "        .attr(\"stroke\", \"black\")\n",
+       "        .attr(\"class\", \"selectable\")\n",
+       "        .attr(\"transform\", \"translate(\" + (-node_size/2) + \", 0) rotate(-90)\");\n",
+       "\n",
+       "    // draw a square for Z box: d.t == 6\n",
+       "    node.filter(function(d) { return d.t == 6; })\n",
+       "        .append(\"rect\")\n",
+       "        .attr(\"x\", -0.75 * node_size).attr(\"y\", -0.75 * node_size)\n",
+       "        .attr(\"width\", node_size * 1.5).attr(\"height\", node_size * 1.5)\n",
+       "        .attr(\"fill\", function(d) { return nodeColor(d.t); })\n",
+       "        .attr(\"stroke\", \"black\")\n",
+       "        .attr(\"class\", \"selectable\");\n",
+       "\n",
+       "    node.filter(function(d) { return d.phase != ''; })\n",
+       "        .append(\"text\")\n",
+       "        .attr(\"y\", 0.7 * node_size + 14)\n",
+       "        .text(function (d) { return d.phase })\n",
+       "        .attr(\"text-anchor\", \"middle\")\n",
+       "        .attr(\"font-size\", \"12px\")\n",
+       "        .attr(\"font-family\", \"monospace\")\n",
+       "        .attr(\"fill\", \"#00d\")\n",
+       "        .attr('style', 'pointer-events: none; user-select: none;');\n",
+       "\n",
+       "    if (show_labels) {\n",
+       "        node.append(\"text\")\n",
+       "            .attr(\"y\", -0.7 * node_size - 8)\n",
+       "            .text(function (d) { return d.name; })\n",
+       "            .attr(\"text-anchor\", \"middle\")\n",
+       "            .attr(\"font-size\", \"10px\")\n",
+       "            .attr(\"font-family\", \"monospace\")\n",
+       "            .attr(\"fill\", \"#999\")\n",
+       "            .attr('style', 'pointer-events: none; user-select: none;');\n",
+       "    }\n",
+       "\n",
+       "    // Display the chosen data fields over the node.\n",
+       "    node.filter(d => d.vdata.length > 0)\n",
+       "        .append(\"text\")\n",
+       "        .attr(\"y\", d => -0.7 * node_size - 14 - 10 * d.vdata.length)\n",
+       "        .attr(\"text-anchor\", \"middle\")\n",
+       "        .attr(\"font-size\", \"8px\")\n",
+       "        .attr(\"font-family\", \"monospace\")\n",
+       "        .attr(\"fill\", \"#c66\")\n",
+       "        .attr('style', 'pointer-events: none; user-select: none;')\n",
+       "        .selectAll(\"tspan\")\n",
+       "        .data(d => d.vdata)\n",
+       "        .enter()\n",
+       "        .append(\"tspan\")\n",
+       "        .attr(\"x\", \"0\")\n",
+       "        .attr(\"dy\", \"1.2em\")\n",
+       "        .text(x => x.join(\": \"));\n",
+       "\n",
+       "    if (scalar_str != \"\") {\n",
+       "        svg.append(\"text\")\n",
+       "            .text(scalar_str)\n",
+       "            .attr(\"x\", 60).attr(\"y\", 40)\n",
+       "            .attr(\"text-anchor\", \"middle\")\n",
+       "    }\n",
+       "\n",
+       "    function update_hboxes() {\n",
+       "        if (auto_hbox) {\n",
+       "            var pos = {};\n",
+       "            hbox.attr(\"transform\", function(d) {\n",
+       "                // calculate barycenter of non-hbox neighbours, then nudge a bit\n",
+       "                // to the NE.\n",
+       "                var x=0,y=0,sz=0;\n",
+       "                for (var i = 0; i < d.nhd.length; ++i) {\n",
+       "                    if (d.nhd[i].t != 3) {\n",
+       "                        sz++;\n",
+       "                        x += d.nhd[i].x;\n",
+       "                        y += d.nhd[i].y;\n",
+       "                    }\n",
+       "                }\n",
+       "\n",
+       "                offset = 0.25 * scale;\n",
+       "\n",
+       "                if (sz != 0) {\n",
+       "                    x = (x/sz) + offset;\n",
+       "                    y = (y/sz) - offset;\n",
+       "\n",
+       "                    while (pos[[x,y]]) {\n",
+       "                        x += offset;\n",
+       "                    }\n",
+       "                    d.x = x;\n",
+       "                    d.y = y;\n",
+       "                    pos[[x,y]] = true;\n",
+       "                }\n",
+       "\n",
+       "                return \"translate(\"+d.x+\",\"+d.y+\")\";\n",
+       "            });\n",
+       "        }\n",
+       "    }\n",
+       "\n",
+       "    update_hboxes();\n",
+       "\n",
+       "    var link_curve = function(d) {\n",
+       "        var x1 = d.source.x, x2 = d.target.x, y1 = d.source.y, y2 = d.target.y;\n",
+       "        if (d.num_parallel == 1) {\n",
+       "            return `M ${x1} ${y1} L ${x2} ${y2}`;\n",
+       "        } else {\n",
+       "            var dx = x2 - x1, dy = y2 - y1;\n",
+       "            var midx = 0.5 * (x1 + x2), midy = 0.5 * (y1 + y2);\n",
+       "            var pos = (d.index / (d.num_parallel-1)) - 0.5;\n",
+       "            var cx = midx - pos * dy;\n",
+       "            var cy = midy + pos * dx;\n",
+       "            return `M ${x1} ${y1} Q ${cx} ${cy}, ${x2} ${y2}`;\n",
+       "            // return `M ${x1} ${y1} L ${x2} ${y2}`;\n",
+       "        }\n",
+       "    };\n",
+       "    link.attr(\"d\", link_curve);\n",
+       "\n",
+       "\n",
+       "    var web_curve = function(d) {\n",
+       "        var x1 = d.source.x, x2 = (x1 + d.target.x)/2, y1 = d.source.y, y2 = (y1 + d.target.y)/2;\n",
+       "        return `M ${x1} ${y1} L ${x2} ${y2}`;\n",
+       "    }\n",
+       "    web.attr(\"d\", web_curve);\n",
+       "\n",
+       "    // EVENTS FOR DRAGGING AND SELECTION\n",
+       "\n",
+       "    node.on(\"mousedown\", function(d) {\n",
+       "        if (shiftKey) {\n",
+       "            d3.select(this).selectAll(\".selectable\").attr(\"style\", nodeStyle(d.selected = !d.selected));\n",
+       "            d3.event.stopImmediatePropagation();\n",
+       "        } else if (!d.selected) {\n",
+       "            node.selectAll(\".selectable\").attr(\"style\", function(p) { return nodeStyle(p.selected = d === p); });\n",
+       "        }\n",
+       "    })\n",
+       "        .call(d3.drag().on(\"drag\", function(d) {\n",
+       "            var dx = d3.event.dx;\n",
+       "            var dy = d3.event.dy;\n",
+       "            // node.filter(function(d) { return d.selected; })\n",
+       "            //     .attr(\"cx\", function(d) { return d.x += dx; })\n",
+       "            //     .attr(\"cy\", function(d) { return d.y += dy; });\n",
+       "            node.filter(function(d) { return d.selected; })\n",
+       "                .attr(\"transform\", function(d) {\n",
+       "                    d.x += dx;\n",
+       "                    d.y += dy;\n",
+       "                    return \"translate(\" + d.x + \",\" + d.y +\")\";\n",
+       "                });\n",
+       "\n",
+       "            update_hboxes();\n",
+       "\n",
+       "            link.filter(function(d) { return d.source.selected || d.target.selected ||\n",
+       "                    (auto_hbox && d.source.t == 3); })\n",
+       "                .attr(\"d\", link_curve);\n",
+       "            web.filter(function(d) { return d.source.selected || d.target.selected; })\n",
+       "                .attr(\"d\", web_curve);\n",
+       "        }));\n",
+       "\n",
+       "    brush.call(d3.brush().keyModifiers(false)\n",
+       "        .extent([[0, 0], [width, height]])\n",
+       "        .on(\"start\", function() {\n",
+       "            if (d3.event.sourceEvent.type !== \"end\") {\n",
+       "                node.selectAll(\".selectable\").attr(\"style\", function(d) {\n",
+       "                    return nodeStyle(\n",
+       "                        d.selected = d.previouslySelected = shiftKey &&\n",
+       "                        d.selected);\n",
+       "                });\n",
+       "            }\n",
+       "        })\n",
+       "        .on(\"brush\", function() {\n",
+       "            if (d3.event.sourceEvent.type !== \"end\") {\n",
+       "                var selection = d3.event.selection;\n",
+       "                node.selectAll(\".selectable\").attr(\"style\", function(d) {\n",
+       "                    return nodeStyle(d.selected = d.previouslySelected ^\n",
+       "                        (selection != null\n",
+       "                            && selection[0][0] <= d.x && d.x < selection[1][0]\n",
+       "                            && selection[0][1] <= d.y && d.y < selection[1][1]));\n",
+       "                });\n",
+       "            }\n",
+       "        })\n",
+       "        .on(\"end\", function() {\n",
+       "            if (d3.event.selection != null) {\n",
+       "                d3.select(this).call(d3.event.target.move, null);\n",
+       "            }\n",
+       "        }));\n",
+       "}\n",
+       "\n",
+       "\n",
+       "showGraph('#graph-output-Q6uJmpId',\n",
+       "  JSON.parse('{\"nodes\": [{\"name\": \"0\", \"x\": 50, \"y\": 100, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"1\", \"x\": 50, \"y\": 150, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"2\", \"x\": 50, \"y\": 200, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"3\", \"x\": 100, \"y\": 150, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"4\", \"x\": 100, \"y\": 100, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"5\", \"x\": 150, \"y\": 200, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"6\", \"x\": 150, \"y\": 150, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"7\", \"x\": 200, \"y\": 200, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"8\", \"x\": 200, \"y\": 100, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"9\", \"x\": 250, \"y\": 150, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"10\", \"x\": 250, \"y\": 200, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"11\", \"x\": 300, \"y\": 150, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"12\", \"x\": 300, \"y\": 100, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"13\", \"x\": 350, \"y\": 100, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"14\", \"x\": 350, \"y\": 200, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"15\", \"x\": 400, \"y\": 100, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"16\", \"x\": 400, \"y\": 150, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"17\", \"x\": 400, \"y\": 200, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}], \"links\": [{\"source\": \"0\", \"target\": \"4\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"1\", \"target\": \"3\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"2\", \"target\": \"5\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"3\", \"target\": \"4\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"3\", \"target\": \"6\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"4\", \"target\": \"8\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"5\", \"target\": \"6\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"5\", \"target\": \"7\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"6\", \"target\": \"9\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"7\", \"target\": \"8\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"7\", \"target\": \"10\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"8\", \"target\": \"12\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"9\", \"target\": \"10\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"9\", \"target\": \"11\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"10\", \"target\": \"14\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"11\", \"target\": \"12\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"11\", \"target\": \"16\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"12\", \"target\": \"13\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"13\", \"target\": \"14\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"13\", \"target\": \"15\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"14\", \"target\": \"17\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}], \"pauli_web\": [{\"source\": 4, \"target\": 0, \"t\": \"X\"}, {\"source\": 0, \"target\": 4, \"t\": \"X\"}, {\"source\": 4, \"target\": 3, \"t\": \"X\"}, {\"source\": 3, \"target\": 4, \"t\": \"X\"}, {\"source\": 4, \"target\": 8, \"t\": \"X\"}, {\"source\": 8, \"target\": 4, \"t\": \"X\"}, {\"source\": 8, \"target\": 7, \"t\": \"X\"}, {\"source\": 7, \"target\": 8, \"t\": \"X\"}, {\"source\": 8, \"target\": 12, \"t\": \"X\"}, {\"source\": 10, \"target\": 7, \"t\": \"X\"}, {\"source\": 7, \"target\": 10, \"t\": \"X\"}, {\"source\": 10, \"target\": 9, \"t\": \"X\"}, {\"source\": 9, \"target\": 10, \"t\": \"X\"}, {\"source\": 10, \"target\": 14, \"t\": \"X\"}, {\"source\": 12, \"target\": 8, \"t\": \"X\"}, {\"source\": 12, \"target\": 11, \"t\": \"X\"}, {\"source\": 11, \"target\": 12, \"t\": \"X\"}, {\"source\": 12, \"target\": 13, \"t\": \"X\"}, {\"source\": 13, \"target\": 12, \"t\": \"X\"}, {\"source\": 14, \"target\": 10, \"t\": \"X\"}, {\"source\": 14, \"target\": 13, \"t\": \"X\"}, {\"source\": 13, \"target\": 14, \"t\": \"X\"}, {\"source\": 14, \"target\": 17, \"t\": \"X\"}, {\"source\": 17, \"target\": 14, \"t\": \"X\"}, {\"source\": 9, \"target\": 11, \"t\": \"X\"}, {\"source\": 11, \"target\": 9, \"t\": \"X\"}, {\"source\": 1, \"target\": 3, \"t\": \"X\"}, {\"source\": 3, \"target\": 1, \"t\": \"X\"}]}'), 450, 250, 50,\n",
+       "  10.0, false, true, '');\n",
+       "</script>"
+      ],
+      "text/plain": [
+       "<IPython.core.display.HTML object>"
+      ]
+     },
+     "metadata": {},
+     "output_type": "display_data"
+    },
+    {
+     "data": {
+      "text/html": [
+       "<div style=\"overflow:auto; background-color: white\" id=\"graph-output-FygEjZM0\"></div>\n",
+       "<script type=\"module\">\n",
+       "var d3;\n",
+       "if (d3 == null) { d3 = await import(\"https://cdn.skypack.dev/d3@5\"); }\n",
+       "// PyZX - Python library for quantum circuit rewriting \n",
+       "//        and optimisation using the ZX-calculus\n",
+       "// Copyright (C) 2018 - Aleks Kissinger and John van de Wetering\n",
+       "\n",
+       "// Licensed under the Apache License, Version 2.0 (the \"License\");\n",
+       "// you may not use this file except in compliance with the License.\n",
+       "// You may obtain a copy of the License at\n",
+       "\n",
+       "//    http://www.apache.org/licenses/LICENSE-2.0\n",
+       "\n",
+       "// Unless required by applicable law or agreed to in writing, software\n",
+       "// distributed under the License is distributed on an \"AS IS\" BASIS,\n",
+       "// WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.\n",
+       "// See the License for the specific language governing permissions and\n",
+       "// limitations under the License.\n",
+       "\n",
+       "// styling functions\n",
+       "function nodeColor(t) {\n",
+       "    if (t == 0) return \"black\";\n",
+       "    else if (t == 1) return \"#ccffcc\";\n",
+       "    else if (t == 2) return \"#ff8888\";\n",
+       "    else if (t == 3) return \"yellow\";\n",
+       "    else if (t == 4) return \"black\";\n",
+       "    else if (t == 5) return \"black\";\n",
+       "    else if (t == 6) return \"#ccffcc\";\n",
+       "}\n",
+       "\n",
+       "function edgeColor(t) {\n",
+       "    if (t == 1) return \"black\";\n",
+       "    else if (t == 2) return \"#08f\";\n",
+       "    else if (t == 3) return \"gray\";\n",
+       "}\n",
+       "\n",
+       "function webColor(t) {\n",
+       "    if (t == 'X') return \"#ff8888\";\n",
+       "    else if (t == 'Y') return \"#eebb00\";\n",
+       "    else if (t == 'Z') return \"#66cc66\";\n",
+       "    else if (t == 'I') return \"#eee\";\n",
+       "}\n",
+       "\n",
+       "function nodeStyle(selected) {\n",
+       "    return selected ? \"stroke-width: 2px; stroke: #00f\" : \"stroke-width: 1.5px\";\n",
+       "}\n",
+       "\n",
+       "var symbolGround = {\n",
+       "    draw: function(context, size){\n",
+       "        let s = size/2;\n",
+       "\n",
+       "        context.moveTo(0,-s);\n",
+       "        context.lineTo(0,0);\n",
+       "\n",
+       "        context.moveTo(-s,0);\n",
+       "        context.lineTo(s,0);\n",
+       "\n",
+       "        context.moveTo(-2*s/3,s/3);\n",
+       "        context.lineTo(2*s/3,s/3);\n",
+       "\n",
+       "        context.moveTo(-s/3,2*s/3);\n",
+       "        context.lineTo(s/3,2*s/3);\n",
+       "    }\n",
+       "}\n",
+       "\n",
+       "function showGraph(tag, graph, width, height, scale, node_size, auto_hbox, show_labels, scalar_str) {\n",
+       "    var ntab = {};\n",
+       "\n",
+       "    var groundOffset = 2.5 * node_size;\n",
+       "\n",
+       "    graph.nodes.forEach(function(d) {\n",
+       "        ntab[d.name] = d;\n",
+       "        d.selected = false;\n",
+       "        d.previouslySelected = false;\n",
+       "        d.nhd = [];\n",
+       "    });\n",
+       "\n",
+       "    var spiders_and_boundaries = graph.nodes.filter(function(d) {\n",
+       "        return d.t != 3;\n",
+       "    });\n",
+       "\n",
+       "    graph.links.forEach(function(d) {\n",
+       "        var s = ntab[d.source];\n",
+       "        var t = ntab[d.target];\n",
+       "        d.source = s;\n",
+       "        d.target = t;\n",
+       "        s.nhd.push(t);\n",
+       "        t.nhd.push(s);\n",
+       "    });\n",
+       "\n",
+       "    graph.pauli_web.forEach(function(d) {\n",
+       "        var s = ntab[d.source];\n",
+       "        var t = ntab[d.target];\n",
+       "        d.source = s;\n",
+       "        d.target = t;\n",
+       "    });\n",
+       "\n",
+       "    var shiftKey;\n",
+       "\n",
+       "    // SETUP SVG ITEMS\n",
+       "\n",
+       "    var svg = d3.select(tag)\n",
+       "    //.attr(\"tabindex\", 1)\n",
+       "        .on(\"keydown.brush\", function() {shiftKey = d3.event.shiftKey || d3.event.metaKey;})\n",
+       "        .on(\"keyup.brush\", function() {shiftKey = d3.event.shiftKey || d3.event.metaKey;})\n",
+       "    //.each(function() { this.focus(); })\n",
+       "        .append(\"svg\")\n",
+       "        .attr(\"style\", \"max-width: none; max-height: none\")\n",
+       "        .attr(\"width\", width)\n",
+       "        .attr(\"height\", height);\n",
+       "\n",
+       "    var web = svg.append(\"g\")\n",
+       "        .attr(\"class\", \"web\")\n",
+       "        .selectAll(\"line\")\n",
+       "        .data(graph.pauli_web)\n",
+       "        .enter().append(\"path\")\n",
+       "        .attr(\"stroke\", function(d) { return webColor(d.t); })\n",
+       "        .attr(\"fill\", \"transparent\")\n",
+       "        .attr(\"style\", \"stroke-width: 7px\");\n",
+       "\n",
+       "    var link = svg.append(\"g\")\n",
+       "        .attr(\"class\", \"link\")\n",
+       "        .selectAll(\"line\")\n",
+       "        .data(graph.links)\n",
+       "        .enter().append(\"path\")\n",
+       "        .attr(\"stroke\", function(d) { return edgeColor(d.t); })\n",
+       "        .attr(\"fill\", \"transparent\")\n",
+       "        .attr(\"style\", \"stroke-width: 1.5px\");\n",
+       "\n",
+       "    var brush = svg.append(\"g\")\n",
+       "        .attr(\"class\", \"brush\");\n",
+       "\n",
+       "    var node = svg.append(\"g\")\n",
+       "        .attr(\"class\", \"node\")\n",
+       "        .selectAll(\"g\")\n",
+       "        .data(graph.nodes)\n",
+       "        .enter().append(\"g\")\n",
+       "        .attr(\"transform\", function(d) {\n",
+       "            return \"translate(\" + d.x + \",\" + d.y +\")\";\n",
+       "        });\n",
+       "\n",
+       "    // Draw a ground symbol connected to the node.\n",
+       "    node.filter(function(d) { return d.ground; })\n",
+       "        .append(\"path\")\n",
+       "        .attr(\"stroke\", \"black\")\n",
+       "        .attr(\"style\", \"stroke-width: 1.5px\")\n",
+       "        .attr(\"fill\", \"none\")\n",
+       "        .attr(\"d\", \"M 0 0 L 0 \"+(groundOffset))\n",
+       "        .attr(\"class\", \"selectable\");\n",
+       "    node.filter(function(d) { return d.ground; })\n",
+       "        .append(\"path\")\n",
+       "        .attr(\"stroke\", \"black\")\n",
+       "        .attr(\"style\", \"stroke-width: 1.5px\")\n",
+       "        .attr(\"fill\", \"none\")\n",
+       "        .attr(\"d\", d3.symbol().type(symbolGround).size(node_size*1.5))\n",
+       "        .attr(\"transform\", \"translate(0,\"+groundOffset+\")\")\n",
+       "        .attr(\"class\", \"selectable\");\n",
+       "\n",
+       "    node.filter(function(d) { return d.t != 3 && d.t != 5 && d.t != 6; })\n",
+       "        .append(\"circle\")\n",
+       "        .attr(\"r\", function(d) {\n",
+       "            if (d.t == 0) return 0.5 * node_size;\n",
+       "            else if (d.t == 4) return 0.25 * node_size;\n",
+       "            else return node_size;\n",
+       "        })\n",
+       "        .attr(\"fill\", function(d) { return nodeColor(d.t); })\n",
+       "        .attr(\"stroke\", \"black\")\n",
+       "        .attr(\"class\", \"selectable\");\n",
+       "\n",
+       "    var hbox = node.filter(function(d) { return d.t == 3; });\n",
+       "\n",
+       "    hbox.append(\"rect\")\n",
+       "        .attr(\"x\", -0.75 * node_size).attr(\"y\", -0.75 * node_size)\n",
+       "        .attr(\"width\", node_size * 1.5).attr(\"height\", node_size * 1.5)\n",
+       "        .attr(\"fill\", function(d) { return nodeColor(d.t); })\n",
+       "        .attr(\"stroke\", \"black\")\n",
+       "        .attr(\"class\", \"selectable\");\n",
+       "\n",
+       "    // draw a triangle for d.t == 5\n",
+       "    node.filter(function(d) { return d.t == 5; })\n",
+       "        .append(\"path\")\n",
+       "        .attr(\"d\", \"M 0 0 L \"+node_size+\" \"+node_size+\" L -\"+node_size+\" \"+node_size+\" Z\")\n",
+       "        .attr(\"fill\", function(d) { return nodeColor(d.t); })\n",
+       "        .attr(\"stroke\", \"black\")\n",
+       "        .attr(\"class\", \"selectable\")\n",
+       "        .attr(\"transform\", \"translate(\" + (-node_size/2) + \", 0) rotate(-90)\");\n",
+       "\n",
+       "    // draw a square for Z box: d.t == 6\n",
+       "    node.filter(function(d) { return d.t == 6; })\n",
+       "        .append(\"rect\")\n",
+       "        .attr(\"x\", -0.75 * node_size).attr(\"y\", -0.75 * node_size)\n",
+       "        .attr(\"width\", node_size * 1.5).attr(\"height\", node_size * 1.5)\n",
+       "        .attr(\"fill\", function(d) { return nodeColor(d.t); })\n",
+       "        .attr(\"stroke\", \"black\")\n",
+       "        .attr(\"class\", \"selectable\");\n",
+       "\n",
+       "    node.filter(function(d) { return d.phase != ''; })\n",
+       "        .append(\"text\")\n",
+       "        .attr(\"y\", 0.7 * node_size + 14)\n",
+       "        .text(function (d) { return d.phase })\n",
+       "        .attr(\"text-anchor\", \"middle\")\n",
+       "        .attr(\"font-size\", \"12px\")\n",
+       "        .attr(\"font-family\", \"monospace\")\n",
+       "        .attr(\"fill\", \"#00d\")\n",
+       "        .attr('style', 'pointer-events: none; user-select: none;');\n",
+       "\n",
+       "    if (show_labels) {\n",
+       "        node.append(\"text\")\n",
+       "            .attr(\"y\", -0.7 * node_size - 8)\n",
+       "            .text(function (d) { return d.name; })\n",
+       "            .attr(\"text-anchor\", \"middle\")\n",
+       "            .attr(\"font-size\", \"10px\")\n",
+       "            .attr(\"font-family\", \"monospace\")\n",
+       "            .attr(\"fill\", \"#999\")\n",
+       "            .attr('style', 'pointer-events: none; user-select: none;');\n",
+       "    }\n",
+       "\n",
+       "    // Display the chosen data fields over the node.\n",
+       "    node.filter(d => d.vdata.length > 0)\n",
+       "        .append(\"text\")\n",
+       "        .attr(\"y\", d => -0.7 * node_size - 14 - 10 * d.vdata.length)\n",
+       "        .attr(\"text-anchor\", \"middle\")\n",
+       "        .attr(\"font-size\", \"8px\")\n",
+       "        .attr(\"font-family\", \"monospace\")\n",
+       "        .attr(\"fill\", \"#c66\")\n",
+       "        .attr('style', 'pointer-events: none; user-select: none;')\n",
+       "        .selectAll(\"tspan\")\n",
+       "        .data(d => d.vdata)\n",
+       "        .enter()\n",
+       "        .append(\"tspan\")\n",
+       "        .attr(\"x\", \"0\")\n",
+       "        .attr(\"dy\", \"1.2em\")\n",
+       "        .text(x => x.join(\": \"));\n",
+       "\n",
+       "    if (scalar_str != \"\") {\n",
+       "        svg.append(\"text\")\n",
+       "            .text(scalar_str)\n",
+       "            .attr(\"x\", 60).attr(\"y\", 40)\n",
+       "            .attr(\"text-anchor\", \"middle\")\n",
+       "    }\n",
+       "\n",
+       "    function update_hboxes() {\n",
+       "        if (auto_hbox) {\n",
+       "            var pos = {};\n",
+       "            hbox.attr(\"transform\", function(d) {\n",
+       "                // calculate barycenter of non-hbox neighbours, then nudge a bit\n",
+       "                // to the NE.\n",
+       "                var x=0,y=0,sz=0;\n",
+       "                for (var i = 0; i < d.nhd.length; ++i) {\n",
+       "                    if (d.nhd[i].t != 3) {\n",
+       "                        sz++;\n",
+       "                        x += d.nhd[i].x;\n",
+       "                        y += d.nhd[i].y;\n",
+       "                    }\n",
+       "                }\n",
+       "\n",
+       "                offset = 0.25 * scale;\n",
+       "\n",
+       "                if (sz != 0) {\n",
+       "                    x = (x/sz) + offset;\n",
+       "                    y = (y/sz) - offset;\n",
+       "\n",
+       "                    while (pos[[x,y]]) {\n",
+       "                        x += offset;\n",
+       "                    }\n",
+       "                    d.x = x;\n",
+       "                    d.y = y;\n",
+       "                    pos[[x,y]] = true;\n",
+       "                }\n",
+       "\n",
+       "                return \"translate(\"+d.x+\",\"+d.y+\")\";\n",
+       "            });\n",
+       "        }\n",
+       "    }\n",
+       "\n",
+       "    update_hboxes();\n",
+       "\n",
+       "    var link_curve = function(d) {\n",
+       "        var x1 = d.source.x, x2 = d.target.x, y1 = d.source.y, y2 = d.target.y;\n",
+       "        if (d.num_parallel == 1) {\n",
+       "            return `M ${x1} ${y1} L ${x2} ${y2}`;\n",
+       "        } else {\n",
+       "            var dx = x2 - x1, dy = y2 - y1;\n",
+       "            var midx = 0.5 * (x1 + x2), midy = 0.5 * (y1 + y2);\n",
+       "            var pos = (d.index / (d.num_parallel-1)) - 0.5;\n",
+       "            var cx = midx - pos * dy;\n",
+       "            var cy = midy + pos * dx;\n",
+       "            return `M ${x1} ${y1} Q ${cx} ${cy}, ${x2} ${y2}`;\n",
+       "            // return `M ${x1} ${y1} L ${x2} ${y2}`;\n",
+       "        }\n",
+       "    };\n",
+       "    link.attr(\"d\", link_curve);\n",
+       "\n",
+       "\n",
+       "    var web_curve = function(d) {\n",
+       "        var x1 = d.source.x, x2 = (x1 + d.target.x)/2, y1 = d.source.y, y2 = (y1 + d.target.y)/2;\n",
+       "        return `M ${x1} ${y1} L ${x2} ${y2}`;\n",
+       "    }\n",
+       "    web.attr(\"d\", web_curve);\n",
+       "\n",
+       "    // EVENTS FOR DRAGGING AND SELECTION\n",
+       "\n",
+       "    node.on(\"mousedown\", function(d) {\n",
+       "        if (shiftKey) {\n",
+       "            d3.select(this).selectAll(\".selectable\").attr(\"style\", nodeStyle(d.selected = !d.selected));\n",
+       "            d3.event.stopImmediatePropagation();\n",
+       "        } else if (!d.selected) {\n",
+       "            node.selectAll(\".selectable\").attr(\"style\", function(p) { return nodeStyle(p.selected = d === p); });\n",
+       "        }\n",
+       "    })\n",
+       "        .call(d3.drag().on(\"drag\", function(d) {\n",
+       "            var dx = d3.event.dx;\n",
+       "            var dy = d3.event.dy;\n",
+       "            // node.filter(function(d) { return d.selected; })\n",
+       "            //     .attr(\"cx\", function(d) { return d.x += dx; })\n",
+       "            //     .attr(\"cy\", function(d) { return d.y += dy; });\n",
+       "            node.filter(function(d) { return d.selected; })\n",
+       "                .attr(\"transform\", function(d) {\n",
+       "                    d.x += dx;\n",
+       "                    d.y += dy;\n",
+       "                    return \"translate(\" + d.x + \",\" + d.y +\")\";\n",
+       "                });\n",
+       "\n",
+       "            update_hboxes();\n",
+       "\n",
+       "            link.filter(function(d) { return d.source.selected || d.target.selected ||\n",
+       "                    (auto_hbox && d.source.t == 3); })\n",
+       "                .attr(\"d\", link_curve);\n",
+       "            web.filter(function(d) { return d.source.selected || d.target.selected; })\n",
+       "                .attr(\"d\", web_curve);\n",
+       "        }));\n",
+       "\n",
+       "    brush.call(d3.brush().keyModifiers(false)\n",
+       "        .extent([[0, 0], [width, height]])\n",
+       "        .on(\"start\", function() {\n",
+       "            if (d3.event.sourceEvent.type !== \"end\") {\n",
+       "                node.selectAll(\".selectable\").attr(\"style\", function(d) {\n",
+       "                    return nodeStyle(\n",
+       "                        d.selected = d.previouslySelected = shiftKey &&\n",
+       "                        d.selected);\n",
+       "                });\n",
+       "            }\n",
+       "        })\n",
+       "        .on(\"brush\", function() {\n",
+       "            if (d3.event.sourceEvent.type !== \"end\") {\n",
+       "                var selection = d3.event.selection;\n",
+       "                node.selectAll(\".selectable\").attr(\"style\", function(d) {\n",
+       "                    return nodeStyle(d.selected = d.previouslySelected ^\n",
+       "                        (selection != null\n",
+       "                            && selection[0][0] <= d.x && d.x < selection[1][0]\n",
+       "                            && selection[0][1] <= d.y && d.y < selection[1][1]));\n",
+       "                });\n",
+       "            }\n",
+       "        })\n",
+       "        .on(\"end\", function() {\n",
+       "            if (d3.event.selection != null) {\n",
+       "                d3.select(this).call(d3.event.target.move, null);\n",
+       "            }\n",
+       "        }));\n",
+       "}\n",
+       "\n",
+       "\n",
+       "showGraph('#graph-output-FygEjZM0',\n",
+       "  JSON.parse('{\"nodes\": [{\"name\": \"0\", \"x\": 50, \"y\": 100, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"1\", \"x\": 50, \"y\": 150, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"2\", \"x\": 50, \"y\": 200, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"3\", \"x\": 100, \"y\": 150, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"4\", \"x\": 100, \"y\": 100, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"5\", \"x\": 150, \"y\": 200, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"6\", \"x\": 150, \"y\": 150, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"7\", \"x\": 200, \"y\": 200, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"8\", \"x\": 200, \"y\": 100, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"9\", \"x\": 250, \"y\": 150, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"10\", \"x\": 250, \"y\": 200, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"11\", \"x\": 300, \"y\": 150, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"12\", \"x\": 300, \"y\": 100, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"13\", \"x\": 350, \"y\": 100, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"14\", \"x\": 350, \"y\": 200, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"15\", \"x\": 400, \"y\": 100, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"16\", \"x\": 400, \"y\": 150, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"17\", \"x\": 400, \"y\": 200, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}], \"links\": [{\"source\": \"0\", \"target\": \"4\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"1\", \"target\": \"3\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"2\", \"target\": \"5\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"3\", \"target\": \"4\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"3\", \"target\": \"6\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"4\", \"target\": \"8\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"5\", \"target\": \"6\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"5\", \"target\": \"7\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"6\", \"target\": \"9\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"7\", \"target\": \"8\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"7\", \"target\": \"10\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"8\", \"target\": \"12\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"9\", \"target\": \"10\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"9\", \"target\": \"11\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"10\", \"target\": \"14\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"11\", \"target\": \"12\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"11\", \"target\": \"16\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"12\", \"target\": \"13\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"13\", \"target\": \"14\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"13\", \"target\": \"15\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"14\", \"target\": \"17\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}], \"pauli_web\": [{\"source\": 17, \"target\": 14, \"t\": \"Z\"}, {\"source\": 14, \"target\": 17, \"t\": \"Z\"}, {\"source\": 3, \"target\": 1, \"t\": \"Z\"}, {\"source\": 1, \"target\": 3, \"t\": \"Z\"}, {\"source\": 3, \"target\": 4, \"t\": \"Z\"}, {\"source\": 4, \"target\": 3, \"t\": \"Z\"}, {\"source\": 3, \"target\": 6, \"t\": \"Z\"}, {\"source\": 6, \"target\": 3, \"t\": \"Z\"}, {\"source\": 5, \"target\": 2, \"t\": \"Z\"}, {\"source\": 2, \"target\": 5, \"t\": \"Z\"}, {\"source\": 5, \"target\": 6, \"t\": \"Z\"}, {\"source\": 6, \"target\": 5, \"t\": \"Z\"}, {\"source\": 5, \"target\": 7, \"t\": \"Z\"}, {\"source\": 7, \"target\": 5, \"t\": \"Z\"}, {\"source\": 7, \"target\": 8, \"t\": \"Z\"}, {\"source\": 8, \"target\": 7, \"t\": \"Z\"}, {\"source\": 7, \"target\": 10, \"t\": \"Z\"}, {\"source\": 10, \"target\": 7, \"t\": \"Z\"}, {\"source\": 4, \"target\": 8, \"t\": \"Z\"}, {\"source\": 8, \"target\": 4, \"t\": \"Z\"}, {\"source\": 10, \"target\": 14, \"t\": \"Z\"}, {\"source\": 14, \"target\": 10, \"t\": \"Z\"}]}'), 450, 250, 50,\n",
+       "  10.0, false, true, '');\n",
+       "</script>"
+      ],
+      "text/plain": [
+       "<IPython.core.display.HTML object>"
+      ]
+     },
+     "metadata": {},
+     "output_type": "display_data"
+    }
+   ],
+   "source": [
+    "order, zwebs, xwebs = compute_pauli_webs(g)\n",
+    "zx.draw(g, labels=True, pauli_web=xwebs[15])\n",
+    "zx.draw(g, labels=True, pauli_web=zwebs[15])\n",
+    "zx.draw(g, labels=True, pauli_web=xwebs[16])\n",
+    "zx.draw(g, labels=True, pauli_web=zwebs[16])\n",
+    "zx.draw(g, labels=True, pauli_web=xwebs[17])\n",
+    "zx.draw(g, labels=True, pauli_web=zwebs[17])"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 7,
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "text/html": [
+       "<div style=\"overflow:auto; background-color: white\" id=\"graph-output-h8yGb4hV\"></div>\n",
+       "<script type=\"module\">\n",
+       "var d3;\n",
+       "if (d3 == null) { d3 = await import(\"https://cdn.skypack.dev/d3@5\"); }\n",
+       "// PyZX - Python library for quantum circuit rewriting \n",
+       "//        and optimisation using the ZX-calculus\n",
+       "// Copyright (C) 2018 - Aleks Kissinger and John van de Wetering\n",
+       "\n",
+       "// Licensed under the Apache License, Version 2.0 (the \"License\");\n",
+       "// you may not use this file except in compliance with the License.\n",
+       "// You may obtain a copy of the License at\n",
+       "\n",
+       "//    http://www.apache.org/licenses/LICENSE-2.0\n",
+       "\n",
+       "// Unless required by applicable law or agreed to in writing, software\n",
+       "// distributed under the License is distributed on an \"AS IS\" BASIS,\n",
+       "// WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.\n",
+       "// See the License for the specific language governing permissions and\n",
+       "// limitations under the License.\n",
+       "\n",
+       "// styling functions\n",
+       "function nodeColor(t) {\n",
+       "    if (t == 0) return \"black\";\n",
+       "    else if (t == 1) return \"#ccffcc\";\n",
+       "    else if (t == 2) return \"#ff8888\";\n",
+       "    else if (t == 3) return \"yellow\";\n",
+       "    else if (t == 4) return \"black\";\n",
+       "    else if (t == 5) return \"black\";\n",
+       "    else if (t == 6) return \"#ccffcc\";\n",
+       "}\n",
+       "\n",
+       "function edgeColor(t) {\n",
+       "    if (t == 1) return \"black\";\n",
+       "    else if (t == 2) return \"#08f\";\n",
+       "    else if (t == 3) return \"gray\";\n",
+       "}\n",
+       "\n",
+       "function webColor(t) {\n",
+       "    if (t == 'X') return \"#ff8888\";\n",
+       "    else if (t == 'Y') return \"#eebb00\";\n",
+       "    else if (t == 'Z') return \"#66cc66\";\n",
+       "    else if (t == 'I') return \"#eee\";\n",
+       "}\n",
+       "\n",
+       "function nodeStyle(selected) {\n",
+       "    return selected ? \"stroke-width: 2px; stroke: #00f\" : \"stroke-width: 1.5px\";\n",
+       "}\n",
+       "\n",
+       "var symbolGround = {\n",
+       "    draw: function(context, size){\n",
+       "        let s = size/2;\n",
+       "\n",
+       "        context.moveTo(0,-s);\n",
+       "        context.lineTo(0,0);\n",
+       "\n",
+       "        context.moveTo(-s,0);\n",
+       "        context.lineTo(s,0);\n",
+       "\n",
+       "        context.moveTo(-2*s/3,s/3);\n",
+       "        context.lineTo(2*s/3,s/3);\n",
+       "\n",
+       "        context.moveTo(-s/3,2*s/3);\n",
+       "        context.lineTo(s/3,2*s/3);\n",
+       "    }\n",
+       "}\n",
+       "\n",
+       "function showGraph(tag, graph, width, height, scale, node_size, auto_hbox, show_labels, scalar_str) {\n",
+       "    var ntab = {};\n",
+       "\n",
+       "    var groundOffset = 2.5 * node_size;\n",
+       "\n",
+       "    graph.nodes.forEach(function(d) {\n",
+       "        ntab[d.name] = d;\n",
+       "        d.selected = false;\n",
+       "        d.previouslySelected = false;\n",
+       "        d.nhd = [];\n",
+       "    });\n",
+       "\n",
+       "    var spiders_and_boundaries = graph.nodes.filter(function(d) {\n",
+       "        return d.t != 3;\n",
+       "    });\n",
+       "\n",
+       "    graph.links.forEach(function(d) {\n",
+       "        var s = ntab[d.source];\n",
+       "        var t = ntab[d.target];\n",
+       "        d.source = s;\n",
+       "        d.target = t;\n",
+       "        s.nhd.push(t);\n",
+       "        t.nhd.push(s);\n",
+       "    });\n",
+       "\n",
+       "    graph.pauli_web.forEach(function(d) {\n",
+       "        var s = ntab[d.source];\n",
+       "        var t = ntab[d.target];\n",
+       "        d.source = s;\n",
+       "        d.target = t;\n",
+       "    });\n",
+       "\n",
+       "    var shiftKey;\n",
+       "\n",
+       "    // SETUP SVG ITEMS\n",
+       "\n",
+       "    var svg = d3.select(tag)\n",
+       "    //.attr(\"tabindex\", 1)\n",
+       "        .on(\"keydown.brush\", function() {shiftKey = d3.event.shiftKey || d3.event.metaKey;})\n",
+       "        .on(\"keyup.brush\", function() {shiftKey = d3.event.shiftKey || d3.event.metaKey;})\n",
+       "    //.each(function() { this.focus(); })\n",
+       "        .append(\"svg\")\n",
+       "        .attr(\"style\", \"max-width: none; max-height: none\")\n",
+       "        .attr(\"width\", width)\n",
+       "        .attr(\"height\", height);\n",
+       "\n",
+       "    var web = svg.append(\"g\")\n",
+       "        .attr(\"class\", \"web\")\n",
+       "        .selectAll(\"line\")\n",
+       "        .data(graph.pauli_web)\n",
+       "        .enter().append(\"path\")\n",
+       "        .attr(\"stroke\", function(d) { return webColor(d.t); })\n",
+       "        .attr(\"fill\", \"transparent\")\n",
+       "        .attr(\"style\", \"stroke-width: 7px\");\n",
+       "\n",
+       "    var link = svg.append(\"g\")\n",
+       "        .attr(\"class\", \"link\")\n",
+       "        .selectAll(\"line\")\n",
+       "        .data(graph.links)\n",
+       "        .enter().append(\"path\")\n",
+       "        .attr(\"stroke\", function(d) { return edgeColor(d.t); })\n",
+       "        .attr(\"fill\", \"transparent\")\n",
+       "        .attr(\"style\", \"stroke-width: 1.5px\");\n",
+       "\n",
+       "    var brush = svg.append(\"g\")\n",
+       "        .attr(\"class\", \"brush\");\n",
+       "\n",
+       "    var node = svg.append(\"g\")\n",
+       "        .attr(\"class\", \"node\")\n",
+       "        .selectAll(\"g\")\n",
+       "        .data(graph.nodes)\n",
+       "        .enter().append(\"g\")\n",
+       "        .attr(\"transform\", function(d) {\n",
+       "            return \"translate(\" + d.x + \",\" + d.y +\")\";\n",
+       "        });\n",
+       "\n",
+       "    // Draw a ground symbol connected to the node.\n",
+       "    node.filter(function(d) { return d.ground; })\n",
+       "        .append(\"path\")\n",
+       "        .attr(\"stroke\", \"black\")\n",
+       "        .attr(\"style\", \"stroke-width: 1.5px\")\n",
+       "        .attr(\"fill\", \"none\")\n",
+       "        .attr(\"d\", \"M 0 0 L 0 \"+(groundOffset))\n",
+       "        .attr(\"class\", \"selectable\");\n",
+       "    node.filter(function(d) { return d.ground; })\n",
+       "        .append(\"path\")\n",
+       "        .attr(\"stroke\", \"black\")\n",
+       "        .attr(\"style\", \"stroke-width: 1.5px\")\n",
+       "        .attr(\"fill\", \"none\")\n",
+       "        .attr(\"d\", d3.symbol().type(symbolGround).size(node_size*1.5))\n",
+       "        .attr(\"transform\", \"translate(0,\"+groundOffset+\")\")\n",
+       "        .attr(\"class\", \"selectable\");\n",
+       "\n",
+       "    node.filter(function(d) { return d.t != 3 && d.t != 5 && d.t != 6; })\n",
+       "        .append(\"circle\")\n",
+       "        .attr(\"r\", function(d) {\n",
+       "            if (d.t == 0) return 0.5 * node_size;\n",
+       "            else if (d.t == 4) return 0.25 * node_size;\n",
+       "            else return node_size;\n",
+       "        })\n",
+       "        .attr(\"fill\", function(d) { return nodeColor(d.t); })\n",
+       "        .attr(\"stroke\", \"black\")\n",
+       "        .attr(\"class\", \"selectable\");\n",
+       "\n",
+       "    var hbox = node.filter(function(d) { return d.t == 3; });\n",
+       "\n",
+       "    hbox.append(\"rect\")\n",
+       "        .attr(\"x\", -0.75 * node_size).attr(\"y\", -0.75 * node_size)\n",
+       "        .attr(\"width\", node_size * 1.5).attr(\"height\", node_size * 1.5)\n",
+       "        .attr(\"fill\", function(d) { return nodeColor(d.t); })\n",
+       "        .attr(\"stroke\", \"black\")\n",
+       "        .attr(\"class\", \"selectable\");\n",
+       "\n",
+       "    // draw a triangle for d.t == 5\n",
+       "    node.filter(function(d) { return d.t == 5; })\n",
+       "        .append(\"path\")\n",
+       "        .attr(\"d\", \"M 0 0 L \"+node_size+\" \"+node_size+\" L -\"+node_size+\" \"+node_size+\" Z\")\n",
+       "        .attr(\"fill\", function(d) { return nodeColor(d.t); })\n",
+       "        .attr(\"stroke\", \"black\")\n",
+       "        .attr(\"class\", \"selectable\")\n",
+       "        .attr(\"transform\", \"translate(\" + (-node_size/2) + \", 0) rotate(-90)\");\n",
+       "\n",
+       "    // draw a square for Z box: d.t == 6\n",
+       "    node.filter(function(d) { return d.t == 6; })\n",
+       "        .append(\"rect\")\n",
+       "        .attr(\"x\", -0.75 * node_size).attr(\"y\", -0.75 * node_size)\n",
+       "        .attr(\"width\", node_size * 1.5).attr(\"height\", node_size * 1.5)\n",
+       "        .attr(\"fill\", function(d) { return nodeColor(d.t); })\n",
+       "        .attr(\"stroke\", \"black\")\n",
+       "        .attr(\"class\", \"selectable\");\n",
+       "\n",
+       "    node.filter(function(d) { return d.phase != ''; })\n",
+       "        .append(\"text\")\n",
+       "        .attr(\"y\", 0.7 * node_size + 14)\n",
+       "        .text(function (d) { return d.phase })\n",
+       "        .attr(\"text-anchor\", \"middle\")\n",
+       "        .attr(\"font-size\", \"12px\")\n",
+       "        .attr(\"font-family\", \"monospace\")\n",
+       "        .attr(\"fill\", \"#00d\")\n",
+       "        .attr('style', 'pointer-events: none; user-select: none;');\n",
+       "\n",
+       "    if (show_labels) {\n",
+       "        node.append(\"text\")\n",
+       "            .attr(\"y\", -0.7 * node_size - 8)\n",
+       "            .text(function (d) { return d.name; })\n",
+       "            .attr(\"text-anchor\", \"middle\")\n",
+       "            .attr(\"font-size\", \"10px\")\n",
+       "            .attr(\"font-family\", \"monospace\")\n",
+       "            .attr(\"fill\", \"#999\")\n",
+       "            .attr('style', 'pointer-events: none; user-select: none;');\n",
+       "    }\n",
+       "\n",
+       "    // Display the chosen data fields over the node.\n",
+       "    node.filter(d => d.vdata.length > 0)\n",
+       "        .append(\"text\")\n",
+       "        .attr(\"y\", d => -0.7 * node_size - 14 - 10 * d.vdata.length)\n",
+       "        .attr(\"text-anchor\", \"middle\")\n",
+       "        .attr(\"font-size\", \"8px\")\n",
+       "        .attr(\"font-family\", \"monospace\")\n",
+       "        .attr(\"fill\", \"#c66\")\n",
+       "        .attr('style', 'pointer-events: none; user-select: none;')\n",
+       "        .selectAll(\"tspan\")\n",
+       "        .data(d => d.vdata)\n",
+       "        .enter()\n",
+       "        .append(\"tspan\")\n",
+       "        .attr(\"x\", \"0\")\n",
+       "        .attr(\"dy\", \"1.2em\")\n",
+       "        .text(x => x.join(\": \"));\n",
+       "\n",
+       "    if (scalar_str != \"\") {\n",
+       "        svg.append(\"text\")\n",
+       "            .text(scalar_str)\n",
+       "            .attr(\"x\", 60).attr(\"y\", 40)\n",
+       "            .attr(\"text-anchor\", \"middle\")\n",
+       "    }\n",
+       "\n",
+       "    function update_hboxes() {\n",
+       "        if (auto_hbox) {\n",
+       "            var pos = {};\n",
+       "            hbox.attr(\"transform\", function(d) {\n",
+       "                // calculate barycenter of non-hbox neighbours, then nudge a bit\n",
+       "                // to the NE.\n",
+       "                var x=0,y=0,sz=0;\n",
+       "                for (var i = 0; i < d.nhd.length; ++i) {\n",
+       "                    if (d.nhd[i].t != 3) {\n",
+       "                        sz++;\n",
+       "                        x += d.nhd[i].x;\n",
+       "                        y += d.nhd[i].y;\n",
+       "                    }\n",
+       "                }\n",
+       "\n",
+       "                offset = 0.25 * scale;\n",
+       "\n",
+       "                if (sz != 0) {\n",
+       "                    x = (x/sz) + offset;\n",
+       "                    y = (y/sz) - offset;\n",
+       "\n",
+       "                    while (pos[[x,y]]) {\n",
+       "                        x += offset;\n",
+       "                    }\n",
+       "                    d.x = x;\n",
+       "                    d.y = y;\n",
+       "                    pos[[x,y]] = true;\n",
+       "                }\n",
+       "\n",
+       "                return \"translate(\"+d.x+\",\"+d.y+\")\";\n",
+       "            });\n",
+       "        }\n",
+       "    }\n",
+       "\n",
+       "    update_hboxes();\n",
+       "\n",
+       "    var link_curve = function(d) {\n",
+       "        var x1 = d.source.x, x2 = d.target.x, y1 = d.source.y, y2 = d.target.y;\n",
+       "        if (d.num_parallel == 1) {\n",
+       "            return `M ${x1} ${y1} L ${x2} ${y2}`;\n",
+       "        } else {\n",
+       "            var dx = x2 - x1, dy = y2 - y1;\n",
+       "            var midx = 0.5 * (x1 + x2), midy = 0.5 * (y1 + y2);\n",
+       "            var pos = (d.index / (d.num_parallel-1)) - 0.5;\n",
+       "            var cx = midx - pos * dy;\n",
+       "            var cy = midy + pos * dx;\n",
+       "            return `M ${x1} ${y1} Q ${cx} ${cy}, ${x2} ${y2}`;\n",
+       "            // return `M ${x1} ${y1} L ${x2} ${y2}`;\n",
+       "        }\n",
+       "    };\n",
+       "    link.attr(\"d\", link_curve);\n",
+       "\n",
+       "\n",
+       "    var web_curve = function(d) {\n",
+       "        var x1 = d.source.x, x2 = (x1 + d.target.x)/2, y1 = d.source.y, y2 = (y1 + d.target.y)/2;\n",
+       "        return `M ${x1} ${y1} L ${x2} ${y2}`;\n",
+       "    }\n",
+       "    web.attr(\"d\", web_curve);\n",
+       "\n",
+       "    // EVENTS FOR DRAGGING AND SELECTION\n",
+       "\n",
+       "    node.on(\"mousedown\", function(d) {\n",
+       "        if (shiftKey) {\n",
+       "            d3.select(this).selectAll(\".selectable\").attr(\"style\", nodeStyle(d.selected = !d.selected));\n",
+       "            d3.event.stopImmediatePropagation();\n",
+       "        } else if (!d.selected) {\n",
+       "            node.selectAll(\".selectable\").attr(\"style\", function(p) { return nodeStyle(p.selected = d === p); });\n",
+       "        }\n",
+       "    })\n",
+       "        .call(d3.drag().on(\"drag\", function(d) {\n",
+       "            var dx = d3.event.dx;\n",
+       "            var dy = d3.event.dy;\n",
+       "            // node.filter(function(d) { return d.selected; })\n",
+       "            //     .attr(\"cx\", function(d) { return d.x += dx; })\n",
+       "            //     .attr(\"cy\", function(d) { return d.y += dy; });\n",
+       "            node.filter(function(d) { return d.selected; })\n",
+       "                .attr(\"transform\", function(d) {\n",
+       "                    d.x += dx;\n",
+       "                    d.y += dy;\n",
+       "                    return \"translate(\" + d.x + \",\" + d.y +\")\";\n",
+       "                });\n",
+       "\n",
+       "            update_hboxes();\n",
+       "\n",
+       "            link.filter(function(d) { return d.source.selected || d.target.selected ||\n",
+       "                    (auto_hbox && d.source.t == 3); })\n",
+       "                .attr(\"d\", link_curve);\n",
+       "            web.filter(function(d) { return d.source.selected || d.target.selected; })\n",
+       "                .attr(\"d\", web_curve);\n",
+       "        }));\n",
+       "\n",
+       "    brush.call(d3.brush().keyModifiers(false)\n",
+       "        .extent([[0, 0], [width, height]])\n",
+       "        .on(\"start\", function() {\n",
+       "            if (d3.event.sourceEvent.type !== \"end\") {\n",
+       "                node.selectAll(\".selectable\").attr(\"style\", function(d) {\n",
+       "                    return nodeStyle(\n",
+       "                        d.selected = d.previouslySelected = shiftKey &&\n",
+       "                        d.selected);\n",
+       "                });\n",
+       "            }\n",
+       "        })\n",
+       "        .on(\"brush\", function() {\n",
+       "            if (d3.event.sourceEvent.type !== \"end\") {\n",
+       "                var selection = d3.event.selection;\n",
+       "                node.selectAll(\".selectable\").attr(\"style\", function(d) {\n",
+       "                    return nodeStyle(d.selected = d.previouslySelected ^\n",
+       "                        (selection != null\n",
+       "                            && selection[0][0] <= d.x && d.x < selection[1][0]\n",
+       "                            && selection[0][1] <= d.y && d.y < selection[1][1]));\n",
+       "                });\n",
+       "            }\n",
+       "        })\n",
+       "        .on(\"end\", function() {\n",
+       "            if (d3.event.selection != null) {\n",
+       "                d3.select(this).call(d3.event.target.move, null);\n",
+       "            }\n",
+       "        }));\n",
+       "}\n",
+       "\n",
+       "\n",
+       "showGraph('#graph-output-h8yGb4hV',\n",
+       "  JSON.parse('{\"nodes\": [{\"name\": \"0\", \"x\": 50, \"y\": 100, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"1\", \"x\": 50, \"y\": 150, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"2\", \"x\": 50, \"y\": 200, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"3\", \"x\": 100, \"y\": 150, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"4\", \"x\": 100, \"y\": 100, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"5\", \"x\": 150, \"y\": 200, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"6\", \"x\": 150, \"y\": 150, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"9\", \"x\": 250, \"y\": 150, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"10\", \"x\": 250, \"y\": 200, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"13\", \"x\": 350, \"y\": 100, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"15\", \"x\": 400, \"y\": 100, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"16\", \"x\": 400, \"y\": 150, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"17\", \"x\": 400, \"y\": 200, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}], \"links\": [{\"source\": \"0\", \"target\": \"4\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"1\", \"target\": \"3\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"2\", \"target\": \"5\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"3\", \"target\": \"4\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"3\", \"target\": \"6\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"4\", \"target\": \"13\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"4\", \"target\": \"5\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"4\", \"target\": \"9\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"5\", \"target\": \"6\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"5\", \"target\": \"10\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"6\", \"target\": \"9\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"9\", \"target\": \"10\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"9\", \"target\": \"16\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"10\", \"target\": \"13\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"10\", \"target\": \"17\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"13\", \"target\": \"15\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}], \"pauli_web\": []}'), 450, 250, 50,\n",
        "  10.0, false, true, '');\n",
        "</script>"
       ],
@@ -2728,21 +6565,21 @@
     }
    ],
    "source": [
-    "# The CNOT example also has 2 more bounded Pauli webs, with boundaries on the two interior spiders.\n",
-    "# Since we aren't injecting magic states at these locations, we don't need these Pauli webs.\n",
-    "zx.draw(g, labels=True, pauli_web=zwebs[3])\n",
-    "zx.draw(g, labels=True, pauli_web=xwebs[2])"
+    "# Alternatively, we can fuse all the spiders of the same color together to get a\n",
+    "# more interesting diagram.\n",
+    "zx.spider_simp(g)\n",
+    "zx.draw(g, labels=True)"
    ]
   },
   {
    "cell_type": "code",
-   "execution_count": 5,
+   "execution_count": 8,
    "metadata": {},
    "outputs": [
     {
      "data": {
       "text/html": [
-       "<div style=\"overflow:auto; background-color: white\" id=\"graph-output-W49FAkVj\"></div>\n",
+       "<div style=\"overflow:auto; background-color: white\" id=\"graph-output-bJnRs36u\"></div>\n",
        "<script type=\"module\">\n",
        "var d3;\n",
        "if (d3 == null) { d3 = await import(\"https://cdn.skypack.dev/d3@5\"); }\n",
@@ -3105,8 +6942,8 @@
        "}\n",
        "\n",
        "\n",
-       "showGraph('#graph-output-W49FAkVj',\n",
-       "  JSON.parse('{\"nodes\": [{\"name\": \"0\", \"x\": 50, \"y\": 100, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"1\", \"x\": 50, \"y\": 150, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"2\", \"x\": 50, \"y\": 200, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"3\", \"x\": 100, \"y\": 150, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"4\", \"x\": 100, \"y\": 100, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"5\", \"x\": 150, \"y\": 200, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"6\", \"x\": 150, \"y\": 150, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"7\", \"x\": 200, \"y\": 200, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"8\", \"x\": 200, \"y\": 100, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"9\", \"x\": 250, \"y\": 150, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"10\", \"x\": 250, \"y\": 200, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"11\", \"x\": 300, \"y\": 150, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"12\", \"x\": 300, \"y\": 100, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"13\", \"x\": 350, \"y\": 100, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"14\", \"x\": 350, \"y\": 200, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"15\", \"x\": 400, \"y\": 100, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"16\", \"x\": 400, \"y\": 150, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"17\", \"x\": 400, \"y\": 200, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}], \"links\": [{\"source\": \"0\", \"target\": \"4\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"1\", \"target\": \"3\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"2\", \"target\": \"5\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"3\", \"target\": \"4\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"3\", \"target\": \"6\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"4\", \"target\": \"8\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"5\", \"target\": \"6\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"5\", \"target\": \"7\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"6\", \"target\": \"9\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"7\", \"target\": \"8\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"7\", \"target\": \"10\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"8\", \"target\": \"12\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"9\", \"target\": \"10\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"9\", \"target\": \"11\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"10\", \"target\": \"14\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"11\", \"target\": \"12\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"11\", \"target\": \"16\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"12\", \"target\": \"13\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"13\", \"target\": \"14\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"13\", \"target\": \"15\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"14\", \"target\": \"17\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}], \"pauli_web\": []}'), 450, 250, 50,\n",
+       "showGraph('#graph-output-bJnRs36u',\n",
+       "  JSON.parse('{\"nodes\": [{\"name\": \"0\", \"x\": 50, \"y\": 100, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"1\", \"x\": 50, \"y\": 150, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"2\", \"x\": 50, \"y\": 200, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"3\", \"x\": 100, \"y\": 150, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"4\", \"x\": 100, \"y\": 100, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"5\", \"x\": 150, \"y\": 200, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"6\", \"x\": 150, \"y\": 150, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"9\", \"x\": 250, \"y\": 150, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"10\", \"x\": 250, \"y\": 200, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"13\", \"x\": 350, \"y\": 100, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"15\", \"x\": 400, \"y\": 100, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"16\", \"x\": 400, \"y\": 150, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"17\", \"x\": 400, \"y\": 200, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}], \"links\": [{\"source\": \"0\", \"target\": \"4\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"1\", \"target\": \"3\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"2\", \"target\": \"5\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"3\", \"target\": \"4\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"3\", \"target\": \"6\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"4\", \"target\": \"13\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"4\", \"target\": \"5\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"4\", \"target\": \"9\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"5\", \"target\": \"6\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"5\", \"target\": \"10\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"6\", \"target\": \"9\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"9\", \"target\": \"10\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"9\", \"target\": \"16\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"10\", \"target\": \"13\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"10\", \"target\": \"17\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"13\", \"target\": \"15\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}], \"pauli_web\": [{\"source\": 15, \"target\": 13, \"t\": \"X\"}, {\"source\": 13, \"target\": 15, \"t\": \"X\"}, {\"source\": 4, \"target\": 0, \"t\": \"X\"}, {\"source\": 0, \"target\": 4, \"t\": \"X\"}, {\"source\": 4, \"target\": 3, \"t\": \"X\"}, {\"source\": 3, \"target\": 4, \"t\": \"X\"}, {\"source\": 4, \"target\": 13, \"t\": \"X\"}, {\"source\": 13, \"target\": 4, \"t\": \"X\"}, {\"source\": 4, \"target\": 5, \"t\": \"X\"}, {\"source\": 5, \"target\": 4, \"t\": \"X\"}, {\"source\": 4, \"target\": 9, \"t\": \"X\"}, {\"source\": 9, \"target\": 4, \"t\": \"X\"}, {\"source\": 6, \"target\": 3, \"t\": \"X\"}, {\"source\": 3, \"target\": 6, \"t\": \"X\"}, {\"source\": 6, \"target\": 5, \"t\": \"X\"}, {\"source\": 5, \"target\": 6, \"t\": \"X\"}, {\"source\": 6, \"target\": 9, \"t\": \"X\"}, {\"source\": 9, \"target\": 6, \"t\": \"X\"}]}'), 450, 250, 50,\n",
        "  10.0, false, true, '');\n",
        "</script>"
       ],
@@ -3116,34 +6953,11 @@
      },
      "metadata": {},
      "output_type": "display_data"
-    }
-   ],
-   "source": [
-    "# Here's an example with more CNOT gates\n",
-    "\n",
-    "c = zx.qasm(\"\"\"\n",
-    "qreg q[3];\n",
-    "cx q[0], q[1];\n",
-    "cx q[1], q[2];\n",
-    "cx q[0], q[2];\n",
-    "cx q[2], q[1];\n",
-    "cx q[0], q[1];\n",
-    "cx q[2], q[0];\n",
-    "\"\"\")\n",
-    "g = c.to_graph()\n",
-    "\n",
-    "zx.draw(g, labels=True)"
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": 6,
-   "metadata": {},
-   "outputs": [
+    },
     {
      "data": {
       "text/html": [
-       "<div style=\"overflow:auto; background-color: white\" id=\"graph-output-y0W9ULX7\"></div>\n",
+       "<div style=\"overflow:auto; background-color: white\" id=\"graph-output-KiTgNX9P\"></div>\n",
        "<script type=\"module\">\n",
        "var d3;\n",
        "if (d3 == null) { d3 = await import(\"https://cdn.skypack.dev/d3@5\"); }\n",
@@ -3506,8 +7320,8 @@
        "}\n",
        "\n",
        "\n",
-       "showGraph('#graph-output-y0W9ULX7',\n",
-       "  JSON.parse('{\"nodes\": [{\"name\": \"0\", \"x\": 50, \"y\": 100, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"1\", \"x\": 50, \"y\": 150, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"2\", \"x\": 50, \"y\": 200, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"3\", \"x\": 100, \"y\": 150, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"4\", \"x\": 100, \"y\": 100, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"5\", \"x\": 150, \"y\": 200, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"6\", \"x\": 150, \"y\": 150, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"7\", \"x\": 200, \"y\": 200, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"8\", \"x\": 200, \"y\": 100, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"9\", \"x\": 250, \"y\": 150, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"10\", \"x\": 250, \"y\": 200, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"11\", \"x\": 300, \"y\": 150, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"12\", \"x\": 300, \"y\": 100, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"13\", \"x\": 350, \"y\": 100, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"14\", \"x\": 350, \"y\": 200, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"15\", \"x\": 400, \"y\": 100, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"16\", \"x\": 400, \"y\": 150, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"17\", \"x\": 400, \"y\": 200, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}], \"links\": [{\"source\": \"0\", \"target\": \"4\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"1\", \"target\": \"3\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"2\", \"target\": \"5\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"3\", \"target\": \"4\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"3\", \"target\": \"6\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"4\", \"target\": \"8\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"5\", \"target\": \"6\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"5\", \"target\": \"7\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"6\", \"target\": \"9\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"7\", \"target\": \"8\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"7\", \"target\": \"10\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"8\", \"target\": \"12\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"9\", \"target\": \"10\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"9\", \"target\": \"11\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"10\", \"target\": \"14\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"11\", \"target\": \"12\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"11\", \"target\": \"16\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"12\", \"target\": \"13\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"13\", \"target\": \"14\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"13\", \"target\": \"15\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"14\", \"target\": \"17\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}], \"pauli_web\": [{\"source\": 4, \"target\": 0, \"t\": \"X\"}, {\"source\": 0, \"target\": 4, \"t\": \"X\"}, {\"source\": 4, \"target\": 3, \"t\": \"X\"}, {\"source\": 3, \"target\": 4, \"t\": \"X\"}, {\"source\": 4, \"target\": 8, \"t\": \"X\"}, {\"source\": 6, \"target\": 3, \"t\": \"X\"}, {\"source\": 3, \"target\": 6, \"t\": \"X\"}, {\"source\": 6, \"target\": 5, \"t\": \"X\"}, {\"source\": 5, \"target\": 6, \"t\": \"X\"}, {\"source\": 6, \"target\": 9, \"t\": \"X\"}, {\"source\": 9, \"target\": 6, \"t\": \"X\"}, {\"source\": 8, \"target\": 4, \"t\": \"X\"}, {\"source\": 8, \"target\": 7, \"t\": \"X\"}, {\"source\": 7, \"target\": 8, \"t\": \"X\"}, {\"source\": 8, \"target\": 12, \"t\": \"X\"}, {\"source\": 12, \"target\": 8, \"t\": \"X\"}, {\"source\": 12, \"target\": 11, \"t\": \"X\"}, {\"source\": 11, \"target\": 12, \"t\": \"X\"}, {\"source\": 12, \"target\": 13, \"t\": \"X\"}, {\"source\": 13, \"target\": 12, \"t\": \"X\"}, {\"source\": 5, \"target\": 7, \"t\": \"X\"}, {\"source\": 7, \"target\": 5, \"t\": \"X\"}, {\"source\": 9, \"target\": 11, \"t\": \"X\"}, {\"source\": 11, \"target\": 9, \"t\": \"X\"}, {\"source\": 15, \"target\": 13, \"t\": \"X\"}, {\"source\": 13, \"target\": 15, \"t\": \"X\"}]}'), 450, 250, 50,\n",
+       "showGraph('#graph-output-KiTgNX9P',\n",
+       "  JSON.parse('{\"nodes\": [{\"name\": \"0\", \"x\": 50, \"y\": 100, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"1\", \"x\": 50, \"y\": 150, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"2\", \"x\": 50, \"y\": 200, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"3\", \"x\": 100, \"y\": 150, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"4\", \"x\": 100, \"y\": 100, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"5\", \"x\": 150, \"y\": 200, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"6\", \"x\": 150, \"y\": 150, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"9\", \"x\": 250, \"y\": 150, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"10\", \"x\": 250, \"y\": 200, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"13\", \"x\": 350, \"y\": 100, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"15\", \"x\": 400, \"y\": 100, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"16\", \"x\": 400, \"y\": 150, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"17\", \"x\": 400, \"y\": 200, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}], \"links\": [{\"source\": \"0\", \"target\": \"4\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"1\", \"target\": \"3\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"2\", \"target\": \"5\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"3\", \"target\": \"4\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"3\", \"target\": \"6\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"4\", \"target\": \"13\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"4\", \"target\": \"5\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"4\", \"target\": \"9\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"5\", \"target\": \"6\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"5\", \"target\": \"10\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"6\", \"target\": \"9\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"9\", \"target\": \"10\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"9\", \"target\": \"16\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"10\", \"target\": \"13\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"10\", \"target\": \"17\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"13\", \"target\": \"15\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}], \"pauli_web\": [{\"source\": 3, \"target\": 1, \"t\": \"Z\"}, {\"source\": 1, \"target\": 3, \"t\": \"Z\"}, {\"source\": 3, \"target\": 4, \"t\": \"Z\"}, {\"source\": 4, \"target\": 3, \"t\": \"Z\"}, {\"source\": 3, \"target\": 6, \"t\": \"Z\"}, {\"source\": 6, \"target\": 3, \"t\": \"Z\"}, {\"source\": 5, \"target\": 2, \"t\": \"Z\"}, {\"source\": 2, \"target\": 5, \"t\": \"Z\"}, {\"source\": 5, \"target\": 6, \"t\": \"Z\"}, {\"source\": 6, \"target\": 5, \"t\": \"Z\"}, {\"source\": 5, \"target\": 10, \"t\": \"Z\"}, {\"source\": 10, \"target\": 5, \"t\": \"Z\"}, {\"source\": 5, \"target\": 4, \"t\": \"Z\"}, {\"source\": 4, \"target\": 5, \"t\": \"Z\"}, {\"source\": 13, \"target\": 15, \"t\": \"Z\"}, {\"source\": 15, \"target\": 13, \"t\": \"Z\"}, {\"source\": 13, \"target\": 10, \"t\": \"Z\"}, {\"source\": 10, \"target\": 13, \"t\": \"Z\"}, {\"source\": 13, \"target\": 4, \"t\": \"Z\"}, {\"source\": 4, \"target\": 13, \"t\": \"Z\"}, {\"source\": 0, \"target\": 4, \"t\": \"Z\"}, {\"source\": 4, \"target\": 0, \"t\": \"Z\"}]}'), 450, 250, 50,\n",
        "  10.0, false, true, '');\n",
        "</script>"
       ],
@@ -3521,7 +7335,7 @@
     {
      "data": {
       "text/html": [
-       "<div style=\"overflow:auto; background-color: white\" id=\"graph-output-IQBiiaAK\"></div>\n",
+       "<div style=\"overflow:auto; background-color: white\" id=\"graph-output-16cPyoFw\"></div>\n",
        "<script type=\"module\">\n",
        "var d3;\n",
        "if (d3 == null) { d3 = await import(\"https://cdn.skypack.dev/d3@5\"); }\n",
@@ -3884,8 +7698,8 @@
        "}\n",
        "\n",
        "\n",
-       "showGraph('#graph-output-IQBiiaAK',\n",
-       "  JSON.parse('{\"nodes\": [{\"name\": \"0\", \"x\": 50, \"y\": 100, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"1\", \"x\": 50, \"y\": 150, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"2\", \"x\": 50, \"y\": 200, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"3\", \"x\": 100, \"y\": 150, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"4\", \"x\": 100, \"y\": 100, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"5\", \"x\": 150, \"y\": 200, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"6\", \"x\": 150, \"y\": 150, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"7\", \"x\": 200, \"y\": 200, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"8\", \"x\": 200, \"y\": 100, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"9\", \"x\": 250, 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\"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"9\", \"target\": \"11\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"10\", \"target\": \"14\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"11\", \"target\": \"12\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"11\", \"target\": \"16\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"12\", \"target\": \"13\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"13\", \"target\": \"14\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"13\", \"target\": \"15\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"14\", \"target\": \"17\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}], \"pauli_web\": [{\"source\": 3, \"target\": 1, \"t\": \"Z\"}, {\"source\": 1, \"target\": 3, \"t\": \"Z\"}, {\"source\": 3, \"target\": 4, \"t\": \"Z\"}, {\"source\": 4, \"target\": 3, \"t\": \"Z\"}, {\"source\": 3, \"target\": 6, \"t\": \"Z\"}, {\"source\": 6, \"target\": 3, \"t\": \"Z\"}, {\"source\": 5, \"target\": 2, \"t\": \"Z\"}, {\"source\": 2, \"target\": 5, \"t\": \"Z\"}, {\"source\": 5, \"target\": 6, \"t\": \"Z\"}, {\"source\": 6, \"target\": 5, \"t\": \"Z\"}, {\"source\": 5, \"target\": 7, \"t\": \"Z\"}, {\"source\": 7, \"target\": 5, \"t\": \"Z\"}, {\"source\": 7, \"target\": 8, \"t\": \"Z\"}, {\"source\": 8, \"target\": 7, \"t\": \"Z\"}, {\"source\": 7, \"target\": 10, \"t\": \"Z\"}, {\"source\": 10, \"target\": 7, \"t\": \"Z\"}, {\"source\": 13, \"target\": 12, \"t\": \"Z\"}, {\"source\": 12, \"target\": 13, \"t\": \"Z\"}, {\"source\": 13, \"target\": 14, \"t\": \"Z\"}, {\"source\": 14, \"target\": 13, \"t\": \"Z\"}, {\"source\": 13, \"target\": 15, \"t\": \"Z\"}, {\"source\": 15, \"target\": 13, \"t\": \"Z\"}, {\"source\": 8, \"target\": 12, \"t\": \"Z\"}, {\"source\": 12, \"target\": 8, \"t\": \"Z\"}, {\"source\": 10, \"target\": 14, \"t\": \"Z\"}, {\"source\": 14, \"target\": 10, \"t\": \"Z\"}, {\"source\": 4, \"target\": 0, \"t\": \"Z\"}, {\"source\": 0, \"target\": 4, \"t\": \"Z\"}]}'), 450, 250, 50,\n",
+       "showGraph('#graph-output-16cPyoFw',\n",
+       "  JSON.parse('{\"nodes\": [{\"name\": \"0\", \"x\": 50, \"y\": 100, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"1\", \"x\": 50, \"y\": 150, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"2\", \"x\": 50, \"y\": 200, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"3\", \"x\": 100, \"y\": 150, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"4\", \"x\": 100, \"y\": 100, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"5\", \"x\": 150, \"y\": 200, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"6\", \"x\": 150, \"y\": 150, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"9\", \"x\": 250, \"y\": 150, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"10\", \"x\": 250, \"y\": 200, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"13\", \"x\": 350, \"y\": 100, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"15\", \"x\": 400, \"y\": 100, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"16\", \"x\": 400, \"y\": 150, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"17\", \"x\": 400, \"y\": 200, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}], \"links\": [{\"source\": \"0\", \"target\": \"4\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"1\", \"target\": \"3\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"2\", \"target\": \"5\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"3\", \"target\": \"4\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"3\", \"target\": \"6\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"4\", \"target\": \"13\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"4\", \"target\": \"5\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"4\", \"target\": \"9\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"5\", \"target\": \"6\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"5\", \"target\": \"10\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"6\", \"target\": \"9\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"9\", \"target\": \"10\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"9\", \"target\": \"16\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"10\", \"target\": \"13\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"10\", \"target\": \"17\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"13\", \"target\": \"15\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}], \"pauli_web\": [{\"source\": 16, \"target\": 9, \"t\": \"X\"}, {\"source\": 9, \"target\": 16, \"t\": \"X\"}, {\"source\": 1, \"target\": 3, \"t\": \"X\"}, {\"source\": 3, \"target\": 1, \"t\": \"X\"}, {\"source\": 6, \"target\": 3, \"t\": \"X\"}, {\"source\": 3, \"target\": 6, \"t\": \"X\"}, {\"source\": 6, \"target\": 5, \"t\": \"X\"}, {\"source\": 5, \"target\": 6, \"t\": \"X\"}, {\"source\": 6, \"target\": 9, \"t\": \"X\"}, {\"source\": 9, \"target\": 6, \"t\": \"X\"}, {\"source\": 2, \"target\": 5, \"t\": \"X\"}, {\"source\": 5, \"target\": 2, \"t\": \"X\"}]}'), 450, 250, 50,\n",
        "  10.0, false, true, '');\n",
        "</script>"
       ],
@@ -3899,7 +7713,7 @@
     {
      "data": {
       "text/html": [
-       "<div style=\"overflow:auto; background-color: white\" id=\"graph-output-LokrMgvn\"></div>\n",
+       "<div style=\"overflow:auto; background-color: white\" id=\"graph-output-0FPsNEi7\"></div>\n",
        "<script type=\"module\">\n",
        "var d3;\n",
        "if (d3 == null) { d3 = await import(\"https://cdn.skypack.dev/d3@5\"); }\n",
@@ -4262,8 +8076,8 @@
        "}\n",
        "\n",
        "\n",
-       "showGraph('#graph-output-LokrMgvn',\n",
-       "  JSON.parse('{\"nodes\": [{\"name\": \"0\", \"x\": 50, \"y\": 100, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"1\", \"x\": 50, \"y\": 150, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"2\", \"x\": 50, \"y\": 200, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"3\", \"x\": 100, \"y\": 150, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"4\", \"x\": 100, \"y\": 100, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"5\", \"x\": 150, \"y\": 200, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"6\", \"x\": 150, \"y\": 150, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"7\", \"x\": 200, \"y\": 200, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"8\", \"x\": 200, \"y\": 100, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"9\", \"x\": 250, \"y\": 150, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"10\", \"x\": 250, \"y\": 200, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"11\", \"x\": 300, \"y\": 150, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"12\", \"x\": 300, \"y\": 100, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"13\", \"x\": 350, \"y\": 100, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"14\", \"x\": 350, \"y\": 200, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"15\", \"x\": 400, \"y\": 100, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"16\", \"x\": 400, \"y\": 150, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"17\", \"x\": 400, \"y\": 200, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}], \"links\": [{\"source\": \"0\", \"target\": \"4\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"1\", \"target\": \"3\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"2\", \"target\": \"5\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"3\", \"target\": \"4\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"3\", \"target\": \"6\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"4\", \"target\": \"8\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"5\", \"target\": \"6\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"5\", \"target\": \"7\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"6\", \"target\": \"9\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"7\", \"target\": \"8\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"7\", \"target\": \"10\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"8\", \"target\": \"12\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"9\", \"target\": \"10\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"9\", \"target\": \"11\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"10\", \"target\": \"14\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"11\", \"target\": \"12\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"11\", \"target\": \"16\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"12\", \"target\": \"13\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"13\", \"target\": \"14\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"13\", \"target\": \"15\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"14\", \"target\": \"17\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}], \"pauli_web\": [{\"source\": 16, \"target\": 11, \"t\": \"X\"}, {\"source\": 11, \"target\": 16, \"t\": \"X\"}, {\"source\": 9, \"target\": 11, \"t\": \"X\"}, {\"source\": 11, \"target\": 9, \"t\": \"X\"}, {\"source\": 6, \"target\": 3, \"t\": \"X\"}, {\"source\": 3, \"target\": 6, \"t\": \"X\"}, {\"source\": 6, \"target\": 5, \"t\": \"X\"}, {\"source\": 5, \"target\": 6, \"t\": \"X\"}, {\"source\": 6, \"target\": 9, \"t\": \"X\"}, {\"source\": 9, \"target\": 6, \"t\": \"X\"}, {\"source\": 3, \"target\": 1, \"t\": \"X\"}, {\"source\": 1, \"target\": 3, \"t\": \"X\"}, {\"source\": 5, \"target\": 2, \"t\": \"X\"}, {\"source\": 2, \"target\": 5, \"t\": \"X\"}]}'), 450, 250, 50,\n",
+       "showGraph('#graph-output-0FPsNEi7',\n",
+       "  JSON.parse('{\"nodes\": [{\"name\": \"0\", \"x\": 50, \"y\": 100, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"1\", \"x\": 50, \"y\": 150, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"2\", \"x\": 50, \"y\": 200, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"3\", \"x\": 100, \"y\": 150, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"4\", \"x\": 100, \"y\": 100, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"5\", \"x\": 150, \"y\": 200, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"6\", \"x\": 150, \"y\": 150, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"9\", \"x\": 250, \"y\": 150, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"10\", \"x\": 250, \"y\": 200, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"13\", \"x\": 350, \"y\": 100, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"15\", \"x\": 400, \"y\": 100, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"16\", \"x\": 400, \"y\": 150, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"17\", \"x\": 400, \"y\": 200, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}], \"links\": [{\"source\": \"0\", \"target\": \"4\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"1\", \"target\": \"3\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"2\", \"target\": \"5\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"3\", \"target\": \"4\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"3\", \"target\": \"6\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"4\", \"target\": \"13\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"4\", \"target\": \"5\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"4\", \"target\": \"9\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"5\", \"target\": \"6\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"5\", \"target\": \"10\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"6\", \"target\": \"9\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"9\", \"target\": \"10\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"9\", \"target\": \"16\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"10\", \"target\": \"13\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"10\", \"target\": \"17\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"13\", \"target\": \"15\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}], \"pauli_web\": [{\"source\": 9, \"target\": 6, \"t\": \"Z\"}, {\"source\": 6, \"target\": 9, \"t\": \"Z\"}, {\"source\": 9, \"target\": 10, \"t\": \"Z\"}, {\"source\": 10, \"target\": 9, \"t\": \"Z\"}, {\"source\": 9, \"target\": 16, \"t\": \"Z\"}, {\"source\": 16, \"target\": 9, \"t\": \"Z\"}, {\"source\": 9, \"target\": 4, \"t\": \"Z\"}, {\"source\": 4, \"target\": 9, \"t\": \"Z\"}, {\"source\": 5, \"target\": 2, \"t\": \"Z\"}, {\"source\": 2, \"target\": 5, \"t\": \"Z\"}, {\"source\": 5, \"target\": 6, \"t\": \"Z\"}, {\"source\": 6, \"target\": 5, \"t\": \"Z\"}, {\"source\": 5, \"target\": 10, \"t\": \"Z\"}, {\"source\": 10, \"target\": 5, \"t\": \"Z\"}, {\"source\": 5, \"target\": 4, \"t\": \"Z\"}, {\"source\": 4, \"target\": 5, \"t\": \"Z\"}]}'), 450, 250, 50,\n",
        "  10.0, false, true, '');\n",
        "</script>"
       ],
@@ -4277,7 +8091,7 @@
     {
      "data": {
       "text/html": [
-       "<div style=\"overflow:auto; background-color: white\" id=\"graph-output-XoQyxcXc\"></div>\n",
+       "<div style=\"overflow:auto; background-color: white\" id=\"graph-output-JQM4XVNE\"></div>\n",
        "<script type=\"module\">\n",
        "var d3;\n",
        "if (d3 == null) { d3 = await import(\"https://cdn.skypack.dev/d3@5\"); }\n",
@@ -4640,8 +8454,8 @@
        "}\n",
        "\n",
        "\n",
-       "showGraph('#graph-output-XoQyxcXc',\n",
-       "  JSON.parse('{\"nodes\": [{\"name\": \"0\", \"x\": 50, \"y\": 100, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"1\", \"x\": 50, \"y\": 150, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"2\", \"x\": 50, \"y\": 200, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"3\", \"x\": 100, \"y\": 150, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"4\", \"x\": 100, \"y\": 100, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"5\", \"x\": 150, \"y\": 200, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"6\", \"x\": 150, \"y\": 150, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"7\", \"x\": 200, \"y\": 200, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"8\", \"x\": 200, \"y\": 100, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"9\", \"x\": 250, \"y\": 150, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"10\", \"x\": 250, \"y\": 200, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"11\", \"x\": 300, \"y\": 150, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"12\", \"x\": 300, \"y\": 100, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"13\", \"x\": 350, \"y\": 100, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"14\", \"x\": 350, \"y\": 200, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"15\", \"x\": 400, \"y\": 100, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"16\", \"x\": 400, \"y\": 150, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"17\", \"x\": 400, \"y\": 200, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}], \"links\": [{\"source\": \"0\", \"target\": \"4\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"1\", \"target\": \"3\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"2\", \"target\": \"5\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"3\", \"target\": \"4\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"3\", \"target\": \"6\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"4\", \"target\": \"8\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"5\", \"target\": \"6\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"5\", \"target\": \"7\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"6\", \"target\": \"9\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"7\", \"target\": \"8\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"7\", \"target\": \"10\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"8\", \"target\": \"12\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"9\", \"target\": \"10\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"9\", \"target\": \"11\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"10\", \"target\": \"14\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"11\", \"target\": \"12\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"11\", \"target\": \"16\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"12\", \"target\": \"13\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"13\", \"target\": \"14\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"13\", \"target\": \"15\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"14\", \"target\": \"17\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}], \"pauli_web\": [{\"source\": 5, \"target\": 2, \"t\": \"Z\"}, {\"source\": 2, \"target\": 5, \"t\": \"Z\"}, {\"source\": 5, \"target\": 6, \"t\": \"Z\"}, {\"source\": 6, \"target\": 5, \"t\": \"Z\"}, {\"source\": 5, \"target\": 7, \"t\": \"Z\"}, {\"source\": 7, \"target\": 5, \"t\": \"Z\"}, {\"source\": 7, \"target\": 8, \"t\": \"Z\"}, {\"source\": 8, \"target\": 7, \"t\": \"Z\"}, {\"source\": 7, \"target\": 10, \"t\": \"Z\"}, {\"source\": 10, \"target\": 7, \"t\": \"Z\"}, {\"source\": 9, \"target\": 6, \"t\": \"Z\"}, {\"source\": 6, \"target\": 9, \"t\": \"Z\"}, {\"source\": 9, \"target\": 10, \"t\": \"Z\"}, {\"source\": 10, \"target\": 9, \"t\": \"Z\"}, {\"source\": 9, \"target\": 11, \"t\": \"Z\"}, {\"source\": 11, \"target\": 9, \"t\": \"Z\"}, {\"source\": 11, \"target\": 12, \"t\": \"Z\"}, {\"source\": 12, \"target\": 11, \"t\": \"Z\"}, {\"source\": 11, \"target\": 16, \"t\": \"Z\"}, {\"source\": 16, \"target\": 11, \"t\": \"Z\"}, {\"source\": 8, \"target\": 12, \"t\": \"Z\"}, {\"source\": 12, \"target\": 8, \"t\": \"Z\"}]}'), 450, 250, 50,\n",
+       "showGraph('#graph-output-JQM4XVNE',\n",
+       "  JSON.parse('{\"nodes\": [{\"name\": \"0\", \"x\": 50, \"y\": 100, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"1\", \"x\": 50, \"y\": 150, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"2\", \"x\": 50, \"y\": 200, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"3\", \"x\": 100, \"y\": 150, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"4\", \"x\": 100, \"y\": 100, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"5\", \"x\": 150, \"y\": 200, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"6\", \"x\": 150, \"y\": 150, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"9\", \"x\": 250, \"y\": 150, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"10\", \"x\": 250, \"y\": 200, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"13\", \"x\": 350, \"y\": 100, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"15\", \"x\": 400, \"y\": 100, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"16\", \"x\": 400, \"y\": 150, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"17\", \"x\": 400, \"y\": 200, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}], \"links\": [{\"source\": \"0\", \"target\": \"4\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"1\", \"target\": \"3\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"2\", \"target\": \"5\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"3\", \"target\": \"4\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"3\", \"target\": \"6\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"4\", \"target\": \"13\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"4\", \"target\": \"5\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"4\", \"target\": \"9\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"5\", \"target\": \"6\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"5\", \"target\": \"10\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"6\", \"target\": \"9\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"9\", \"target\": \"10\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"9\", \"target\": \"16\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"10\", \"target\": \"13\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"10\", \"target\": \"17\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"13\", \"target\": \"15\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}], \"pauli_web\": [{\"source\": 10, \"target\": 9, \"t\": \"X\"}, {\"source\": 9, \"target\": 10, \"t\": \"X\"}, {\"source\": 10, \"target\": 13, \"t\": \"X\"}, {\"source\": 13, \"target\": 10, \"t\": \"X\"}, {\"source\": 10, \"target\": 17, \"t\": \"X\"}, {\"source\": 17, \"target\": 10, \"t\": \"X\"}, {\"source\": 10, \"target\": 5, \"t\": \"X\"}, {\"source\": 5, \"target\": 10, \"t\": \"X\"}, {\"source\": 4, \"target\": 0, \"t\": \"X\"}, {\"source\": 0, \"target\": 4, \"t\": \"X\"}, {\"source\": 4, \"target\": 3, \"t\": \"X\"}, {\"source\": 3, \"target\": 4, \"t\": \"X\"}, {\"source\": 4, \"target\": 13, \"t\": \"X\"}, {\"source\": 13, \"target\": 4, \"t\": \"X\"}, {\"source\": 4, \"target\": 5, \"t\": \"X\"}, {\"source\": 5, \"target\": 4, \"t\": \"X\"}, {\"source\": 4, \"target\": 9, \"t\": \"X\"}, {\"source\": 9, \"target\": 4, \"t\": \"X\"}, {\"source\": 1, \"target\": 3, \"t\": \"X\"}, {\"source\": 3, \"target\": 1, \"t\": \"X\"}]}'), 450, 250, 50,\n",
        "  10.0, false, true, '');\n",
        "</script>"
       ],
@@ -4655,7 +8469,7 @@
     {
      "data": {
       "text/html": [
-       "<div style=\"overflow:auto; background-color: white\" id=\"graph-output-bbP90f0R\"></div>\n",
+       "<div style=\"overflow:auto; background-color: white\" id=\"graph-output-T4IbsS5N\"></div>\n",
        "<script type=\"module\">\n",
        "var d3;\n",
        "if (d3 == null) { d3 = await import(\"https://cdn.skypack.dev/d3@5\"); }\n",
@@ -5018,8 +8832,8 @@
        "}\n",
        "\n",
        "\n",
-       "showGraph('#graph-output-bbP90f0R',\n",
-       "  JSON.parse('{\"nodes\": [{\"name\": \"0\", \"x\": 50, \"y\": 100, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"1\", \"x\": 50, \"y\": 150, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"2\", \"x\": 50, \"y\": 200, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"3\", \"x\": 100, \"y\": 150, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"4\", \"x\": 100, \"y\": 100, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"5\", \"x\": 150, \"y\": 200, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"6\", \"x\": 150, \"y\": 150, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"7\", \"x\": 200, \"y\": 200, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"8\", \"x\": 200, \"y\": 100, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"9\", \"x\": 250, 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\"num_parallel\": 1}, {\"source\": \"1\", \"target\": \"3\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"2\", \"target\": \"5\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"3\", \"target\": \"4\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"3\", \"target\": \"6\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"4\", \"target\": \"8\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"5\", \"target\": \"6\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"5\", \"target\": \"7\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"6\", \"target\": \"9\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"7\", \"target\": \"8\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"7\", \"target\": \"10\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"8\", \"target\": \"12\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"9\", \"target\": \"10\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"9\", \"target\": \"11\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"10\", \"target\": \"14\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"11\", \"target\": \"12\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"11\", \"target\": \"16\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"12\", \"target\": \"13\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"13\", \"target\": \"14\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"13\", \"target\": \"15\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"14\", \"target\": \"17\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}], \"pauli_web\": [{\"source\": 4, \"target\": 0, \"t\": \"X\"}, {\"source\": 0, \"target\": 4, \"t\": \"X\"}, {\"source\": 4, \"target\": 3, \"t\": \"X\"}, {\"source\": 3, \"target\": 4, \"t\": \"X\"}, {\"source\": 4, \"target\": 8, \"t\": \"X\"}, {\"source\": 8, \"target\": 4, \"t\": \"X\"}, {\"source\": 8, \"target\": 7, \"t\": \"X\"}, {\"source\": 7, \"target\": 8, \"t\": \"X\"}, {\"source\": 8, \"target\": 12, \"t\": \"X\"}, {\"source\": 10, \"target\": 7, \"t\": \"X\"}, {\"source\": 7, \"target\": 10, \"t\": \"X\"}, {\"source\": 10, \"target\": 9, \"t\": \"X\"}, {\"source\": 9, \"target\": 10, \"t\": \"X\"}, {\"source\": 10, \"target\": 14, \"t\": \"X\"}, {\"source\": 12, \"target\": 8, \"t\": \"X\"}, {\"source\": 12, \"target\": 11, \"t\": \"X\"}, {\"source\": 11, \"target\": 12, \"t\": \"X\"}, {\"source\": 12, \"target\": 13, \"t\": \"X\"}, {\"source\": 13, \"target\": 12, \"t\": \"X\"}, {\"source\": 14, \"target\": 10, \"t\": \"X\"}, {\"source\": 14, \"target\": 13, \"t\": \"X\"}, {\"source\": 13, \"target\": 14, \"t\": \"X\"}, {\"source\": 14, \"target\": 17, \"t\": \"X\"}, {\"source\": 17, \"target\": 14, \"t\": \"X\"}, {\"source\": 9, \"target\": 11, \"t\": \"X\"}, {\"source\": 11, \"target\": 9, \"t\": \"X\"}, {\"source\": 3, \"target\": 1, \"t\": \"X\"}, {\"source\": 1, \"target\": 3, \"t\": \"X\"}]}'), 450, 250, 50,\n",
+       "showGraph('#graph-output-T4IbsS5N',\n",
+       "  JSON.parse('{\"nodes\": [{\"name\": \"0\", \"x\": 50, \"y\": 100, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"1\", \"x\": 50, \"y\": 150, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"2\", \"x\": 50, \"y\": 200, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"3\", \"x\": 100, \"y\": 150, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"4\", \"x\": 100, \"y\": 100, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"5\", \"x\": 150, \"y\": 200, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"6\", \"x\": 150, \"y\": 150, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"9\", \"x\": 250, \"y\": 150, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"10\", \"x\": 250, \"y\": 200, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"13\", \"x\": 350, \"y\": 100, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"15\", \"x\": 400, \"y\": 100, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"16\", \"x\": 400, \"y\": 150, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"17\", \"x\": 400, \"y\": 200, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}], \"links\": [{\"source\": \"0\", \"target\": \"4\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"1\", \"target\": \"3\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"2\", \"target\": \"5\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"3\", \"target\": \"4\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"3\", \"target\": \"6\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"4\", \"target\": \"13\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"4\", \"target\": \"5\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"4\", \"target\": \"9\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"5\", \"target\": \"6\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"5\", \"target\": \"10\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"6\", \"target\": \"9\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"9\", \"target\": \"10\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"9\", \"target\": \"16\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"10\", \"target\": \"13\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"10\", \"target\": \"17\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"13\", \"target\": \"15\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}], \"pauli_web\": [{\"source\": 3, \"target\": 1, \"t\": \"Z\"}, {\"source\": 1, \"target\": 3, \"t\": \"Z\"}, {\"source\": 3, \"target\": 4, \"t\": \"Z\"}, {\"source\": 4, \"target\": 3, \"t\": \"Z\"}, {\"source\": 3, \"target\": 6, \"t\": \"Z\"}, {\"source\": 6, \"target\": 3, \"t\": \"Z\"}, {\"source\": 5, \"target\": 2, \"t\": \"Z\"}, {\"source\": 2, \"target\": 5, \"t\": \"Z\"}, {\"source\": 5, \"target\": 6, \"t\": \"Z\"}, {\"source\": 6, \"target\": 5, \"t\": \"Z\"}, {\"source\": 5, \"target\": 10, \"t\": \"Z\"}, {\"source\": 10, \"target\": 5, \"t\": \"Z\"}, {\"source\": 5, \"target\": 4, \"t\": \"Z\"}, {\"source\": 4, \"target\": 5, \"t\": \"Z\"}, {\"source\": 17, \"target\": 10, \"t\": \"Z\"}, {\"source\": 10, \"target\": 17, \"t\": \"Z\"}]}'), 450, 250, 50,\n",
        "  10.0, false, true, '');\n",
        "</script>"
       ],
@@ -5029,11 +8843,36 @@
      },
      "metadata": {},
      "output_type": "display_data"
-    },
+    }
+   ],
+   "source": [
+    "# Since this describes the same unitary, the 6 output webs should all have the same support on\n",
+    "# the inputs as before.\n",
+    "order, zwebs, xwebs = compute_pauli_webs(g)\n",
+    "zx.draw(g, labels=True, pauli_web=xwebs[15])\n",
+    "zx.draw(g, labels=True, pauli_web=zwebs[15])\n",
+    "zx.draw(g, labels=True, pauli_web=xwebs[16])\n",
+    "zx.draw(g, labels=True, pauli_web=zwebs[16])\n",
+    "zx.draw(g, labels=True, pauli_web=xwebs[17])\n",
+    "zx.draw(g, labels=True, pauli_web=zwebs[17])"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "# Clifford examples"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 9,
+   "metadata": {},
+   "outputs": [
     {
      "data": {
       "text/html": [
-       "<div style=\"overflow:auto; background-color: white\" id=\"graph-output-bGysltxu\"></div>\n",
+       "<div style=\"overflow:auto; background-color: white\" id=\"graph-output-cS5x8s30\"></div>\n",
        "<script type=\"module\">\n",
        "var d3;\n",
        "if (d3 == null) { d3 = await import(\"https://cdn.skypack.dev/d3@5\"); }\n",
@@ -5396,9 +9235,9 @@
        "}\n",
        "\n",
        "\n",
-       "showGraph('#graph-output-bGysltxu',\n",
-       "  JSON.parse('{\"nodes\": [{\"name\": \"0\", \"x\": 50, \"y\": 100, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"1\", \"x\": 50, \"y\": 150, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"2\", \"x\": 50, \"y\": 200, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"3\", \"x\": 100, \"y\": 150, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"4\", \"x\": 100, \"y\": 100, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"5\", \"x\": 150, \"y\": 200, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"6\", \"x\": 150, \"y\": 150, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"7\", \"x\": 200, \"y\": 200, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"8\", \"x\": 200, \"y\": 100, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"9\", \"x\": 250, \"y\": 150, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"10\", \"x\": 250, \"y\": 200, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"11\", \"x\": 300, \"y\": 150, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"12\", \"x\": 300, \"y\": 100, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"13\", \"x\": 350, \"y\": 100, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"14\", \"x\": 350, \"y\": 200, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"15\", \"x\": 400, \"y\": 100, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"16\", \"x\": 400, \"y\": 150, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"17\", \"x\": 400, \"y\": 200, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}], \"links\": [{\"source\": \"0\", \"target\": \"4\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"1\", \"target\": \"3\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"2\", \"target\": \"5\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"3\", \"target\": \"4\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"3\", \"target\": \"6\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"4\", \"target\": \"8\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"5\", \"target\": \"6\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"5\", \"target\": \"7\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"6\", \"target\": \"9\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"7\", \"target\": \"8\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"7\", \"target\": \"10\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"8\", \"target\": \"12\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"9\", \"target\": \"10\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"9\", \"target\": \"11\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"10\", \"target\": \"14\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"11\", \"target\": \"12\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"11\", \"target\": \"16\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"12\", \"target\": \"13\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"13\", \"target\": \"14\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"13\", \"target\": \"15\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"14\", \"target\": \"17\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}], \"pauli_web\": [{\"source\": 3, \"target\": 1, \"t\": \"Z\"}, {\"source\": 1, \"target\": 3, \"t\": \"Z\"}, {\"source\": 3, \"target\": 4, \"t\": \"Z\"}, {\"source\": 4, \"target\": 3, \"t\": \"Z\"}, {\"source\": 3, \"target\": 6, \"t\": \"Z\"}, {\"source\": 6, \"target\": 3, \"t\": \"Z\"}, {\"source\": 5, \"target\": 2, \"t\": \"Z\"}, {\"source\": 2, \"target\": 5, \"t\": \"Z\"}, {\"source\": 5, \"target\": 6, \"t\": \"Z\"}, {\"source\": 6, \"target\": 5, \"t\": \"Z\"}, {\"source\": 5, \"target\": 7, \"t\": \"Z\"}, {\"source\": 7, \"target\": 5, \"t\": \"Z\"}, {\"source\": 7, \"target\": 8, \"t\": \"Z\"}, {\"source\": 8, \"target\": 7, \"t\": \"Z\"}, {\"source\": 7, \"target\": 10, \"t\": \"Z\"}, {\"source\": 10, \"target\": 7, \"t\": \"Z\"}, {\"source\": 4, \"target\": 8, \"t\": \"Z\"}, {\"source\": 8, \"target\": 4, \"t\": \"Z\"}, {\"source\": 10, \"target\": 14, \"t\": \"Z\"}, {\"source\": 14, \"target\": 10, \"t\": \"Z\"}, {\"source\": 17, \"target\": 14, \"t\": \"Z\"}, {\"source\": 14, \"target\": 17, \"t\": \"Z\"}]}'), 450, 250, 50,\n",
-       "  10.0, false, true, '');\n",
+       "showGraph('#graph-output-cS5x8s30',\n",
+       "  JSON.parse('{\"nodes\": [{\"name\": \"0\", \"x\": 50, \"y\": 100, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"1\", \"x\": 100, \"y\": 100, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"2\", \"x\": 150, \"y\": 100, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}], \"links\": [{\"source\": \"0\", \"target\": \"1\", \"t\": 2, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"1\", \"target\": \"2\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}], \"pauli_web\": []}'), 200, 150, 50,\n",
+       "  10.0, false, false, '');\n",
        "</script>"
       ],
       "text/plain": [
@@ -5407,27 +9246,11 @@
      },
      "metadata": {},
      "output_type": "display_data"
-    }
-   ],
-   "source": [
-    "order, zwebs, xwebs = compute_pauli_webs(g)\n",
-    "zx.draw(g, labels=True, pauli_web=xwebs[15])\n",
-    "zx.draw(g, labels=True, pauli_web=zwebs[15])\n",
-    "zx.draw(g, labels=True, pauli_web=xwebs[16])\n",
-    "zx.draw(g, labels=True, pauli_web=zwebs[16])\n",
-    "zx.draw(g, labels=True, pauli_web=xwebs[17])\n",
-    "zx.draw(g, labels=True, pauli_web=zwebs[17])"
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": 7,
-   "metadata": {},
-   "outputs": [
+    },
     {
      "data": {
       "text/html": [
-       "<div style=\"overflow:auto; background-color: white\" id=\"graph-output-eZ1ry73c\"></div>\n",
+       "<div style=\"overflow:auto; background-color: white\" id=\"graph-output-l5r0Kc4I\"></div>\n",
        "<script type=\"module\">\n",
        "var d3;\n",
        "if (d3 == null) { d3 = await import(\"https://cdn.skypack.dev/d3@5\"); }\n",
@@ -5790,8 +9613,8 @@
        "}\n",
        "\n",
        "\n",
-       "showGraph('#graph-output-eZ1ry73c',\n",
-       "  JSON.parse('{\"nodes\": [{\"name\": \"0\", \"x\": 50, \"y\": 100, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"1\", \"x\": 50, \"y\": 150, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"2\", \"x\": 50, \"y\": 200, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"3\", \"x\": 100, \"y\": 150, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"4\", \"x\": 100, \"y\": 100, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"5\", \"x\": 150, \"y\": 200, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"6\", \"x\": 150, \"y\": 150, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"9\", \"x\": 250, \"y\": 150, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"10\", \"x\": 250, \"y\": 200, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"13\", \"x\": 350, \"y\": 100, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"15\", \"x\": 400, \"y\": 100, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"16\", \"x\": 400, \"y\": 150, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"17\", \"x\": 400, \"y\": 200, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}], \"links\": [{\"source\": \"0\", \"target\": \"4\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"1\", \"target\": \"3\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"2\", \"target\": \"5\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"3\", \"target\": \"4\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"3\", \"target\": \"6\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"4\", \"target\": \"13\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"4\", \"target\": \"5\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"4\", \"target\": \"9\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"5\", \"target\": \"6\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"5\", \"target\": \"10\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"6\", \"target\": \"9\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"9\", \"target\": \"10\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"9\", \"target\": \"16\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"10\", \"target\": \"13\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"10\", \"target\": \"17\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"13\", \"target\": \"15\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}], \"pauli_web\": []}'), 450, 250, 50,\n",
+       "showGraph('#graph-output-l5r0Kc4I',\n",
+       "  JSON.parse('{\"nodes\": [{\"name\": \"0\", \"x\": 50, \"y\": 100, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"2\", \"x\": 150, \"y\": 100, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}], \"links\": [{\"source\": \"0\", \"target\": \"2\", \"t\": 2, \"index\": 0, \"num_parallel\": 1}], \"pauli_web\": []}'), 200, 150, 50,\n",
        "  10.0, false, true, '');\n",
        "</script>"
       ],
@@ -5804,21 +9627,32 @@
     }
    ],
    "source": [
-    "# Alternatively, we can fuse all the spiders of the same color together to get a\n",
-    "# more interesting diagram.\n",
-    "zx.spider_simp(g)\n",
-    "zx.draw(g, labels=True)"
+    "# Next, we'll look at a single H gate\n",
+    "c = zx.qasm(\"\"\"\n",
+    "qreg q[1];\n",
+    "h q[0];\n",
+    "\"\"\")\n",
+    "g = c.to_graph()\n",
+    "\n",
+    "# PyZX renders this as a single H-edge connected to an identity spider\n",
+    "zx.draw(g)\n",
+    "\n",
+    "# we don't really need this id-spider, so we can remove it with id_simp\n",
+    "zx.id_simp(g)\n",
+    "zx.draw(g, labels=True)\n",
+    "\n",
+    "order, zwebs, xwebs = compute_pauli_webs(g)"
    ]
   },
   {
    "cell_type": "code",
-   "execution_count": 8,
+   "execution_count": 10,
    "metadata": {},
    "outputs": [
     {
      "data": {
       "text/html": [
-       "<div style=\"overflow:auto; background-color: white\" id=\"graph-output-99rDmDYX\"></div>\n",
+       "<div style=\"overflow:auto; background-color: white\" id=\"graph-output-lYGPpKdv\"></div>\n",
        "<script type=\"module\">\n",
        "var d3;\n",
        "if (d3 == null) { d3 = await import(\"https://cdn.skypack.dev/d3@5\"); }\n",
@@ -6181,8 +10015,8 @@
        "}\n",
        "\n",
        "\n",
-       "showGraph('#graph-output-99rDmDYX',\n",
-       "  JSON.parse('{\"nodes\": [{\"name\": \"0\", \"x\": 50, \"y\": 100, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"1\", \"x\": 50, \"y\": 150, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"2\", \"x\": 50, \"y\": 200, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"3\", \"x\": 100, \"y\": 150, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"4\", \"x\": 100, \"y\": 100, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"5\", \"x\": 150, \"y\": 200, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"6\", \"x\": 150, \"y\": 150, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"9\", \"x\": 250, \"y\": 150, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"10\", \"x\": 250, \"y\": 200, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"13\", \"x\": 350, \"y\": 100, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"15\", \"x\": 400, \"y\": 100, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"16\", \"x\": 400, \"y\": 150, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"17\", \"x\": 400, \"y\": 200, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}], \"links\": [{\"source\": \"0\", \"target\": \"4\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"1\", \"target\": \"3\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"2\", \"target\": \"5\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"3\", \"target\": \"4\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"3\", \"target\": \"6\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"4\", \"target\": \"13\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"4\", \"target\": \"5\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"4\", \"target\": \"9\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"5\", \"target\": \"6\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"5\", \"target\": \"10\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"6\", \"target\": \"9\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"9\", \"target\": \"10\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"9\", \"target\": \"16\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"10\", \"target\": \"13\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"10\", \"target\": \"17\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"13\", \"target\": \"15\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}], \"pauli_web\": [{\"source\": 15, \"target\": 13, \"t\": \"X\"}, {\"source\": 13, \"target\": 15, \"t\": \"X\"}, {\"source\": 4, \"target\": 0, \"t\": \"X\"}, {\"source\": 0, \"target\": 4, \"t\": \"X\"}, {\"source\": 4, \"target\": 3, \"t\": \"X\"}, {\"source\": 3, \"target\": 4, \"t\": \"X\"}, {\"source\": 4, \"target\": 13, \"t\": \"X\"}, {\"source\": 13, \"target\": 4, \"t\": \"X\"}, {\"source\": 4, \"target\": 5, \"t\": \"X\"}, {\"source\": 5, \"target\": 4, \"t\": \"X\"}, {\"source\": 4, \"target\": 9, \"t\": \"X\"}, {\"source\": 9, \"target\": 4, \"t\": \"X\"}, {\"source\": 6, \"target\": 3, \"t\": \"X\"}, {\"source\": 3, \"target\": 6, \"t\": \"X\"}, {\"source\": 6, \"target\": 5, \"t\": \"X\"}, {\"source\": 5, \"target\": 6, \"t\": \"X\"}, {\"source\": 6, \"target\": 9, \"t\": \"X\"}, {\"source\": 9, \"target\": 6, \"t\": \"X\"}]}'), 450, 250, 50,\n",
+       "showGraph('#graph-output-lYGPpKdv',\n",
+       "  JSON.parse('{\"nodes\": [{\"name\": \"0\", \"x\": 50, \"y\": 100, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"2\", \"x\": 150, \"y\": 100, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}], \"links\": [{\"source\": \"0\", \"target\": \"2\", \"t\": 2, \"index\": 0, \"num_parallel\": 1}], \"pauli_web\": [{\"source\": 0, \"target\": 2, \"t\": \"X\"}, {\"source\": 2, \"target\": 0, \"t\": \"Z\"}]}'), 200, 150, 50,\n",
        "  10.0, false, true, '');\n",
        "</script>"
       ],
@@ -6196,7 +10030,7 @@
     {
      "data": {
       "text/html": [
-       "<div style=\"overflow:auto; background-color: white\" id=\"graph-output-9vvXpbiV\"></div>\n",
+       "<div style=\"overflow:auto; background-color: white\" id=\"graph-output-dO4G2dzc\"></div>\n",
        "<script type=\"module\">\n",
        "var d3;\n",
        "if (d3 == null) { d3 = await import(\"https://cdn.skypack.dev/d3@5\"); }\n",
@@ -6559,8 +10393,8 @@
        "}\n",
        "\n",
        "\n",
-       "showGraph('#graph-output-9vvXpbiV',\n",
-       "  JSON.parse('{\"nodes\": [{\"name\": \"0\", \"x\": 50, \"y\": 100, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"1\", \"x\": 50, \"y\": 150, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"2\", \"x\": 50, \"y\": 200, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"3\", \"x\": 100, \"y\": 150, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"4\", \"x\": 100, \"y\": 100, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"5\", \"x\": 150, \"y\": 200, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"6\", \"x\": 150, \"y\": 150, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"9\", \"x\": 250, \"y\": 150, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"10\", \"x\": 250, \"y\": 200, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"13\", \"x\": 350, \"y\": 100, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"15\", \"x\": 400, \"y\": 100, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"16\", \"x\": 400, \"y\": 150, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"17\", \"x\": 400, \"y\": 200, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}], \"links\": [{\"source\": \"0\", \"target\": \"4\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"1\", \"target\": \"3\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"2\", \"target\": \"5\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"3\", \"target\": \"4\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"3\", \"target\": \"6\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"4\", \"target\": \"13\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"4\", \"target\": \"5\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"4\", \"target\": \"9\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"5\", \"target\": \"6\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"5\", \"target\": \"10\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"6\", \"target\": \"9\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"9\", \"target\": \"10\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"9\", \"target\": \"16\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"10\", \"target\": \"13\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"10\", \"target\": \"17\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"13\", \"target\": \"15\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}], \"pauli_web\": [{\"source\": 13, \"target\": 15, \"t\": \"Z\"}, {\"source\": 15, \"target\": 13, \"t\": \"Z\"}, {\"source\": 13, \"target\": 10, \"t\": \"Z\"}, {\"source\": 10, \"target\": 13, \"t\": \"Z\"}, {\"source\": 13, \"target\": 4, \"t\": \"Z\"}, {\"source\": 4, \"target\": 13, \"t\": \"Z\"}, {\"source\": 3, \"target\": 1, \"t\": \"Z\"}, {\"source\": 1, \"target\": 3, \"t\": \"Z\"}, {\"source\": 3, \"target\": 4, \"t\": \"Z\"}, {\"source\": 4, \"target\": 3, \"t\": \"Z\"}, {\"source\": 3, \"target\": 6, \"t\": \"Z\"}, {\"source\": 6, \"target\": 3, \"t\": \"Z\"}, {\"source\": 5, \"target\": 2, \"t\": \"Z\"}, {\"source\": 2, \"target\": 5, \"t\": \"Z\"}, {\"source\": 5, \"target\": 6, \"t\": \"Z\"}, {\"source\": 6, \"target\": 5, \"t\": \"Z\"}, {\"source\": 5, \"target\": 10, \"t\": \"Z\"}, {\"source\": 10, \"target\": 5, \"t\": \"Z\"}, {\"source\": 5, \"target\": 4, \"t\": \"Z\"}, {\"source\": 4, \"target\": 5, \"t\": \"Z\"}, {\"source\": 4, \"target\": 0, \"t\": \"Z\"}, {\"source\": 0, \"target\": 4, \"t\": \"Z\"}]}'), 450, 250, 50,\n",
+       "showGraph('#graph-output-dO4G2dzc',\n",
+       "  JSON.parse('{\"nodes\": [{\"name\": \"0\", \"x\": 50, \"y\": 100, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"2\", \"x\": 150, \"y\": 100, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}], \"links\": [{\"source\": \"0\", \"target\": \"2\", \"t\": 2, \"index\": 0, \"num_parallel\": 1}], \"pauli_web\": [{\"source\": 2, \"target\": 3, \"t\": \"X\"}, {\"source\": 3, \"target\": 2, \"t\": \"Z\"}]}'), 200, 150, 50,\n",
        "  10.0, false, true, '');\n",
        "</script>"
       ],
@@ -6571,10 +10405,35 @@
      "metadata": {},
      "output_type": "display_data"
     },
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "{(0, 2): 'X', (2, 0): 'Z'}\n"
+     ]
+    }
+   ],
+   "source": [
+    "# The single output has two Pauli webs\n",
+    "zx.draw(g, labels=True, pauli_web=zwebs[2])\n",
+    "zx.draw(g, labels=True, pauli_web=xwebs[2])\n",
+    "\n",
+    "# Note that on a hadamard edge, the color changes in the middle of the edge. To handle this, Pauli webs actually label every\n",
+    "# \"half-edge\" with a Pauli. (v,w) means the half of the edge touching v, whereas (w,v) means the half of the edge touching w.\n",
+    "\n",
+    "#  See for example:\n",
+    "print(zwebs[2].half_edges())"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 11,
+   "metadata": {},
+   "outputs": [
     {
      "data": {
       "text/html": [
-       "<div style=\"overflow:auto; background-color: white\" id=\"graph-output-IRwSW6c3\"></div>\n",
+       "<div style=\"overflow:auto; background-color: white\" id=\"graph-output-2B25BzzK\"></div>\n",
        "<script type=\"module\">\n",
        "var d3;\n",
        "if (d3 == null) { d3 = await import(\"https://cdn.skypack.dev/d3@5\"); }\n",
@@ -6937,8 +10796,8 @@
        "}\n",
        "\n",
        "\n",
-       "showGraph('#graph-output-IRwSW6c3',\n",
-       "  JSON.parse('{\"nodes\": [{\"name\": \"0\", \"x\": 50, \"y\": 100, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"1\", \"x\": 50, \"y\": 150, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"2\", \"x\": 50, \"y\": 200, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"3\", \"x\": 100, \"y\": 150, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"4\", \"x\": 100, \"y\": 100, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"5\", \"x\": 150, \"y\": 200, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"6\", \"x\": 150, \"y\": 150, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"9\", \"x\": 250, \"y\": 150, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"10\", \"x\": 250, \"y\": 200, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"13\", \"x\": 350, \"y\": 100, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"15\", \"x\": 400, \"y\": 100, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"16\", \"x\": 400, \"y\": 150, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"17\", \"x\": 400, \"y\": 200, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}], \"links\": [{\"source\": \"0\", \"target\": \"4\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"1\", \"target\": \"3\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"2\", \"target\": \"5\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"3\", \"target\": \"4\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"3\", \"target\": \"6\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"4\", \"target\": \"13\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"4\", \"target\": \"5\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"4\", \"target\": \"9\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"5\", \"target\": \"6\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"5\", \"target\": \"10\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"6\", \"target\": \"9\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"9\", \"target\": \"10\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"9\", \"target\": \"16\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"10\", \"target\": \"13\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"10\", \"target\": \"17\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"13\", \"target\": \"15\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}], \"pauli_web\": [{\"source\": 16, \"target\": 9, \"t\": \"X\"}, {\"source\": 9, \"target\": 16, \"t\": \"X\"}, {\"source\": 6, \"target\": 3, \"t\": \"X\"}, {\"source\": 3, \"target\": 6, \"t\": \"X\"}, {\"source\": 6, \"target\": 5, \"t\": \"X\"}, {\"source\": 5, \"target\": 6, \"t\": \"X\"}, {\"source\": 6, \"target\": 9, \"t\": \"X\"}, {\"source\": 9, \"target\": 6, \"t\": \"X\"}, {\"source\": 3, \"target\": 1, \"t\": \"X\"}, {\"source\": 1, \"target\": 3, \"t\": \"X\"}, {\"source\": 5, \"target\": 2, \"t\": \"X\"}, {\"source\": 2, \"target\": 5, \"t\": \"X\"}]}'), 450, 250, 50,\n",
+       "showGraph('#graph-output-2B25BzzK',\n",
+       "  JSON.parse('{\"nodes\": [{\"name\": \"0\", \"x\": 50, \"y\": 100, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"1\", \"x\": 50, \"y\": 150, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"2\", \"x\": 50, \"y\": 200, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"3\", \"x\": 100, \"y\": 150, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"4\", \"x\": 100, \"y\": 100, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"5\", \"x\": 150, \"y\": 150, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"6\", \"x\": 100, \"y\": 200, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"7\", \"x\": 200, \"y\": 200, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"8\", \"x\": 200, \"y\": 150, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"9\", \"x\": 250, \"y\": 200, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"10\", \"x\": 250, \"y\": 100, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"11\", \"x\": 300, \"y\": 100, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"12\", \"x\": 300, \"y\": 150, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"13\", \"x\": 300, \"y\": 200, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"14\", \"x\": 350, \"y\": 150, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"15\", \"x\": 400, \"y\": 150, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"16\", \"x\": 450, \"y\": 150, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"17\", \"x\": 450, \"y\": 100, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"18\", \"x\": 500, \"y\": 100, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"19\", \"x\": 500, \"y\": 200, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"20\", \"x\": 550, \"y\": 100, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"21\", \"x\": 550, \"y\": 150, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"22\", \"x\": 550, \"y\": 200, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}], \"links\": [{\"source\": \"0\", \"target\": \"4\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"1\", \"target\": \"3\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"2\", \"target\": \"6\", \"t\": 2, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"3\", \"target\": \"4\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"3\", \"target\": \"5\", \"t\": 2, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"4\", \"target\": \"10\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"5\", \"target\": \"8\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"6\", \"target\": \"7\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"7\", \"target\": \"8\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"7\", \"target\": \"9\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"8\", \"target\": \"12\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"9\", \"target\": \"10\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"9\", \"target\": \"13\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"10\", \"target\": \"11\", \"t\": 2, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"11\", \"target\": \"17\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"12\", \"target\": \"13\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"12\", \"target\": \"14\", \"t\": 2, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"13\", \"target\": \"19\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"14\", \"target\": \"15\", \"t\": 2, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"15\", \"target\": \"16\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"16\", \"target\": \"17\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"16\", \"target\": \"21\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"17\", \"target\": \"18\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"18\", \"target\": \"19\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"18\", \"target\": \"20\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"19\", \"target\": \"22\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}], \"pauli_web\": []}'), 600, 250, 50,\n",
        "  10.0, false, true, '');\n",
        "</script>"
       ],
@@ -6948,11 +10807,39 @@
      },
      "metadata": {},
      "output_type": "display_data"
-    },
+    }
+   ],
+   "source": [
+    "# Here's an example mixing CNOT and H gates\n",
+    "\n",
+    "c = zx.qasm(\"\"\"\n",
+    "qreg q[3];\n",
+    "cx q[0], q[1];\n",
+    "h q[1];\n",
+    "h q[2];\n",
+    "cx q[1], q[2];\n",
+    "cx q[0], q[2];\n",
+    "h q[0];\n",
+    "cx q[2], q[1];\n",
+    "h q[1];\n",
+    "h q[1];\n",
+    "cx q[0], q[1];\n",
+    "cx q[2], q[0];\n",
+    "\"\"\")\n",
+    "g = c.to_graph()\n",
+    "\n",
+    "zx.draw(g, labels=True)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 12,
+   "metadata": {},
+   "outputs": [
     {
      "data": {
       "text/html": [
-       "<div style=\"overflow:auto; background-color: white\" id=\"graph-output-8WzIx3Oi\"></div>\n",
+       "<div style=\"overflow:auto; background-color: white\" id=\"graph-output-z76X6TiP\"></div>\n",
        "<script type=\"module\">\n",
        "var d3;\n",
        "if (d3 == null) { d3 = await import(\"https://cdn.skypack.dev/d3@5\"); }\n",
@@ -7315,8 +11202,8 @@
        "}\n",
        "\n",
        "\n",
-       "showGraph('#graph-output-8WzIx3Oi',\n",
-       "  JSON.parse('{\"nodes\": [{\"name\": \"0\", \"x\": 50, \"y\": 100, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"1\", \"x\": 50, \"y\": 150, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"2\", \"x\": 50, \"y\": 200, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"3\", \"x\": 100, \"y\": 150, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"4\", \"x\": 100, \"y\": 100, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"5\", \"x\": 150, \"y\": 200, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"6\", \"x\": 150, \"y\": 150, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"9\", \"x\": 250, \"y\": 150, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"10\", \"x\": 250, \"y\": 200, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"13\", \"x\": 350, \"y\": 100, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"15\", \"x\": 400, \"y\": 100, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"16\", \"x\": 400, \"y\": 150, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"17\", \"x\": 400, \"y\": 200, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}], \"links\": [{\"source\": \"0\", \"target\": \"4\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"1\", \"target\": \"3\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"2\", \"target\": \"5\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"3\", \"target\": \"4\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"3\", \"target\": \"6\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"4\", \"target\": \"13\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"4\", \"target\": \"5\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"4\", \"target\": \"9\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"5\", \"target\": \"6\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"5\", \"target\": \"10\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"6\", \"target\": \"9\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"9\", \"target\": \"10\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"9\", \"target\": \"16\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"10\", \"target\": \"13\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"10\", \"target\": \"17\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"13\", \"target\": \"15\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}], \"pauli_web\": [{\"source\": 9, \"target\": 6, \"t\": \"Z\"}, {\"source\": 6, \"target\": 9, \"t\": \"Z\"}, {\"source\": 9, \"target\": 10, \"t\": \"Z\"}, {\"source\": 10, \"target\": 9, \"t\": \"Z\"}, {\"source\": 9, \"target\": 16, \"t\": \"Z\"}, {\"source\": 16, \"target\": 9, \"t\": \"Z\"}, {\"source\": 9, \"target\": 4, \"t\": \"Z\"}, {\"source\": 4, \"target\": 9, \"t\": \"Z\"}, {\"source\": 5, \"target\": 2, \"t\": \"Z\"}, {\"source\": 2, \"target\": 5, \"t\": \"Z\"}, {\"source\": 5, \"target\": 6, \"t\": \"Z\"}, {\"source\": 6, \"target\": 5, \"t\": \"Z\"}, {\"source\": 5, \"target\": 10, \"t\": \"Z\"}, {\"source\": 10, \"target\": 5, \"t\": \"Z\"}, {\"source\": 5, \"target\": 4, \"t\": \"Z\"}, {\"source\": 4, \"target\": 5, \"t\": \"Z\"}]}'), 450, 250, 50,\n",
+       "showGraph('#graph-output-z76X6TiP',\n",
+       "  JSON.parse('{\"nodes\": [{\"name\": \"0\", \"x\": 50, \"y\": 100, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"1\", \"x\": 50, \"y\": 150, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"2\", \"x\": 50, \"y\": 200, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"3\", \"x\": 100, \"y\": 150, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"4\", \"x\": 100, \"y\": 100, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"5\", \"x\": 150, \"y\": 150, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"6\", \"x\": 100, \"y\": 200, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"7\", \"x\": 200, \"y\": 200, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"8\", \"x\": 200, \"y\": 150, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"9\", \"x\": 250, \"y\": 200, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"10\", \"x\": 250, \"y\": 100, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"11\", \"x\": 300, \"y\": 100, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"12\", \"x\": 300, \"y\": 150, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"13\", \"x\": 300, \"y\": 200, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"14\", \"x\": 350, \"y\": 150, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"15\", \"x\": 400, \"y\": 150, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"16\", \"x\": 450, \"y\": 150, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"17\", \"x\": 450, \"y\": 100, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"18\", \"x\": 500, \"y\": 100, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"19\", \"x\": 500, \"y\": 200, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"20\", \"x\": 550, \"y\": 100, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"21\", \"x\": 550, \"y\": 150, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"22\", \"x\": 550, \"y\": 200, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}], \"links\": [{\"source\": \"0\", \"target\": \"4\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"1\", \"target\": \"3\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"2\", \"target\": \"6\", \"t\": 2, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"3\", \"target\": \"4\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"3\", \"target\": \"5\", \"t\": 2, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"4\", \"target\": \"10\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"5\", \"target\": 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{\"source\": \"14\", \"target\": \"15\", \"t\": 2, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"15\", \"target\": \"16\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"16\", \"target\": \"17\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"16\", \"target\": \"21\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"17\", \"target\": \"18\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"18\", \"target\": \"19\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"18\", \"target\": \"20\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"19\", \"target\": \"22\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}], \"pauli_web\": [{\"source\": 3, \"target\": 1, \"t\": \"Z\"}, {\"source\": 1, \"target\": 3, \"t\": \"Z\"}, {\"source\": 3, \"target\": 4, \"t\": \"Z\"}, {\"source\": 4, \"target\": 3, \"t\": \"Z\"}, {\"source\": 3, \"target\": 5, \"t\": \"Z\"}, {\"source\": 5, \"target\": 3, \"t\": \"X\"}, {\"source\": 5, \"target\": 8, \"t\": \"X\"}, {\"source\": 6, \"target\": 2, \"t\": \"X\"}, {\"source\": 2, \"target\": 6, \"t\": \"Z\"}, {\"source\": 6, \"target\": 7, \"t\": \"X\"}, {\"source\": 7, \"target\": 6, \"t\": \"X\"}, {\"source\": 20, \"target\": 18, \"t\": \"X\"}, {\"source\": 18, \"target\": 20, \"t\": \"X\"}, {\"source\": 8, \"target\": 5, \"t\": \"X\"}, {\"source\": 8, \"target\": 7, \"t\": \"X\"}, {\"source\": 7, \"target\": 8, \"t\": \"X\"}, {\"source\": 8, \"target\": 12, \"t\": \"X\"}, {\"source\": 12, \"target\": 8, \"t\": \"X\"}, {\"source\": 11, \"target\": 10, \"t\": \"X\"}, {\"source\": 10, \"target\": 11, \"t\": \"Z\"}, {\"source\": 11, \"target\": 17, \"t\": \"X\"}, {\"source\": 15, \"target\": 14, \"t\": \"X\"}, {\"source\": 14, \"target\": 15, \"t\": \"Z\"}, {\"source\": 15, \"target\": 16, \"t\": \"X\"}, {\"source\": 16, \"target\": 15, \"t\": \"X\"}, {\"source\": 17, \"target\": 11, \"t\": \"X\"}, {\"source\": 17, \"target\": 16, \"t\": \"X\"}, {\"source\": 16, \"target\": 17, \"t\": \"X\"}, {\"source\": 17, \"target\": 18, \"t\": \"X\"}, {\"source\": 18, \"target\": 17, \"t\": \"X\"}, {\"source\": 4, \"target\": 10, \"t\": \"Z\"}, {\"source\": 10, \"target\": 4, \"t\": \"Z\"}, {\"source\": 12, \"target\": 14, \"t\": \"X\"}, {\"source\": 14, \"target\": 12, \"t\": \"Z\"}]}'), 600, 250, 50,\n",
        "  10.0, false, true, '');\n",
        "</script>"
       ],
@@ -7330,7 +11217,7 @@
     {
      "data": {
       "text/html": [
-       "<div style=\"overflow:auto; background-color: white\" id=\"graph-output-1GhTD6P3\"></div>\n",
+       "<div style=\"overflow:auto; background-color: white\" id=\"graph-output-lFGeZ5WQ\"></div>\n",
        "<script type=\"module\">\n",
        "var d3;\n",
        "if (d3 == null) { d3 = await import(\"https://cdn.skypack.dev/d3@5\"); }\n",
@@ -7693,8 +11580,8 @@
        "}\n",
        "\n",
        "\n",
-       "showGraph('#graph-output-1GhTD6P3',\n",
-       "  JSON.parse('{\"nodes\": [{\"name\": \"0\", \"x\": 50, \"y\": 100, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"1\", \"x\": 50, \"y\": 150, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"2\", \"x\": 50, \"y\": 200, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"3\", \"x\": 100, \"y\": 150, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"4\", \"x\": 100, \"y\": 100, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"5\", \"x\": 150, \"y\": 200, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"6\", \"x\": 150, \"y\": 150, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"9\", \"x\": 250, \"y\": 150, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"10\", \"x\": 250, \"y\": 200, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"13\", \"x\": 350, \"y\": 100, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"15\", \"x\": 400, \"y\": 100, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"16\", \"x\": 400, \"y\": 150, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"17\", \"x\": 400, \"y\": 200, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}], \"links\": [{\"source\": \"0\", \"target\": \"4\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"1\", \"target\": \"3\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"2\", \"target\": \"5\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"3\", \"target\": \"4\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"3\", \"target\": \"6\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"4\", \"target\": \"13\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"4\", \"target\": \"5\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"4\", \"target\": \"9\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"5\", \"target\": \"6\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"5\", \"target\": \"10\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"6\", \"target\": \"9\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"9\", \"target\": \"10\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"9\", \"target\": \"16\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"10\", \"target\": \"13\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"10\", \"target\": \"17\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"13\", \"target\": \"15\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}], \"pauli_web\": [{\"source\": 10, \"target\": 9, \"t\": \"X\"}, {\"source\": 9, \"target\": 10, \"t\": \"X\"}, {\"source\": 10, \"target\": 13, \"t\": \"X\"}, {\"source\": 13, \"target\": 10, \"t\": \"X\"}, {\"source\": 10, \"target\": 17, \"t\": \"X\"}, {\"source\": 17, \"target\": 10, \"t\": \"X\"}, {\"source\": 10, \"target\": 5, \"t\": \"X\"}, {\"source\": 5, \"target\": 10, \"t\": \"X\"}, {\"source\": 4, \"target\": 0, \"t\": \"X\"}, {\"source\": 0, \"target\": 4, \"t\": \"X\"}, {\"source\": 4, \"target\": 3, \"t\": \"X\"}, {\"source\": 3, \"target\": 4, \"t\": \"X\"}, {\"source\": 4, \"target\": 13, \"t\": \"X\"}, {\"source\": 13, \"target\": 4, \"t\": \"X\"}, {\"source\": 4, \"target\": 5, \"t\": \"X\"}, {\"source\": 5, \"target\": 4, \"t\": \"X\"}, {\"source\": 4, \"target\": 9, \"t\": \"X\"}, {\"source\": 9, \"target\": 4, \"t\": \"X\"}, {\"source\": 3, \"target\": 1, \"t\": \"X\"}, {\"source\": 1, \"target\": 3, \"t\": \"X\"}]}'), 450, 250, 50,\n",
+       "showGraph('#graph-output-lFGeZ5WQ',\n",
+       "  JSON.parse('{\"nodes\": [{\"name\": \"0\", \"x\": 50, \"y\": 100, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"1\", \"x\": 50, \"y\": 150, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"2\", \"x\": 50, \"y\": 200, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"3\", \"x\": 100, \"y\": 150, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"4\", \"x\": 100, \"y\": 100, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"5\", \"x\": 150, \"y\": 150, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"6\", \"x\": 100, \"y\": 200, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"7\", \"x\": 200, \"y\": 200, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"8\", \"x\": 200, \"y\": 150, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"9\", \"x\": 250, 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{\"source\": \"14\", \"target\": \"15\", \"t\": 2, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"15\", \"target\": \"16\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"16\", \"target\": \"17\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"16\", \"target\": \"21\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"17\", \"target\": \"18\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"18\", \"target\": \"19\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"18\", \"target\": \"20\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"19\", \"target\": \"22\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}], \"pauli_web\": [{\"source\": 2, \"target\": 6, \"t\": \"Y\"}, {\"source\": 6, \"target\": 2, \"t\": \"Y\"}, {\"source\": 4, \"target\": 0, \"t\": \"Y\"}, {\"source\": 0, \"target\": 4, \"t\": \"Y\"}, {\"source\": 4, \"target\": 3, \"t\": \"X\"}, {\"source\": 3, \"target\": 4, \"t\": \"X\"}, {\"source\": 4, \"target\": 10, \"t\": \"Y\"}, {\"source\": 6, \"target\": 7, \"t\": \"Y\"}, {\"source\": 7, \"target\": 6, \"t\": \"Y\"}, {\"source\": 7, \"target\": 8, \"t\": \"Z\"}, {\"source\": 8, \"target\": 7, \"t\": \"Z\"}, {\"source\": 7, \"target\": 9, \"t\": \"Y\"}, {\"source\": 9, \"target\": 7, \"t\": \"Y\"}, {\"source\": 9, \"target\": 10, \"t\": \"Y\"}, {\"source\": 10, \"target\": 9, \"t\": \"Y\"}, {\"source\": 9, \"target\": 13, \"t\": \"Z\"}, {\"source\": 13, \"target\": 9, \"t\": \"Z\"}, {\"source\": 10, \"target\": 4, \"t\": \"Y\"}, {\"source\": 10, \"target\": 11, \"t\": \"X\"}, {\"source\": 11, \"target\": 10, \"t\": \"Z\"}, {\"source\": 18, \"target\": 17, \"t\": \"Z\"}, {\"source\": 17, \"target\": 18, \"t\": \"Z\"}, {\"source\": 18, \"target\": 19, \"t\": \"Z\"}, {\"source\": 19, \"target\": 18, \"t\": \"Z\"}, {\"source\": 18, \"target\": 20, \"t\": \"Z\"}, {\"source\": 20, \"target\": 18, \"t\": \"Z\"}, {\"source\": 3, \"target\": 5, \"t\": \"X\"}, {\"source\": 5, \"target\": 3, \"t\": \"Z\"}, {\"source\": 5, \"target\": 8, \"t\": \"Z\"}, {\"source\": 8, \"target\": 5, \"t\": \"Z\"}, {\"source\": 11, \"target\": 17, \"t\": \"Z\"}, {\"source\": 17, \"target\": 11, \"t\": \"Z\"}, {\"source\": 13, \"target\": 19, \"t\": \"Z\"}, {\"source\": 19, \"target\": 13, \"t\": \"Z\"}]}'), 600, 250, 50,\n",
        "  10.0, false, true, '');\n",
        "</script>"
       ],
@@ -7708,7 +11595,7 @@
     {
      "data": {
       "text/html": [
-       "<div style=\"overflow:auto; background-color: white\" id=\"graph-output-ewXCvE6c\"></div>\n",
+       "<div style=\"overflow:auto; background-color: white\" id=\"graph-output-DuSYlwGD\"></div>\n",
        "<script type=\"module\">\n",
        "var d3;\n",
        "if (d3 == null) { d3 = await import(\"https://cdn.skypack.dev/d3@5\"); }\n",
@@ -8071,8 +11958,8 @@
        "}\n",
        "\n",
        "\n",
-       "showGraph('#graph-output-ewXCvE6c',\n",
-       "  JSON.parse('{\"nodes\": [{\"name\": \"0\", \"x\": 50, \"y\": 100, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"1\", \"x\": 50, \"y\": 150, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"2\", \"x\": 50, \"y\": 200, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"3\", \"x\": 100, \"y\": 150, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"4\", \"x\": 100, \"y\": 100, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"5\", \"x\": 150, \"y\": 200, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"6\", \"x\": 150, \"y\": 150, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"9\", \"x\": 250, \"y\": 150, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"10\", \"x\": 250, \"y\": 200, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"13\", \"x\": 350, \"y\": 100, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"15\", \"x\": 400, \"y\": 100, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"16\", \"x\": 400, \"y\": 150, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"17\", \"x\": 400, \"y\": 200, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}], \"links\": [{\"source\": \"0\", \"target\": \"4\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"1\", \"target\": \"3\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"2\", \"target\": \"5\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"3\", \"target\": \"4\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"3\", \"target\": \"6\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"4\", \"target\": \"13\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"4\", \"target\": \"5\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"4\", \"target\": \"9\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"5\", \"target\": \"6\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"5\", \"target\": \"10\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"6\", \"target\": \"9\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"9\", \"target\": \"10\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"9\", \"target\": \"16\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"10\", \"target\": \"13\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"10\", \"target\": \"17\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"13\", \"target\": \"15\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}], \"pauli_web\": [{\"source\": 3, \"target\": 1, \"t\": \"Z\"}, {\"source\": 1, \"target\": 3, \"t\": \"Z\"}, {\"source\": 3, \"target\": 4, \"t\": \"Z\"}, {\"source\": 4, \"target\": 3, \"t\": \"Z\"}, {\"source\": 3, \"target\": 6, \"t\": \"Z\"}, {\"source\": 6, \"target\": 3, \"t\": \"Z\"}, {\"source\": 5, \"target\": 2, \"t\": \"Z\"}, {\"source\": 2, \"target\": 5, \"t\": \"Z\"}, {\"source\": 5, \"target\": 6, \"t\": \"Z\"}, {\"source\": 6, \"target\": 5, \"t\": \"Z\"}, {\"source\": 5, \"target\": 10, \"t\": \"Z\"}, {\"source\": 10, \"target\": 5, \"t\": \"Z\"}, {\"source\": 5, \"target\": 4, \"t\": \"Z\"}, {\"source\": 4, \"target\": 5, \"t\": \"Z\"}, {\"source\": 17, \"target\": 10, \"t\": \"Z\"}, {\"source\": 10, \"target\": 17, \"t\": \"Z\"}]}'), 450, 250, 50,\n",
+       "showGraph('#graph-output-DuSYlwGD',\n",
+       "  JSON.parse('{\"nodes\": [{\"name\": \"0\", \"x\": 50, \"y\": 100, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"1\", \"x\": 50, \"y\": 150, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"2\", \"x\": 50, \"y\": 200, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"3\", \"x\": 100, \"y\": 150, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"4\", \"x\": 100, \"y\": 100, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"5\", \"x\": 150, \"y\": 150, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"6\", \"x\": 100, \"y\": 200, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"7\", \"x\": 200, \"y\": 200, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"8\", \"x\": 200, \"y\": 150, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"9\", \"x\": 250, \"y\": 200, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"10\", \"x\": 250, \"y\": 100, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"11\", \"x\": 300, \"y\": 100, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"12\", \"x\": 300, \"y\": 150, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"13\", \"x\": 300, \"y\": 200, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"14\", \"x\": 350, \"y\": 150, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"15\", \"x\": 400, \"y\": 150, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"16\", \"x\": 450, \"y\": 150, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"17\", \"x\": 450, \"y\": 100, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"18\", \"x\": 500, \"y\": 100, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"19\", \"x\": 500, \"y\": 200, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"20\", \"x\": 550, \"y\": 100, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"21\", \"x\": 550, \"y\": 150, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"22\", \"x\": 550, \"y\": 200, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}], \"links\": [{\"source\": \"0\", \"target\": \"4\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"1\", \"target\": \"3\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"2\", \"target\": \"6\", \"t\": 2, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"3\", \"target\": \"4\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"3\", \"target\": \"5\", \"t\": 2, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"4\", \"target\": \"10\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"5\", \"target\": \"8\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"6\", \"target\": \"7\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"7\", \"target\": \"8\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"7\", \"target\": \"9\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"8\", \"target\": \"12\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"9\", \"target\": \"10\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"9\", \"target\": \"13\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"10\", \"target\": \"11\", \"t\": 2, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"11\", \"target\": \"17\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"12\", \"target\": \"13\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"12\", \"target\": \"14\", \"t\": 2, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"13\", \"target\": \"19\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"14\", \"target\": \"15\", \"t\": 2, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"15\", \"target\": \"16\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"16\", \"target\": \"17\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"16\", \"target\": \"21\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"17\", \"target\": \"18\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"18\", \"target\": \"19\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"18\", \"target\": \"20\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"19\", \"target\": \"22\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}], \"pauli_web\": [{\"source\": 3, \"target\": 1, \"t\": \"Z\"}, {\"source\": 1, \"target\": 3, \"t\": \"Z\"}, {\"source\": 3, \"target\": 4, \"t\": \"Z\"}, {\"source\": 4, \"target\": 3, \"t\": \"Z\"}, {\"source\": 3, \"target\": 5, \"t\": \"Z\"}, {\"source\": 5, \"target\": 3, \"t\": \"X\"}, {\"source\": 5, \"target\": 8, \"t\": \"X\"}, {\"source\": 6, \"target\": 2, \"t\": \"X\"}, {\"source\": 2, \"target\": 6, \"t\": \"Z\"}, {\"source\": 6, \"target\": 7, \"t\": \"X\"}, {\"source\": 7, \"target\": 6, \"t\": \"X\"}, {\"source\": 21, \"target\": 16, \"t\": \"X\"}, {\"source\": 16, \"target\": 21, \"t\": \"X\"}, {\"source\": 8, \"target\": 5, \"t\": \"X\"}, {\"source\": 8, \"target\": 7, \"t\": \"X\"}, {\"source\": 7, \"target\": 8, \"t\": \"X\"}, {\"source\": 8, \"target\": 12, \"t\": \"X\"}, {\"source\": 12, \"target\": 8, \"t\": \"X\"}, {\"source\": 15, \"target\": 14, \"t\": \"X\"}, {\"source\": 14, \"target\": 15, \"t\": \"Z\"}, {\"source\": 15, \"target\": 16, \"t\": \"X\"}, {\"source\": 16, \"target\": 15, \"t\": \"X\"}, {\"source\": 12, \"target\": 14, \"t\": \"X\"}, {\"source\": 14, \"target\": 12, \"t\": \"Z\"}, {\"source\": 0, \"target\": 4, \"t\": \"Z\"}, {\"source\": 4, \"target\": 0, \"t\": \"Z\"}]}'), 600, 250, 50,\n",
        "  10.0, false, true, '');\n",
        "</script>"
       ],
@@ -8082,36 +11969,11 @@
      },
      "metadata": {},
      "output_type": "display_data"
-    }
-   ],
-   "source": [
-    "# Since this describes the same unitary, the 6 output webs should all have the same support on\n",
-    "# the inputs as before.\n",
-    "order, zwebs, xwebs = compute_pauli_webs(g)\n",
-    "zx.draw(g, labels=True, pauli_web=xwebs[15])\n",
-    "zx.draw(g, labels=True, pauli_web=zwebs[15])\n",
-    "zx.draw(g, labels=True, pauli_web=xwebs[16])\n",
-    "zx.draw(g, labels=True, pauli_web=zwebs[16])\n",
-    "zx.draw(g, labels=True, pauli_web=xwebs[17])\n",
-    "zx.draw(g, labels=True, pauli_web=zwebs[17])"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {},
-   "source": [
-    "# Clifford examples"
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": 9,
-   "metadata": {},
-   "outputs": [
+    },
     {
      "data": {
       "text/html": [
-       "<div style=\"overflow:auto; background-color: white\" id=\"graph-output-4EeOnyn8\"></div>\n",
+       "<div style=\"overflow:auto; background-color: white\" id=\"graph-output-QfXCdNM4\"></div>\n",
        "<script type=\"module\">\n",
        "var d3;\n",
        "if (d3 == null) { d3 = await import(\"https://cdn.skypack.dev/d3@5\"); }\n",
@@ -8474,9 +12336,9 @@
        "}\n",
        "\n",
        "\n",
-       "showGraph('#graph-output-4EeOnyn8',\n",
-       "  JSON.parse('{\"nodes\": [{\"name\": \"0\", \"x\": 50, \"y\": 100, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"1\", \"x\": 100, \"y\": 100, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"2\", \"x\": 150, \"y\": 100, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}], \"links\": [{\"source\": \"0\", \"target\": \"1\", \"t\": 2, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"1\", \"target\": \"2\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}], \"pauli_web\": []}'), 200, 150, 50,\n",
-       "  10.0, false, false, '');\n",
+       "showGraph('#graph-output-QfXCdNM4',\n",
+       "  JSON.parse('{\"nodes\": [{\"name\": \"0\", \"x\": 50, \"y\": 100, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"1\", \"x\": 50, \"y\": 150, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"2\", \"x\": 50, \"y\": 200, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"3\", \"x\": 100, \"y\": 150, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"4\", \"x\": 100, \"y\": 100, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"5\", \"x\": 150, \"y\": 150, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"6\", \"x\": 100, \"y\": 200, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"7\", \"x\": 200, \"y\": 200, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"8\", \"x\": 200, \"y\": 150, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"9\", \"x\": 250, \"y\": 200, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"10\", \"x\": 250, \"y\": 100, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"11\", \"x\": 300, \"y\": 100, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"12\", \"x\": 300, \"y\": 150, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"13\", \"x\": 300, \"y\": 200, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"14\", \"x\": 350, \"y\": 150, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"15\", \"x\": 400, \"y\": 150, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"16\", \"x\": 450, \"y\": 150, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"17\", \"x\": 450, \"y\": 100, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"18\", \"x\": 500, \"y\": 100, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"19\", \"x\": 500, \"y\": 200, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"20\", \"x\": 550, \"y\": 100, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"21\", \"x\": 550, \"y\": 150, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"22\", \"x\": 550, \"y\": 200, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}], \"links\": [{\"source\": \"0\", \"target\": \"4\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"1\", \"target\": \"3\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"2\", \"target\": \"6\", \"t\": 2, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"3\", \"target\": \"4\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"3\", \"target\": \"5\", \"t\": 2, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"4\", \"target\": \"10\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"5\", \"target\": \"8\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"6\", \"target\": \"7\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"7\", \"target\": \"8\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"7\", \"target\": \"9\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"8\", \"target\": \"12\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"9\", \"target\": \"10\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"9\", \"target\": \"13\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"10\", \"target\": \"11\", \"t\": 2, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"11\", \"target\": \"17\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"12\", \"target\": \"13\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"12\", \"target\": \"14\", \"t\": 2, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"13\", \"target\": \"19\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"14\", \"target\": \"15\", \"t\": 2, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"15\", \"target\": \"16\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"16\", \"target\": \"17\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"16\", \"target\": \"21\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"17\", \"target\": \"18\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"18\", \"target\": \"19\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"18\", \"target\": \"20\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"19\", \"target\": \"22\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}], \"pauli_web\": [{\"source\": 1, \"target\": 3, \"t\": \"X\"}, {\"source\": 3, \"target\": 1, \"t\": \"X\"}, {\"source\": 2, \"target\": 6, \"t\": \"Y\"}, {\"source\": 6, \"target\": 2, \"t\": \"Y\"}, {\"source\": 4, \"target\": 0, \"t\": \"Y\"}, {\"source\": 0, \"target\": 4, \"t\": \"Y\"}, {\"source\": 4, \"target\": 3, \"t\": \"X\"}, {\"source\": 3, \"target\": 4, \"t\": \"X\"}, {\"source\": 4, \"target\": 10, \"t\": \"Y\"}, {\"source\": 6, \"target\": 7, \"t\": \"Y\"}, {\"source\": 7, \"target\": 6, \"t\": \"Y\"}, {\"source\": 7, \"target\": 8, \"t\": \"Z\"}, {\"source\": 8, \"target\": 7, \"t\": \"Z\"}, {\"source\": 7, \"target\": 9, \"t\": \"Y\"}, {\"source\": 9, \"target\": 7, \"t\": \"Y\"}, {\"source\": 9, \"target\": 10, \"t\": \"Y\"}, {\"source\": 10, \"target\": 9, \"t\": \"Y\"}, {\"source\": 9, \"target\": 13, \"t\": \"Z\"}, {\"source\": 13, \"target\": 9, \"t\": \"Z\"}, {\"source\": 10, \"target\": 4, \"t\": \"Y\"}, {\"source\": 10, \"target\": 11, \"t\": \"X\"}, {\"source\": 11, \"target\": 10, \"t\": \"Z\"}, {\"source\": 12, \"target\": 8, \"t\": \"Z\"}, {\"source\": 8, \"target\": 12, \"t\": \"Z\"}, {\"source\": 12, \"target\": 13, \"t\": \"Z\"}, {\"source\": 13, \"target\": 12, \"t\": \"Z\"}, {\"source\": 12, \"target\": 14, \"t\": \"Z\"}, {\"source\": 14, \"target\": 12, \"t\": \"X\"}, {\"source\": 14, \"target\": 15, \"t\": \"X\"}, {\"source\": 15, \"target\": 14, \"t\": \"Z\"}, {\"source\": 16, \"target\": 15, \"t\": \"Z\"}, {\"source\": 15, \"target\": 16, \"t\": \"Z\"}, {\"source\": 16, \"target\": 17, \"t\": \"Z\"}, {\"source\": 17, \"target\": 16, \"t\": \"Z\"}, {\"source\": 16, \"target\": 21, \"t\": \"Z\"}, {\"source\": 21, \"target\": 16, \"t\": \"Z\"}, {\"source\": 11, \"target\": 17, \"t\": \"Z\"}, {\"source\": 17, \"target\": 11, \"t\": \"Z\"}]}'), 600, 250, 50,\n",
+       "  10.0, false, true, '');\n",
        "</script>"
       ],
       "text/plain": [
@@ -8489,7 +12351,7 @@
     {
      "data": {
       "text/html": [
-       "<div style=\"overflow:auto; background-color: white\" id=\"graph-output-7Z7J8JnE\"></div>\n",
+       "<div style=\"overflow:auto; background-color: white\" id=\"graph-output-FF9JmJNe\"></div>\n",
        "<script type=\"module\">\n",
        "var d3;\n",
        "if (d3 == null) { d3 = await import(\"https://cdn.skypack.dev/d3@5\"); }\n",
@@ -8852,8 +12714,8 @@
        "}\n",
        "\n",
        "\n",
-       "showGraph('#graph-output-7Z7J8JnE',\n",
-       "  JSON.parse('{\"nodes\": [{\"name\": \"0\", \"x\": 50, \"y\": 100, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"2\", \"x\": 150, \"y\": 100, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}], \"links\": [{\"source\": \"0\", \"target\": \"2\", \"t\": 2, \"index\": 0, \"num_parallel\": 1}], \"pauli_web\": []}'), 200, 150, 50,\n",
+       "showGraph('#graph-output-FF9JmJNe',\n",
+       "  JSON.parse('{\"nodes\": [{\"name\": \"0\", \"x\": 50, \"y\": 100, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"1\", \"x\": 50, \"y\": 150, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"2\", \"x\": 50, \"y\": 200, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"3\", \"x\": 100, \"y\": 150, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"4\", \"x\": 100, \"y\": 100, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"5\", \"x\": 150, \"y\": 150, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"6\", \"x\": 100, \"y\": 200, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"7\", \"x\": 200, \"y\": 200, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"8\", \"x\": 200, \"y\": 150, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"9\", \"x\": 250, \"y\": 200, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"10\", \"x\": 250, \"y\": 100, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"11\", \"x\": 300, \"y\": 100, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"12\", \"x\": 300, \"y\": 150, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"13\", \"x\": 300, \"y\": 200, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"14\", \"x\": 350, \"y\": 150, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"15\", \"x\": 400, \"y\": 150, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"16\", \"x\": 450, \"y\": 150, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"17\", \"x\": 450, \"y\": 100, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"18\", \"x\": 500, \"y\": 100, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"19\", \"x\": 500, \"y\": 200, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"20\", \"x\": 550, \"y\": 100, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"21\", \"x\": 550, \"y\": 150, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"22\", \"x\": 550, \"y\": 200, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}], \"links\": [{\"source\": \"0\", \"target\": \"4\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"1\", \"target\": \"3\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"2\", \"target\": \"6\", \"t\": 2, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"3\", \"target\": \"4\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"3\", \"target\": \"5\", \"t\": 2, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"4\", \"target\": \"10\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"5\", \"target\": \"8\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"6\", \"target\": \"7\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"7\", \"target\": \"8\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"7\", \"target\": \"9\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"8\", \"target\": \"12\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"9\", \"target\": \"10\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"9\", \"target\": \"13\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"10\", \"target\": \"11\", \"t\": 2, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"11\", \"target\": \"17\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"12\", \"target\": \"13\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"12\", \"target\": \"14\", \"t\": 2, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"13\", \"target\": \"19\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"14\", \"target\": \"15\", \"t\": 2, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"15\", \"target\": \"16\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"16\", \"target\": \"17\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"16\", \"target\": \"21\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"17\", \"target\": \"18\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"18\", \"target\": \"19\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"18\", \"target\": \"20\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"19\", \"target\": \"22\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}], \"pauli_web\": [{\"source\": 6, \"target\": 2, \"t\": \"X\"}, {\"source\": 2, \"target\": 6, \"t\": \"Z\"}, {\"source\": 6, \"target\": 7, \"t\": \"X\"}, {\"source\": 7, \"target\": 6, \"t\": \"X\"}, {\"source\": 11, \"target\": 10, \"t\": \"X\"}, {\"source\": 10, \"target\": 11, \"t\": \"Z\"}, {\"source\": 11, \"target\": 17, \"t\": \"X\"}, {\"source\": 13, \"target\": 9, \"t\": \"X\"}, {\"source\": 9, \"target\": 13, \"t\": \"X\"}, {\"source\": 13, \"target\": 12, \"t\": \"X\"}, {\"source\": 12, \"target\": 13, \"t\": \"X\"}, {\"source\": 13, \"target\": 19, \"t\": \"X\"}, {\"source\": 15, \"target\": 14, \"t\": \"X\"}, {\"source\": 14, \"target\": 15, \"t\": \"Z\"}, {\"source\": 15, \"target\": 16, \"t\": \"X\"}, {\"source\": 16, \"target\": 15, \"t\": \"X\"}, {\"source\": 17, \"target\": 11, \"t\": \"X\"}, {\"source\": 17, \"target\": 16, \"t\": \"X\"}, {\"source\": 16, \"target\": 17, \"t\": \"X\"}, {\"source\": 17, \"target\": 18, \"t\": \"X\"}, {\"source\": 18, \"target\": 17, \"t\": \"X\"}, {\"source\": 19, \"target\": 13, \"t\": \"X\"}, {\"source\": 19, \"target\": 18, \"t\": \"X\"}, {\"source\": 18, \"target\": 19, \"t\": \"X\"}, {\"source\": 19, \"target\": 22, \"t\": \"X\"}, {\"source\": 22, \"target\": 19, \"t\": \"X\"}, {\"source\": 4, \"target\": 10, \"t\": \"Z\"}, {\"source\": 10, \"target\": 4, \"t\": \"Z\"}, {\"source\": 7, \"target\": 9, \"t\": \"X\"}, {\"source\": 9, \"target\": 7, \"t\": \"X\"}, {\"source\": 12, \"target\": 14, \"t\": \"X\"}, {\"source\": 14, \"target\": 12, \"t\": \"Z\"}, {\"source\": 0, \"target\": 4, \"t\": \"Z\"}, {\"source\": 4, \"target\": 0, \"t\": \"Z\"}]}'), 600, 250, 50,\n",
        "  10.0, false, true, '');\n",
        "</script>"
       ],
@@ -8863,35 +12725,11 @@
      },
      "metadata": {},
      "output_type": "display_data"
-    }
-   ],
-   "source": [
-    "# Next, we'll look at a single H gate\n",
-    "c = zx.qasm(\"\"\"\n",
-    "qreg q[1];\n",
-    "h q[0];\n",
-    "\"\"\")\n",
-    "g = c.to_graph()\n",
-    "\n",
-    "# PyZX renders this as a single H-edge connected to an identity spider\n",
-    "zx.draw(g)\n",
-    "\n",
-    "# we don't really need this id-spider, so we can remove it with id_simp\n",
-    "zx.id_simp(g)\n",
-    "zx.draw(g, labels=True)\n",
-    "\n",
-    "order, zwebs, xwebs = compute_pauli_webs(g)"
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": 10,
-   "metadata": {},
-   "outputs": [
+    },
     {
      "data": {
       "text/html": [
-       "<div style=\"overflow:auto; background-color: white\" id=\"graph-output-NcdjsjaL\"></div>\n",
+       "<div style=\"overflow:auto; background-color: white\" id=\"graph-output-vDILwiDt\"></div>\n",
        "<script type=\"module\">\n",
        "var d3;\n",
        "if (d3 == null) { d3 = await import(\"https://cdn.skypack.dev/d3@5\"); }\n",
@@ -9254,8 +13092,8 @@
        "}\n",
        "\n",
        "\n",
-       "showGraph('#graph-output-NcdjsjaL',\n",
-       "  JSON.parse('{\"nodes\": [{\"name\": \"0\", \"x\": 50, \"y\": 100, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"2\", \"x\": 150, \"y\": 100, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}], \"links\": [{\"source\": \"0\", \"target\": \"2\", \"t\": 2, \"index\": 0, \"num_parallel\": 1}], \"pauli_web\": [{\"source\": 2, \"target\": 0, \"t\": \"Z\"}, {\"source\": 0, \"target\": 2, \"t\": \"X\"}]}'), 200, 150, 50,\n",
+       "showGraph('#graph-output-vDILwiDt',\n",
+       "  JSON.parse('{\"nodes\": [{\"name\": \"0\", \"x\": 50, \"y\": 100, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"1\", \"x\": 50, \"y\": 150, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"2\", \"x\": 50, \"y\": 200, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"3\", \"x\": 100, \"y\": 150, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"4\", \"x\": 100, \"y\": 100, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"5\", \"x\": 150, \"y\": 150, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"6\", \"x\": 100, \"y\": 200, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"7\", \"x\": 200, \"y\": 200, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"8\", \"x\": 200, \"y\": 150, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"9\", \"x\": 250, \"y\": 200, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"10\", \"x\": 250, \"y\": 100, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"11\", \"x\": 300, \"y\": 100, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"12\", \"x\": 300, \"y\": 150, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"13\", \"x\": 300, \"y\": 200, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"14\", \"x\": 350, \"y\": 150, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"15\", \"x\": 400, \"y\": 150, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"16\", \"x\": 450, \"y\": 150, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"17\", \"x\": 450, \"y\": 100, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"18\", \"x\": 500, \"y\": 100, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"19\", \"x\": 500, \"y\": 200, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"20\", \"x\": 550, \"y\": 100, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"21\", \"x\": 550, \"y\": 150, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"22\", \"x\": 550, \"y\": 200, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}], \"links\": [{\"source\": \"0\", \"target\": \"4\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"1\", \"target\": \"3\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"2\", \"target\": \"6\", \"t\": 2, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"3\", \"target\": \"4\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"3\", \"target\": \"5\", \"t\": 2, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"4\", \"target\": \"10\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"5\", \"target\": \"8\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"6\", \"target\": \"7\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"7\", \"target\": \"8\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"7\", \"target\": \"9\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"8\", \"target\": \"12\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"9\", \"target\": \"10\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"9\", \"target\": \"13\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"10\", \"target\": \"11\", \"t\": 2, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"11\", \"target\": \"17\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"12\", \"target\": \"13\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"12\", \"target\": \"14\", \"t\": 2, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"13\", \"target\": \"19\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"14\", \"target\": \"15\", \"t\": 2, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"15\", \"target\": \"16\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"16\", \"target\": \"17\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"16\", \"target\": \"21\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"17\", \"target\": \"18\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"18\", \"target\": \"19\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"18\", \"target\": \"20\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"19\", \"target\": \"22\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}], \"pauli_web\": [{\"source\": 1, \"target\": 3, \"t\": \"X\"}, {\"source\": 3, \"target\": 1, \"t\": \"X\"}, {\"source\": 2, \"target\": 6, \"t\": \"X\"}, {\"source\": 6, \"target\": 2, \"t\": \"Z\"}, {\"source\": 7, \"target\": 6, \"t\": \"Z\"}, {\"source\": 6, \"target\": 7, \"t\": \"Z\"}, {\"source\": 7, \"target\": 8, \"t\": \"Z\"}, {\"source\": 8, \"target\": 7, \"t\": \"Z\"}, {\"source\": 7, \"target\": 9, \"t\": \"Z\"}, {\"source\": 22, \"target\": 19, \"t\": \"Z\"}, {\"source\": 19, \"target\": 22, \"t\": \"Z\"}, {\"source\": 9, \"target\": 7, \"t\": \"Z\"}, {\"source\": 9, \"target\": 10, \"t\": \"Z\"}, {\"source\": 10, \"target\": 9, \"t\": \"Z\"}, {\"source\": 9, \"target\": 13, \"t\": \"Z\"}, {\"source\": 13, \"target\": 9, \"t\": \"Z\"}, {\"source\": 3, \"target\": 5, \"t\": \"X\"}, {\"source\": 5, \"target\": 3, \"t\": \"Z\"}, {\"source\": 4, \"target\": 10, \"t\": \"Z\"}, {\"source\": 10, \"target\": 4, \"t\": \"Z\"}, {\"source\": 5, \"target\": 8, \"t\": \"Z\"}, {\"source\": 8, \"target\": 5, \"t\": \"Z\"}, {\"source\": 13, \"target\": 19, \"t\": \"Z\"}, {\"source\": 19, \"target\": 13, \"t\": \"Z\"}, {\"source\": 0, \"target\": 4, \"t\": \"Z\"}, {\"source\": 4, \"target\": 0, \"t\": \"Z\"}]}'), 600, 250, 50,\n",
        "  10.0, false, true, '');\n",
        "</script>"
       ],
@@ -9265,11 +13103,28 @@
      },
      "metadata": {},
      "output_type": "display_data"
-    },
+    }
+   ],
+   "source": [
+    "# Note the we get some Y-edges appear, which are shown in a third color\n",
+    "order, zwebs, xwebs = compute_pauli_webs(g)\n",
+    "zx.draw(g, labels=True, pauli_web=xwebs[20])\n",
+    "zx.draw(g, labels=True, pauli_web=zwebs[20])\n",
+    "zx.draw(g, labels=True, pauli_web=xwebs[21])\n",
+    "zx.draw(g, labels=True, pauli_web=zwebs[21])\n",
+    "zx.draw(g, labels=True, pauli_web=xwebs[22])\n",
+    "zx.draw(g, labels=True, pauli_web=zwebs[22])"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 13,
+   "metadata": {},
+   "outputs": [
     {
      "data": {
       "text/html": [
-       "<div style=\"overflow:auto; background-color: white\" id=\"graph-output-MjOGiWnr\"></div>\n",
+       "<div style=\"overflow:auto; background-color: white\" id=\"graph-output-9GxdLQWw\"></div>\n",
        "<script type=\"module\">\n",
        "var d3;\n",
        "if (d3 == null) { d3 = await import(\"https://cdn.skypack.dev/d3@5\"); }\n",
@@ -9632,8 +13487,8 @@
        "}\n",
        "\n",
        "\n",
-       "showGraph('#graph-output-MjOGiWnr',\n",
-       "  JSON.parse('{\"nodes\": [{\"name\": \"0\", \"x\": 50, \"y\": 100, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"2\", \"x\": 150, \"y\": 100, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}], \"links\": [{\"source\": \"0\", \"target\": \"2\", \"t\": 2, \"index\": 0, \"num_parallel\": 1}], \"pauli_web\": [{\"source\": 2, \"target\": 0, \"t\": \"X\"}, {\"source\": 0, \"target\": 2, \"t\": \"Z\"}]}'), 200, 150, 50,\n",
+       "showGraph('#graph-output-9GxdLQWw',\n",
+       "  JSON.parse('{\"nodes\": [{\"name\": \"0\", \"x\": 50, \"y\": 100, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"1\", \"x\": 100, \"y\": 100, \"t\": 1, \"phase\": \"\\u03c0/2\", \"ground\": false, \"vdata\": []}, {\"name\": \"2\", \"x\": 150, \"y\": 100, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}], \"links\": [{\"source\": \"0\", \"target\": \"1\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"1\", \"target\": \"2\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}], \"pauli_web\": []}'), 200, 150, 50,\n",
        "  10.0, false, true, '');\n",
        "</script>"
       ],
@@ -9644,35 +13499,10 @@
      "metadata": {},
      "output_type": "display_data"
     },
-    {
-     "name": "stdout",
-     "output_type": "stream",
-     "text": [
-      "{(2, 0): 'Z', (0, 2): 'X'}\n"
-     ]
-    }
-   ],
-   "source": [
-    "# The single output has two Pauli webs\n",
-    "zx.draw(g, labels=True, pauli_web=zwebs[2])\n",
-    "zx.draw(g, labels=True, pauli_web=xwebs[2])\n",
-    "\n",
-    "# Note that on a hadamard edge, the color changes in the middle of the edge. To handle this, Pauli webs actually label every\n",
-    "# \"half-edge\" with a Pauli. (v,w) means the half of the edge touching v, whereas (w,v) means the half of the edge touching w.\n",
-    "\n",
-    "#  See for example:\n",
-    "print(zwebs[2].half_edges())"
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": 11,
-   "metadata": {},
-   "outputs": [
     {
      "data": {
       "text/html": [
-       "<div style=\"overflow:auto; background-color: white\" id=\"graph-output-BmWY1tRI\"></div>\n",
+       "<div style=\"overflow:auto; background-color: white\" id=\"graph-output-DKdzppIC\"></div>\n",
        "<script type=\"module\">\n",
        "var d3;\n",
        "if (d3 == null) { d3 = await import(\"https://cdn.skypack.dev/d3@5\"); }\n",
@@ -10035,8 +13865,8 @@
        "}\n",
        "\n",
        "\n",
-       "showGraph('#graph-output-BmWY1tRI',\n",
-       "  JSON.parse('{\"nodes\": [{\"name\": \"0\", \"x\": 50, \"y\": 100, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"1\", \"x\": 50, \"y\": 150, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"2\", \"x\": 50, \"y\": 200, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"3\", \"x\": 100, \"y\": 150, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"4\", \"x\": 100, \"y\": 100, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"5\", \"x\": 150, \"y\": 150, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"6\", \"x\": 100, \"y\": 200, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"7\", \"x\": 200, \"y\": 200, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"8\", \"x\": 200, \"y\": 150, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"9\", \"x\": 250, \"y\": 200, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"10\", \"x\": 250, \"y\": 100, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"11\", \"x\": 300, \"y\": 100, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"12\", \"x\": 300, \"y\": 150, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"13\", \"x\": 300, \"y\": 200, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"14\", \"x\": 350, \"y\": 150, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"15\", \"x\": 400, \"y\": 150, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"16\", \"x\": 450, \"y\": 150, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"17\", \"x\": 450, \"y\": 100, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"18\", \"x\": 500, \"y\": 100, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"19\", \"x\": 500, \"y\": 200, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"20\", \"x\": 550, \"y\": 100, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"21\", \"x\": 550, \"y\": 150, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"22\", \"x\": 550, \"y\": 200, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}], \"links\": [{\"source\": \"0\", \"target\": \"4\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"1\", \"target\": \"3\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"2\", \"target\": \"6\", \"t\": 2, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"3\", \"target\": \"4\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"3\", \"target\": \"5\", \"t\": 2, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"4\", \"target\": \"10\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"5\", \"target\": \"8\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"6\", \"target\": \"7\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"7\", \"target\": \"8\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"7\", \"target\": \"9\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"8\", \"target\": \"12\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"9\", \"target\": \"10\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"9\", \"target\": \"13\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"10\", \"target\": \"11\", \"t\": 2, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"11\", \"target\": \"17\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"12\", \"target\": \"13\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"12\", \"target\": \"14\", \"t\": 2, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"13\", \"target\": \"19\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"14\", \"target\": \"15\", \"t\": 2, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"15\", \"target\": \"16\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"16\", \"target\": \"17\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"16\", \"target\": \"21\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"17\", \"target\": \"18\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"18\", \"target\": \"19\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"18\", \"target\": \"20\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"19\", \"target\": \"22\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}], \"pauli_web\": []}'), 600, 250, 50,\n",
+       "showGraph('#graph-output-DKdzppIC',\n",
+       "  JSON.parse('{\"nodes\": [{\"name\": \"0\", \"x\": 50, \"y\": 100, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"1\", \"x\": 100, \"y\": 100, \"t\": 1, \"phase\": \"\\u03c0/2\", \"ground\": false, \"vdata\": []}, {\"name\": \"2\", \"x\": 150, \"y\": 100, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}], \"links\": [{\"source\": \"0\", \"target\": \"1\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"1\", \"target\": \"2\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}], \"pauli_web\": [{\"source\": 1, \"target\": 0, \"t\": \"Y\"}, {\"source\": 0, \"target\": 1, \"t\": \"Y\"}, {\"source\": 1, \"target\": 2, \"t\": \"X\"}, {\"source\": 2, \"target\": 1, \"t\": \"X\"}]}'), 200, 150, 50,\n",
        "  10.0, false, true, '');\n",
        "</script>"
       ],
@@ -10049,36 +13879,27 @@
     }
    ],
    "source": [
-    "# Here's an example mixing CNOT and H gates\n",
-    "\n",
+    "# Next, we'll look at a single S gate\n",
     "c = zx.qasm(\"\"\"\n",
-    "qreg q[3];\n",
-    "cx q[0], q[1];\n",
-    "h q[1];\n",
-    "h q[2];\n",
-    "cx q[1], q[2];\n",
-    "cx q[0], q[2];\n",
-    "h q[0];\n",
-    "cx q[2], q[1];\n",
-    "h q[1];\n",
-    "h q[1];\n",
-    "cx q[0], q[1];\n",
-    "cx q[2], q[0];\n",
+    "qreg q[1];\n",
+    "s q[0];\n",
     "\"\"\")\n",
     "g = c.to_graph()\n",
+    "zx.draw(g, labels=True)\n",
     "\n",
-    "zx.draw(g, labels=True)"
+    "order, zwebs, xwebs = compute_pauli_webs(g)\n",
+    "zx.draw(g, labels=True, pauli_web=xwebs[2])"
    ]
   },
   {
    "cell_type": "code",
-   "execution_count": 12,
+   "execution_count": 22,
    "metadata": {},
    "outputs": [
     {
      "data": {
       "text/html": [
-       "<div style=\"overflow:auto; background-color: white\" id=\"graph-output-T8HhYlpr\"></div>\n",
+       "<div style=\"overflow:auto; background-color: white\" id=\"graph-output-XUTYGp4X\"></div>\n",
        "<script type=\"module\">\n",
        "var d3;\n",
        "if (d3 == null) { d3 = await import(\"https://cdn.skypack.dev/d3@5\"); }\n",
@@ -10441,9 +14262,9 @@
        "}\n",
        "\n",
        "\n",
-       "showGraph('#graph-output-T8HhYlpr',\n",
-       "  JSON.parse('{\"nodes\": [{\"name\": \"0\", \"x\": 50, \"y\": 100, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"1\", \"x\": 50, \"y\": 150, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"2\", \"x\": 50, \"y\": 200, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"3\", \"x\": 100, \"y\": 150, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"4\", \"x\": 100, \"y\": 100, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"5\", \"x\": 150, \"y\": 150, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"6\", \"x\": 100, \"y\": 200, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"7\", \"x\": 200, \"y\": 200, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"8\", \"x\": 200, \"y\": 150, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"9\", \"x\": 250, \"y\": 200, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"10\", \"x\": 250, \"y\": 100, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"11\", \"x\": 300, \"y\": 100, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"12\", \"x\": 300, \"y\": 150, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"13\", \"x\": 300, \"y\": 200, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"14\", \"x\": 350, \"y\": 150, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"15\", \"x\": 400, \"y\": 150, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"16\", \"x\": 450, \"y\": 150, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"17\", \"x\": 450, \"y\": 100, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"18\", \"x\": 500, \"y\": 100, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"19\", \"x\": 500, \"y\": 200, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"20\", \"x\": 550, \"y\": 100, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"21\", \"x\": 550, \"y\": 150, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"22\", \"x\": 550, \"y\": 200, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}], \"links\": [{\"source\": \"0\", \"target\": \"4\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"1\", \"target\": \"3\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"2\", \"target\": \"6\", \"t\": 2, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"3\", \"target\": \"4\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"3\", \"target\": \"5\", \"t\": 2, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"4\", \"target\": \"10\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"5\", \"target\": \"8\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"6\", \"target\": \"7\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"7\", \"target\": \"8\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"7\", \"target\": \"9\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"8\", \"target\": \"12\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"9\", \"target\": \"10\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"9\", \"target\": \"13\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"10\", \"target\": \"11\", \"t\": 2, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"11\", \"target\": \"17\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"12\", \"target\": \"13\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"12\", \"target\": \"14\", \"t\": 2, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"13\", \"target\": \"19\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"14\", \"target\": \"15\", \"t\": 2, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"15\", \"target\": \"16\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"16\", \"target\": \"17\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"16\", \"target\": \"21\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"17\", \"target\": \"18\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"18\", \"target\": \"19\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"18\", \"target\": \"20\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"19\", \"target\": \"22\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}], \"pauli_web\": [{\"source\": 3, \"target\": 1, \"t\": \"Z\"}, {\"source\": 1, \"target\": 3, \"t\": \"Z\"}, {\"source\": 3, \"target\": 4, \"t\": \"Z\"}, {\"source\": 4, \"target\": 3, \"t\": \"Z\"}, {\"source\": 3, \"target\": 5, \"t\": \"Z\"}, {\"source\": 5, \"target\": 3, \"t\": \"X\"}, {\"source\": 5, \"target\": 8, \"t\": \"X\"}, {\"source\": 6, \"target\": 2, \"t\": \"X\"}, {\"source\": 2, \"target\": 6, \"t\": \"Z\"}, {\"source\": 6, \"target\": 7, \"t\": \"X\"}, {\"source\": 7, \"target\": 6, \"t\": \"X\"}, {\"source\": 8, \"target\": 5, \"t\": \"X\"}, {\"source\": 8, \"target\": 7, \"t\": \"X\"}, {\"source\": 7, \"target\": 8, \"t\": \"X\"}, {\"source\": 8, \"target\": 12, \"t\": \"X\"}, {\"source\": 12, \"target\": 8, \"t\": \"X\"}, {\"source\": 11, \"target\": 10, \"t\": \"X\"}, {\"source\": 10, \"target\": 11, \"t\": \"Z\"}, {\"source\": 11, \"target\": 17, \"t\": \"X\"}, {\"source\": 15, \"target\": 14, \"t\": \"X\"}, {\"source\": 14, \"target\": 15, \"t\": \"Z\"}, {\"source\": 15, \"target\": 16, \"t\": \"X\"}, {\"source\": 16, \"target\": 15, \"t\": \"X\"}, {\"source\": 17, \"target\": 11, \"t\": \"X\"}, {\"source\": 17, \"target\": 16, \"t\": \"X\"}, {\"source\": 16, \"target\": 17, \"t\": \"X\"}, {\"source\": 17, \"target\": 18, \"t\": \"X\"}, {\"source\": 18, \"target\": 17, \"t\": \"X\"}, {\"source\": 4, \"target\": 10, \"t\": \"Z\"}, {\"source\": 10, \"target\": 4, \"t\": \"Z\"}, {\"source\": 12, \"target\": 14, \"t\": \"X\"}, {\"source\": 14, \"target\": 12, \"t\": \"Z\"}, {\"source\": 20, \"target\": 18, \"t\": \"X\"}, {\"source\": 18, \"target\": 20, \"t\": \"X\"}]}'), 600, 250, 50,\n",
-       "  10.0, false, true, '');\n",
+       "showGraph('#graph-output-XUTYGp4X',\n",
+       "  JSON.parse('{\"nodes\": [{\"name\": \"0\", \"x\": 28.571428571428573, \"y\": 57.142857142857146, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"1\", \"x\": 28.571428571428573, \"y\": 85.71428571428572, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"2\", \"x\": 28.571428571428573, \"y\": 114.28571428571429, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"3\", \"x\": 28.571428571428573, \"y\": 142.85714285714286, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"4\", \"x\": 57.142857142857146, \"y\": 114.28571428571429, \"t\": 1, \"phase\": \"\\u03c0/2\", \"ground\": false, \"vdata\": []}, {\"name\": \"5\", \"x\": 85.71428571428572, \"y\": 114.28571428571429, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"6\", \"x\": 85.71428571428572, \"y\": 85.71428571428572, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"7\", \"x\": 114.28571428571429, \"y\": 142.85714285714286, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"8\", \"x\": 114.28571428571429, \"y\": 114.28571428571429, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"9\", \"x\": 142.85714285714286, \"y\": 85.71428571428572, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"10\", \"x\": 171.42857142857144, \"y\": 114.28571428571429, \"t\": 1, \"phase\": \"\\u03c0/2\", \"ground\": false, \"vdata\": []}, {\"name\": \"11\", \"x\": 200.0, \"y\": 57.142857142857146, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"12\", \"x\": 200.0, \"y\": 142.85714285714286, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"13\", \"x\": 228.57142857142858, \"y\": 142.85714285714286, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"14\", \"x\": 228.57142857142858, \"y\": 114.28571428571429, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"15\", \"x\": 257.14285714285717, \"y\": 57.142857142857146, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"16\", \"x\": 257.14285714285717, \"y\": 114.28571428571429, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"17\", \"x\": 285.7142857142857, \"y\": 57.142857142857146, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"18\", \"x\": 314.2857142857143, \"y\": 114.28571428571429, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"19\", \"x\": 314.2857142857143, \"y\": 85.71428571428572, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"20\", \"x\": 342.8571428571429, \"y\": 142.85714285714286, \"t\": 1, \"phase\": \"\\u03c0/2\", \"ground\": false, \"vdata\": []}, {\"name\": \"21\", \"x\": 371.42857142857144, \"y\": 85.71428571428572, \"t\": 1, \"phase\": \"\\u03c0/2\", \"ground\": false, \"vdata\": []}, {\"name\": \"22\", \"x\": 400.0, \"y\": 142.85714285714286, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"23\", \"x\": 400.0, \"y\": 57.142857142857146, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"24\", \"x\": 428.5714285714286, \"y\": 57.142857142857146, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"25\", \"x\": 457.14285714285717, \"y\": 85.71428571428572, \"t\": 1, \"phase\": \"\\u03c0/2\", \"ground\": false, \"vdata\": []}, {\"name\": \"26\", \"x\": 485.7142857142857, \"y\": 85.71428571428572, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"27\", \"x\": 485.7142857142857, \"y\": 114.28571428571429, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"28\", \"x\": 514.2857142857143, \"y\": 57.142857142857146, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"29\", \"x\": 514.2857142857143, \"y\": 114.28571428571429, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"30\", \"x\": 542.8571428571429, \"y\": 114.28571428571429, \"t\": 1, \"phase\": \"\\u03c0/2\", \"ground\": false, \"vdata\": []}, {\"name\": \"31\", \"x\": 571.4285714285714, \"y\": 57.142857142857146, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"32\", \"x\": 571.4285714285714, \"y\": 85.71428571428572, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"33\", \"x\": 600.0, \"y\": 114.28571428571429, \"t\": 1, \"phase\": \"\\u03c0/2\", \"ground\": false, \"vdata\": []}, {\"name\": \"34\", \"x\": 628.5714285714286, \"y\": 85.71428571428572, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"35\", \"x\": 657.1428571428572, \"y\": 85.71428571428572, \"t\": 1, \"phase\": \"\\u03c0/2\", \"ground\": false, \"vdata\": []}, {\"name\": \"36\", \"x\": 685.7142857142858, \"y\": 57.142857142857146, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": 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@@ -10456,7 +14277,7 @@
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      "data": {
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-       "<div style=\"overflow:auto; background-color: white\" id=\"graph-output-46xQscEI\"></div>\n",
+       "<div style=\"overflow:auto; background-color: white\" id=\"graph-output-2sdV0X4b\"></div>\n",
        "<script type=\"module\">\n",
        "var d3;\n",
        "if (d3 == null) { d3 = await import(\"https://cdn.skypack.dev/d3@5\"); }\n",
@@ -10819,9 +14640,9 @@
        "}\n",
        "\n",
        "\n",
-       "showGraph('#graph-output-46xQscEI',\n",
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-       "  10.0, false, true, '');\n",
+       "showGraph('#graph-output-2sdV0X4b',\n",
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-       "<div style=\"overflow:auto; background-color: white\" id=\"graph-output-aFG0z1FU\"></div>\n",
+       "<div style=\"overflow:auto; background-color: white\" id=\"graph-output-HBhv2cyB\"></div>\n",
        "<script type=\"module\">\n",
        "var d3;\n",
        "if (d3 == null) { d3 = await import(\"https://cdn.skypack.dev/d3@5\"); }\n",
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\"target\": 8, \"t\": \"X\"}, {\"source\": 10, \"target\": 14, \"t\": \"Y\"}, {\"source\": 12, \"target\": 7, \"t\": \"X\"}, {\"source\": 7, \"target\": 12, \"t\": \"X\"}, {\"source\": 12, \"target\": 11, \"t\": \"X\"}, {\"source\": 11, \"target\": 12, \"t\": \"X\"}, {\"source\": 12, \"target\": 13, \"t\": \"X\"}, {\"source\": 13, \"target\": 12, \"t\": \"X\"}, {\"source\": 14, \"target\": 10, \"t\": \"Y\"}, {\"source\": 14, \"target\": 13, \"t\": \"X\"}, {\"source\": 13, \"target\": 14, \"t\": \"X\"}, {\"source\": 14, \"target\": 16, \"t\": \"Y\"}, {\"source\": 16, \"target\": 14, \"t\": \"Y\"}, {\"source\": 16, \"target\": 15, \"t\": \"X\"}, {\"source\": 15, \"target\": 16, \"t\": \"X\"}, {\"source\": 16, \"target\": 18, \"t\": \"Y\"}, {\"source\": 18, \"target\": 16, \"t\": \"Y\"}, {\"source\": 18, \"target\": 19, \"t\": \"Y\"}, {\"source\": 19, \"target\": 18, \"t\": \"Y\"}, {\"source\": 18, \"target\": 27, \"t\": \"Z\"}, {\"source\": 27, \"target\": 18, \"t\": \"Z\"}, {\"source\": 19, \"target\": 9, \"t\": \"X\"}, {\"source\": 19, \"target\": 21, \"t\": \"Y\"}, {\"source\": 21, \"target\": 19, \"t\": \"Y\"}, {\"source\": 21, \"target\": 25, \"t\": \"X\"}, {\"source\": 25, \"target\": 21, \"t\": \"X\"}, {\"source\": 25, \"target\": 26, \"t\": \"Y\"}, {\"source\": 26, \"target\": 25, \"t\": \"Y\"}, {\"source\": 26, \"target\": 27, \"t\": \"Z\"}, {\"source\": 27, \"target\": 26, \"t\": \"Z\"}, {\"source\": 26, \"target\": 32, \"t\": \"Y\"}, {\"source\": 32, \"target\": 26, \"t\": \"Y\"}, {\"source\": 32, \"target\": 31, \"t\": \"X\"}, {\"source\": 31, \"target\": 32, \"t\": \"X\"}, {\"source\": 32, \"target\": 34, \"t\": \"Y\"}, {\"source\": 34, \"target\": 32, \"t\": \"Y\"}, {\"source\": 34, \"target\": 35, \"t\": \"Y\"}, {\"source\": 35, \"target\": 34, \"t\": \"Y\"}, {\"source\": 35, \"target\": 37, \"t\": \"X\"}, {\"source\": 37, \"target\": 35, \"t\": \"X\"}, {\"source\": 37, \"target\": 36, \"t\": \"X\"}, {\"source\": 36, \"target\": 37, \"t\": \"X\"}, {\"source\": 37, \"target\": 41, \"t\": \"X\"}, {\"source\": 41, \"target\": 37, \"t\": \"X\"}, {\"source\": 11, \"target\": 15, \"t\": \"X\"}, {\"source\": 15, \"target\": 11, \"t\": \"X\"}, {\"source\": 31, \"target\": 36, \"t\": \"X\"}, {\"source\": 36, \"target\": 31, \"t\": \"X\"}, {\"source\": 1, \"target\": 6, \"t\": \"Z\"}, {\"source\": 6, \"target\": 1, \"t\": \"Z\"}]}'), 800.0, 171.42857142857144, 28.571428571428573,\n",
+       "  5.714285714285715, false, false, '');\n",
        "</script>"
       ],
       "text/plain": [
@@ -11208,11 +15029,31 @@
      },
      "metadata": {},
      "output_type": "display_data"
-    },
+    }
+   ],
+   "source": [
+    "# Since we can handle CNOT, H, and S, we can now compute Pauli webs for arbitrary Clifford\n",
+    "# unitaries.\n",
+    "\n",
+    "random.seed(1337)\n",
+    "c = zx.generate.CNOT_HAD_PHASE_circuit(qubits=4, depth=25, clifford=True)\n",
+    "g = c.to_graph(compress_rows=False)\n",
+    "zx.draw(g, labels=True)\n",
+    "\n",
+    "order, zwebs, xwebs = compute_pauli_webs(g)\n",
+    "zx.draw(g, pauli_web=zwebs[41])\n",
+    "zx.draw(g, pauli_web=xwebs[41])"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 36,
+   "metadata": {},
+   "outputs": [
     {
      "data": {
       "text/html": [
-       "<div style=\"overflow:auto; background-color: white\" id=\"graph-output-FP4Hif6E\"></div>\n",
+       "<div style=\"overflow:auto; background-color: white\" id=\"graph-output-PhyhXvSw\"></div>\n",
        "<script type=\"module\">\n",
        "var d3;\n",
        "if (d3 == null) { d3 = await import(\"https://cdn.skypack.dev/d3@5\"); }\n",
@@ -11575,8 +15416,8 @@
        "}\n",
        "\n",
        "\n",
-       "showGraph('#graph-output-FP4Hif6E',\n",
-       "  JSON.parse('{\"nodes\": [{\"name\": \"0\", \"x\": 50, \"y\": 100, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"1\", \"x\": 50, \"y\": 150, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"2\", \"x\": 50, \"y\": 200, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"3\", \"x\": 100, \"y\": 150, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"4\", \"x\": 100, \"y\": 100, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"5\", \"x\": 150, \"y\": 150, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"6\", \"x\": 100, \"y\": 200, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"7\", \"x\": 200, \"y\": 200, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"8\", \"x\": 200, \"y\": 150, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"9\", \"x\": 250, \"y\": 200, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"10\", \"x\": 250, \"y\": 100, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"11\", \"x\": 300, \"y\": 100, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"12\", \"x\": 300, \"y\": 150, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"13\", \"x\": 300, \"y\": 200, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"14\", \"x\": 350, \"y\": 150, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"15\", \"x\": 400, \"y\": 150, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"16\", \"x\": 450, \"y\": 150, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"17\", \"x\": 450, \"y\": 100, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"18\", \"x\": 500, \"y\": 100, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"19\", \"x\": 500, \"y\": 200, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"20\", \"x\": 550, \"y\": 100, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"21\", \"x\": 550, \"y\": 150, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"22\", \"x\": 550, \"y\": 200, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}], \"links\": [{\"source\": \"0\", \"target\": \"4\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"1\", \"target\": \"3\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"2\", \"target\": \"6\", \"t\": 2, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"3\", \"target\": \"4\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"3\", \"target\": \"5\", \"t\": 2, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"4\", \"target\": \"10\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"5\", \"target\": \"8\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"6\", \"target\": \"7\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"7\", \"target\": \"8\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"7\", \"target\": \"9\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"8\", \"target\": \"12\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"9\", \"target\": \"10\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"9\", \"target\": \"13\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"10\", \"target\": \"11\", \"t\": 2, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"11\", \"target\": \"17\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"12\", \"target\": \"13\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"12\", \"target\": \"14\", \"t\": 2, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"13\", \"target\": \"19\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"14\", \"target\": \"15\", \"t\": 2, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"15\", \"target\": \"16\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"16\", \"target\": \"17\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"16\", \"target\": \"21\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"17\", \"target\": \"18\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"18\", \"target\": \"19\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"18\", \"target\": \"20\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"19\", \"target\": \"22\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}], \"pauli_web\": [{\"source\": 4, \"target\": 0, \"t\": \"Y\"}, {\"source\": 0, \"target\": 4, \"t\": \"Y\"}, {\"source\": 4, \"target\": 3, \"t\": \"X\"}, {\"source\": 3, \"target\": 4, \"t\": \"X\"}, {\"source\": 4, \"target\": 10, \"t\": \"Y\"}, {\"source\": 6, \"target\": 2, \"t\": \"Y\"}, {\"source\": 2, \"target\": 6, \"t\": \"Y\"}, {\"source\": 6, \"target\": 7, \"t\": \"Y\"}, {\"source\": 7, \"target\": 6, \"t\": \"Y\"}, {\"source\": 7, \"target\": 8, \"t\": \"Z\"}, {\"source\": 8, \"target\": 7, \"t\": \"Z\"}, {\"source\": 7, \"target\": 9, \"t\": \"Y\"}, {\"source\": 9, \"target\": 7, \"t\": \"Y\"}, {\"source\": 9, \"target\": 10, \"t\": \"Y\"}, {\"source\": 10, \"target\": 9, \"t\": \"Y\"}, {\"source\": 9, \"target\": 13, \"t\": \"Z\"}, {\"source\": 13, \"target\": 9, \"t\": \"Z\"}, {\"source\": 10, \"target\": 4, \"t\": \"Y\"}, {\"source\": 10, \"target\": 11, \"t\": \"X\"}, {\"source\": 11, \"target\": 10, \"t\": \"Z\"}, {\"source\": 12, \"target\": 8, \"t\": \"Z\"}, {\"source\": 8, \"target\": 12, \"t\": \"Z\"}, {\"source\": 12, \"target\": 13, \"t\": \"Z\"}, {\"source\": 13, \"target\": 12, \"t\": \"Z\"}, {\"source\": 12, \"target\": 14, \"t\": \"Z\"}, {\"source\": 14, \"target\": 12, \"t\": \"X\"}, {\"source\": 14, \"target\": 15, \"t\": \"X\"}, {\"source\": 15, \"target\": 14, \"t\": \"Z\"}, {\"source\": 16, \"target\": 15, \"t\": \"Z\"}, {\"source\": 15, \"target\": 16, \"t\": \"Z\"}, {\"source\": 16, \"target\": 17, \"t\": \"Z\"}, {\"source\": 17, \"target\": 16, \"t\": \"Z\"}, {\"source\": 16, \"target\": 21, \"t\": \"Z\"}, {\"source\": 21, \"target\": 16, \"t\": \"Z\"}, {\"source\": 11, \"target\": 17, \"t\": \"Z\"}, {\"source\": 17, \"target\": 11, \"t\": \"Z\"}, {\"source\": 3, \"target\": 1, \"t\": \"X\"}, {\"source\": 1, \"target\": 3, \"t\": \"X\"}]}'), 600, 250, 50,\n",
+       "showGraph('#graph-output-PhyhXvSw',\n",
+       "  JSON.parse('{\"nodes\": [{\"name\": \"0\", \"x\": 50, \"y\": 100, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"1\", \"x\": 50, \"y\": 150, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"2\", \"x\": 50, \"y\": 200, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"3\", \"x\": 50, \"y\": 250, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"4\", \"x\": 150, \"y\": 200, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"5\", \"x\": 150, \"y\": 150, \"t\": 1, \"phase\": \"3\\u03c0/2\", \"ground\": false, \"vdata\": []}, {\"name\": \"11\", \"x\": 100, \"y\": 100, \"t\": 1, \"phase\": \"\\u03c0\", \"ground\": false, \"vdata\": []}, {\"name\": \"13\", \"x\": 150, \"y\": 250, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"27\", \"x\": 200, \"y\": 200, \"t\": 1, \"phase\": \"3\\u03c0/2\", \"ground\": false, \"vdata\": []}, {\"name\": \"34\", \"x\": 200, \"y\": 150, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"38\", \"x\": 200, \"y\": 250, \"t\": 1, \"phase\": \"\\u03c0\", \"ground\": false, \"vdata\": []}, {\"name\": \"40\", \"x\": 250, \"y\": 100, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"41\", \"x\": 250, \"y\": 150, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"42\", \"x\": 250, \"y\": 200, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"43\", \"x\": 250, \"y\": 250, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"44\", \"x\": 200, \"y\": 100, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}], \"links\": [{\"source\": \"0\", \"target\": \"11\", \"t\": 2, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"1\", \"target\": \"5\", \"t\": 2, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"2\", \"target\": \"4\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"3\", \"target\": \"13\", \"t\": 2, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"4\", \"target\": \"34\", \"t\": 2, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"4\", \"target\": \"11\", \"t\": 2, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"5\", \"target\": \"27\", \"t\": 2, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"5\", \"target\": \"34\", \"t\": 2, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"5\", \"target\": \"11\", \"t\": 2, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"11\", \"target\": \"13\", \"t\": 2, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"11\", \"target\": \"44\", \"t\": 2, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"13\", \"target\": \"38\", \"t\": 2, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"27\", \"target\": \"42\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"27\", \"target\": \"34\", \"t\": 2, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"34\", \"target\": \"41\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"38\", \"target\": \"43\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"40\", \"target\": \"44\", \"t\": 2, \"index\": 0, \"num_parallel\": 1}], \"pauli_web\": []}'), 300, 300, 50,\n",
        "  10.0, false, true, '');\n",
        "</script>"
       ],
@@ -11590,7 +15431,7 @@
     {
      "data": {
       "text/html": [
-       "<div style=\"overflow:auto; background-color: white\" id=\"graph-output-yEWMgsWW\"></div>\n",
+       "<div style=\"overflow:auto; background-color: white\" id=\"graph-output-dqsaMAi2\"></div>\n",
        "<script type=\"module\">\n",
        "var d3;\n",
        "if (d3 == null) { d3 = await import(\"https://cdn.skypack.dev/d3@5\"); }\n",
@@ -11953,9 +15794,9 @@
        "}\n",
        "\n",
        "\n",
-       "showGraph('#graph-output-yEWMgsWW',\n",
-       "  JSON.parse('{\"nodes\": [{\"name\": \"0\", \"x\": 50, \"y\": 100, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"1\", \"x\": 50, \"y\": 150, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"2\", \"x\": 50, \"y\": 200, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"3\", \"x\": 100, \"y\": 150, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"4\", \"x\": 100, \"y\": 100, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"5\", \"x\": 150, \"y\": 150, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"6\", \"x\": 100, \"y\": 200, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"7\", \"x\": 200, \"y\": 200, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"8\", \"x\": 200, \"y\": 150, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"9\", \"x\": 250, 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false, \"vdata\": []}, {\"name\": \"19\", \"x\": 500, \"y\": 200, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"20\", \"x\": 550, \"y\": 100, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"21\", \"x\": 550, \"y\": 150, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"22\", \"x\": 550, \"y\": 200, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}], \"links\": [{\"source\": \"0\", \"target\": \"4\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"1\", \"target\": \"3\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"2\", \"target\": \"6\", \"t\": 2, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"3\", \"target\": \"4\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"3\", \"target\": \"5\", \"t\": 2, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"4\", \"target\": \"10\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"5\", \"target\": \"8\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"6\", \"target\": \"7\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"7\", \"target\": \"8\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"7\", \"target\": \"9\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"8\", \"target\": \"12\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"9\", \"target\": \"10\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"9\", \"target\": \"13\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"10\", \"target\": \"11\", \"t\": 2, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"11\", \"target\": \"17\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"12\", \"target\": \"13\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"12\", \"target\": \"14\", \"t\": 2, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"13\", \"target\": \"19\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"14\", \"target\": \"15\", \"t\": 2, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"15\", \"target\": \"16\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"16\", \"target\": \"17\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"16\", \"target\": \"21\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"17\", \"target\": \"18\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"18\", \"target\": \"19\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"18\", \"target\": \"20\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"19\", \"target\": \"22\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}], \"pauli_web\": [{\"source\": 6, \"target\": 2, \"t\": \"X\"}, {\"source\": 2, \"target\": 6, \"t\": \"Z\"}, {\"source\": 6, \"target\": 7, \"t\": \"X\"}, {\"source\": 7, \"target\": 6, \"t\": \"X\"}, {\"source\": 11, \"target\": 10, \"t\": \"X\"}, {\"source\": 10, \"target\": 11, \"t\": \"Z\"}, {\"source\": 11, \"target\": 17, \"t\": \"X\"}, {\"source\": 13, \"target\": 9, \"t\": \"X\"}, {\"source\": 9, \"target\": 13, \"t\": \"X\"}, {\"source\": 13, \"target\": 12, \"t\": \"X\"}, {\"source\": 12, \"target\": 13, \"t\": \"X\"}, {\"source\": 13, \"target\": 19, \"t\": \"X\"}, {\"source\": 15, \"target\": 14, \"t\": \"X\"}, {\"source\": 14, \"target\": 15, \"t\": \"Z\"}, {\"source\": 15, \"target\": 16, \"t\": \"X\"}, {\"source\": 16, \"target\": 15, \"t\": \"X\"}, {\"source\": 17, \"target\": 11, \"t\": \"X\"}, {\"source\": 17, \"target\": 16, \"t\": \"X\"}, {\"source\": 16, \"target\": 17, \"t\": \"X\"}, {\"source\": 17, \"target\": 18, \"t\": \"X\"}, {\"source\": 18, \"target\": 17, \"t\": \"X\"}, {\"source\": 19, \"target\": 13, \"t\": \"X\"}, {\"source\": 19, \"target\": 18, \"t\": \"X\"}, {\"source\": 18, \"target\": 19, \"t\": \"X\"}, {\"source\": 19, \"target\": 22, \"t\": \"X\"}, {\"source\": 22, \"target\": 19, \"t\": \"X\"}, {\"source\": 4, \"target\": 10, \"t\": \"Z\"}, {\"source\": 10, \"target\": 4, \"t\": \"Z\"}, {\"source\": 7, \"target\": 9, \"t\": \"X\"}, {\"source\": 9, \"target\": 7, \"t\": \"X\"}, {\"source\": 12, \"target\": 14, \"t\": \"X\"}, {\"source\": 14, \"target\": 12, \"t\": \"Z\"}, {\"source\": 4, \"target\": 0, \"t\": \"Z\"}, {\"source\": 0, \"target\": 4, \"t\": \"Z\"}]}'), 600, 250, 50,\n",
-       "  10.0, false, true, '');\n",
+       "showGraph('#graph-output-dqsaMAi2',\n",
+       "  JSON.parse('{\"nodes\": [{\"name\": \"0\", \"x\": 50, \"y\": 100, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"1\", \"x\": 50, \"y\": 150, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"2\", \"x\": 50, \"y\": 200, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"3\", \"x\": 50, \"y\": 250, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"4\", \"x\": 150, \"y\": 200, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"5\", \"x\": 150, \"y\": 150, \"t\": 1, \"phase\": \"3\\u03c0/2\", \"ground\": false, \"vdata\": []}, {\"name\": \"11\", \"x\": 100, \"y\": 100, \"t\": 1, \"phase\": \"\\u03c0\", \"ground\": false, \"vdata\": []}, {\"name\": \"13\", \"x\": 150, \"y\": 250, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"27\", \"x\": 200, \"y\": 200, \"t\": 1, \"phase\": \"3\\u03c0/2\", \"ground\": false, \"vdata\": []}, {\"name\": \"34\", \"x\": 200, \"y\": 150, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"38\", \"x\": 200, \"y\": 250, \"t\": 1, \"phase\": \"\\u03c0\", \"ground\": false, \"vdata\": []}, {\"name\": \"40\", \"x\": 250, \"y\": 100, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"41\", \"x\": 250, \"y\": 150, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"42\", \"x\": 250, \"y\": 200, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"43\", \"x\": 250, \"y\": 250, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"44\", \"x\": 200, \"y\": 100, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}], \"links\": [{\"source\": \"0\", \"target\": \"11\", \"t\": 2, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"1\", \"target\": \"5\", \"t\": 2, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"2\", \"target\": \"4\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"3\", \"target\": \"13\", \"t\": 2, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"4\", \"target\": \"34\", \"t\": 2, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"4\", \"target\": \"11\", \"t\": 2, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"5\", \"target\": \"27\", \"t\": 2, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"5\", \"target\": \"34\", \"t\": 2, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"5\", \"target\": \"11\", \"t\": 2, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"11\", \"target\": \"13\", \"t\": 2, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"11\", \"target\": \"44\", \"t\": 2, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"13\", \"target\": \"38\", \"t\": 2, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"27\", \"target\": \"42\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"27\", \"target\": \"34\", \"t\": 2, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"34\", \"target\": \"41\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"38\", \"target\": \"43\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"40\", \"target\": \"44\", \"t\": 2, \"index\": 0, \"num_parallel\": 1}], \"pauli_web\": [{\"source\": 4, \"target\": 2, \"t\": \"X\"}, {\"source\": 2, \"target\": 4, \"t\": \"X\"}, {\"source\": 4, \"target\": 34, \"t\": \"X\"}, {\"source\": 34, \"target\": 4, \"t\": \"Z\"}, {\"source\": 4, \"target\": 11, \"t\": \"X\"}, {\"source\": 11, \"target\": 4, \"t\": \"Z\"}, {\"source\": 0, \"target\": 11, \"t\": \"X\"}, {\"source\": 11, \"target\": 0, \"t\": \"Z\"}, {\"source\": 41, \"target\": 34, \"t\": \"Z\"}, {\"source\": 34, \"target\": 41, \"t\": \"Z\"}]}'), 300, 300, 50,\n",
+       "  10.0, false, false, '');\n",
        "</script>"
       ],
       "text/plain": [
@@ -11968,7 +15809,7 @@
     {
      "data": {
       "text/html": [
-       "<div style=\"overflow:auto; background-color: white\" id=\"graph-output-KDIBSBRU\"></div>\n",
+       "<div style=\"overflow:auto; background-color: white\" id=\"graph-output-5wt6O8yB\"></div>\n",
        "<script type=\"module\">\n",
        "var d3;\n",
        "if (d3 == null) { d3 = await import(\"https://cdn.skypack.dev/d3@5\"); }\n",
@@ -12331,9 +16172,9 @@
        "}\n",
        "\n",
        "\n",
-       "showGraph('#graph-output-KDIBSBRU',\n",
-       "  JSON.parse('{\"nodes\": [{\"name\": \"0\", \"x\": 50, \"y\": 100, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"1\", \"x\": 50, \"y\": 150, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"2\", \"x\": 50, \"y\": 200, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"3\", \"x\": 100, \"y\": 150, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"4\", \"x\": 100, \"y\": 100, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"5\", \"x\": 150, \"y\": 150, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"6\", \"x\": 100, \"y\": 200, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"7\", \"x\": 200, \"y\": 200, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"8\", \"x\": 200, \"y\": 150, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"9\", \"x\": 250, 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{\"source\": \"14\", \"target\": \"15\", \"t\": 2, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"15\", \"target\": \"16\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"16\", \"target\": \"17\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"16\", \"target\": \"21\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"17\", \"target\": \"18\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"18\", \"target\": \"19\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"18\", \"target\": \"20\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"19\", \"target\": \"22\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}], \"pauli_web\": [{\"source\": 22, \"target\": 19, \"t\": \"Z\"}, {\"source\": 19, \"target\": 22, \"t\": \"Z\"}, {\"source\": 7, \"target\": 6, \"t\": \"Z\"}, {\"source\": 6, \"target\": 7, \"t\": \"Z\"}, {\"source\": 7, \"target\": 8, \"t\": \"Z\"}, {\"source\": 8, \"target\": 7, \"t\": \"Z\"}, {\"source\": 7, \"target\": 9, \"t\": \"Z\"}, {\"source\": 9, \"target\": 7, \"t\": \"Z\"}, {\"source\": 9, \"target\": 10, \"t\": \"Z\"}, {\"source\": 10, \"target\": 9, \"t\": \"Z\"}, {\"source\": 9, \"target\": 13, \"t\": \"Z\"}, {\"source\": 13, \"target\": 9, \"t\": \"Z\"}, {\"source\": 3, \"target\": 5, \"t\": \"X\"}, {\"source\": 5, \"target\": 3, \"t\": \"Z\"}, {\"source\": 4, \"target\": 10, \"t\": \"Z\"}, {\"source\": 10, \"target\": 4, \"t\": \"Z\"}, {\"source\": 5, \"target\": 8, \"t\": \"Z\"}, {\"source\": 8, \"target\": 5, \"t\": \"Z\"}, {\"source\": 13, \"target\": 19, \"t\": \"Z\"}, {\"source\": 19, \"target\": 13, \"t\": \"Z\"}, {\"source\": 4, \"target\": 0, \"t\": \"Z\"}, {\"source\": 0, \"target\": 4, \"t\": \"Z\"}, {\"source\": 3, \"target\": 1, \"t\": \"X\"}, {\"source\": 1, \"target\": 3, \"t\": \"X\"}, {\"source\": 6, \"target\": 2, \"t\": \"Z\"}, {\"source\": 2, \"target\": 6, \"t\": \"X\"}]}'), 600, 250, 50,\n",
-       "  10.0, false, true, '');\n",
+       "showGraph('#graph-output-5wt6O8yB',\n",
+       "  JSON.parse('{\"nodes\": [{\"name\": \"0\", \"x\": 50, \"y\": 100, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"1\", \"x\": 50, \"y\": 150, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"2\", \"x\": 50, \"y\": 200, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"3\", \"x\": 50, \"y\": 250, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"4\", \"x\": 150, \"y\": 200, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"5\", \"x\": 150, \"y\": 150, \"t\": 1, \"phase\": \"3\\u03c0/2\", \"ground\": false, \"vdata\": []}, {\"name\": \"11\", \"x\": 100, \"y\": 100, \"t\": 1, \"phase\": \"\\u03c0\", \"ground\": false, \"vdata\": []}, {\"name\": \"13\", \"x\": 150, \"y\": 250, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"27\", \"x\": 200, \"y\": 200, \"t\": 1, \"phase\": \"3\\u03c0/2\", \"ground\": false, \"vdata\": []}, {\"name\": \"34\", \"x\": 200, \"y\": 150, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"38\", \"x\": 200, \"y\": 250, \"t\": 1, \"phase\": \"\\u03c0\", \"ground\": false, \"vdata\": []}, {\"name\": \"40\", \"x\": 250, \"y\": 100, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"41\", \"x\": 250, \"y\": 150, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"42\", \"x\": 250, \"y\": 200, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"43\", \"x\": 250, \"y\": 250, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"44\", \"x\": 200, \"y\": 100, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}], \"links\": [{\"source\": \"0\", \"target\": \"11\", \"t\": 2, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"1\", \"target\": \"5\", \"t\": 2, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"2\", \"target\": \"4\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"3\", \"target\": \"13\", \"t\": 2, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"4\", \"target\": \"34\", \"t\": 2, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"4\", \"target\": \"11\", \"t\": 2, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"5\", \"target\": \"27\", \"t\": 2, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"5\", \"target\": \"34\", \"t\": 2, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"5\", \"target\": \"11\", \"t\": 2, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"11\", \"target\": \"13\", \"t\": 2, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"11\", \"target\": \"44\", \"t\": 2, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"13\", \"target\": \"38\", \"t\": 2, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"27\", \"target\": \"42\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"27\", \"target\": \"34\", \"t\": 2, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"34\", \"target\": \"41\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"38\", \"target\": \"43\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"40\", \"target\": \"44\", \"t\": 2, \"index\": 0, \"num_parallel\": 1}], \"pauli_web\": [{\"source\": 34, \"target\": 41, \"t\": \"X\"}, {\"source\": 41, \"target\": 34, \"t\": \"X\"}, {\"source\": 34, \"target\": 5, \"t\": \"Y\"}, {\"source\": 5, \"target\": 34, \"t\": \"Y\"}, {\"source\": 34, \"target\": 27, \"t\": \"X\"}, {\"source\": 27, \"target\": 34, \"t\": \"Z\"}, {\"source\": 34, \"target\": 4, \"t\": \"Y\"}, {\"source\": 4, \"target\": 34, \"t\": \"Y\"}, {\"source\": 4, \"target\": 2, \"t\": \"Y\"}, {\"source\": 2, \"target\": 4, \"t\": \"Y\"}, {\"source\": 4, \"target\": 11, \"t\": \"X\"}, {\"source\": 11, \"target\": 4, \"t\": \"Z\"}, {\"source\": 5, \"target\": 1, \"t\": \"Y\"}, {\"source\": 1, \"target\": 5, \"t\": \"Y\"}, {\"source\": 5, \"target\": 27, \"t\": \"X\"}, {\"source\": 27, \"target\": 5, \"t\": \"Z\"}, {\"source\": 5, \"target\": 11, \"t\": \"X\"}, {\"source\": 11, \"target\": 5, \"t\": \"Z\"}]}'), 300, 300, 50,\n",
+       "  10.0, false, false, '');\n",
        "</script>"
       ],
       "text/plain": [
@@ -12345,21 +16186,22 @@
     }
    ],
    "source": [
-    "# Note the we get some Y-edges appear, which are shown in a third color\n",
-    "order, zwebs, xwebs = compute_pauli_webs(g)\n",
-    "zx.draw(g, labels=True, pauli_web=xwebs[20])\n",
-    "zx.draw(g, labels=True, pauli_web=zwebs[20])\n",
-    "zx.draw(g, labels=True, pauli_web=xwebs[21])\n",
-    "zx.draw(g, labels=True, pauli_web=zwebs[21])\n",
-    "zx.draw(g, labels=True, pauli_web=xwebs[22])\n",
-    "zx.draw(g, labels=True, pauli_web=zwebs[22])"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {},
-   "source": [
-    "I'm not sure the best way to handle S gates yet. This seems to depend on how you interpret the Pauli web data. The \"MBQC\"-style interpretation, which works for arbitrary circuits, says the Pauli web tells us how to push Paulis through the circuit, but some of the phases (and hence the circuit itself) might change. This makes sense if you are choosing the angles on-the-fly, and can vary them based on previous errors. The \"stabiliser theory\" style, which only works for Clifford circuits, tells us how to push Paulis through the circuit *without* changing it. That is, it visualises the computation of the stabiliser tableau."
+    "# As before, we can simplify the diagram without changing how Pauli webs behave on the boundaries\n",
+    "g1 = g.copy()\n",
+    "\n",
+    "# Calling full_reduce will turn any Clifford circuit into GSLC form, consisting of an input row\n",
+    "# of Clifford Z-spiders connected to an output row of Clifford Z-spiders via Hadamard edges.\n",
+    "# Inputs and outputs can be Hadamard or normal edges.\n",
+    "zx.full_reduce(g1)\n",
+    "g1.normalize()\n",
+    "\n",
+    "# nudge some nodes to left, for visibility\n",
+    "for v in [0,1,2,3,11]: g1.set_row(v, g1.row(v)-1)\n",
+    "zx.draw(g1, labels=True)\n",
+    "\n",
+    "order, zwebs, xwebs = compute_pauli_webs(g1)\n",
+    "zx.draw(g1, pauli_web=zwebs[41])\n",
+    "zx.draw(g1, pauli_web=xwebs[41])"
    ]
   },
   {
@@ -12371,13 +16213,13 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 13,
+   "execution_count": 16,
    "metadata": {},
    "outputs": [
     {
      "data": {
       "text/html": [
-       "<div style=\"overflow:auto; background-color: white\" id=\"graph-output-JYvPrIqx\"></div>\n",
+       "<div style=\"overflow:auto; background-color: white\" id=\"graph-output-NlydUCU0\"></div>\n",
        "<script type=\"module\">\n",
        "var d3;\n",
        "if (d3 == null) { d3 = await import(\"https://cdn.skypack.dev/d3@5\"); }\n",
@@ -12740,7 +16582,7 @@
        "}\n",
        "\n",
        "\n",
-       "showGraph('#graph-output-JYvPrIqx',\n",
+       "showGraph('#graph-output-NlydUCU0',\n",
        "  JSON.parse('{\"nodes\": [{\"name\": \"0\", \"x\": 26.666666666666668, \"y\": 53.333333333333336, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"1\", \"x\": 26.666666666666668, \"y\": 80.0, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"2\", \"x\": 26.666666666666668, \"y\": 106.66666666666667, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"3\", \"x\": 53.333333333333336, \"y\": 80.0, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"4\", \"x\": 53.333333333333336, \"y\": 53.333333333333336, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"5\", \"x\": 80.0, \"y\": 53.333333333333336, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"6\", \"x\": 80.0, \"y\": 80.0, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"7\", \"x\": 106.66666666666667, \"y\": 53.333333333333336, \"t\": 2, \"phase\": 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{\"name\": \"62\", \"x\": 773.3333333333334, \"y\": 53.333333333333336, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"63\", \"x\": 773.3333333333334, \"y\": 80.0, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"64\", \"x\": 773.3333333333334, \"y\": 106.66666666666667, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}], \"links\": [{\"source\": \"0\", \"target\": \"4\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"1\", \"target\": \"3\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"2\", \"target\": \"8\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"3\", \"target\": \"4\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"3\", \"target\": \"6\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"4\", \"target\": \"5\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"5\", \"target\": \"6\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"5\", \"target\": \"7\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"6\", \"target\": \"10\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"7\", \"target\": \"8\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"7\", \"target\": \"13\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"8\", \"target\": \"9\", \"t\": 2, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"9\", \"target\": \"11\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"10\", \"target\": \"11\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"10\", \"target\": \"14\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"11\", \"target\": \"12\", \"t\": 2, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"12\", \"target\": \"15\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"13\", \"target\": \"14\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"13\", \"target\": \"16\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"14\", \"target\": \"18\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"15\", \"target\": \"16\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"15\", \"target\": \"17\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"16\", \"target\": \"19\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"17\", \"target\": \"18\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"17\", \"target\": \"21\", \"t\": 2, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"18\", \"target\": \"20\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"19\", \"target\": \"24\", \"t\": 2, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"20\", \"target\": \"23\", \"t\": 2, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"21\", \"target\": \"22\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"22\", \"target\": \"25\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"23\", \"target\": \"30\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"24\", \"target\": \"26\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"25\", \"target\": \"26\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"25\", \"target\": \"27\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"26\", \"target\": \"28\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"27\", \"target\": \"28\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"27\", \"target\": \"31\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"28\", \"target\": \"29\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"29\", \"target\": \"30\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"29\", \"target\": \"34\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"30\", \"target\": \"33\", \"t\": 2, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"31\", \"target\": \"32\", 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1}, {\"source\": \"40\", \"target\": \"41\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"41\", \"target\": \"42\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"41\", \"target\": \"43\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"42\", \"target\": \"47\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"43\", \"target\": \"44\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"44\", \"target\": \"45\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"45\", \"target\": \"46\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"46\", \"target\": \"47\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"46\", \"target\": \"49\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"47\", \"target\": \"48\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"48\", \"target\": \"49\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"48\", \"target\": \"50\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"49\", \"target\": \"53\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"50\", \"target\": \"58\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"51\", \"target\": \"52\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"52\", \"target\": \"54\", \"t\": 2, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"53\", \"target\": \"56\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"54\", \"target\": \"55\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"55\", \"target\": \"57\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"56\", \"target\": \"57\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"56\", \"target\": \"60\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"57\", \"target\": \"59\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"58\", \"target\": \"59\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"58\", \"target\": \"64\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"59\", \"target\": \"61\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"60\", \"target\": \"61\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"60\", \"target\": \"62\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"61\", \"target\": \"63\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}], \"pauli_web\": []}'), 800.0, 133.33333333333334, 26.666666666666668,\n",
        "  5.333333333333334, false, false, '');\n",
        "</script>"
@@ -12763,13 +16605,13 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 14,
+   "execution_count": 17,
    "metadata": {},
    "outputs": [
     {
      "data": {
       "text/html": [
-       "<div style=\"overflow:auto; background-color: white\" id=\"graph-output-rHt1Qglh\"></div>\n",
+       "<div style=\"overflow:auto; background-color: white\" id=\"graph-output-vUMr2sN0\"></div>\n",
        "<script type=\"module\">\n",
        "var d3;\n",
        "if (d3 == null) { d3 = await import(\"https://cdn.skypack.dev/d3@5\"); }\n",
@@ -13132,7 +16974,7 @@
        "}\n",
        "\n",
        "\n",
-       "showGraph('#graph-output-rHt1Qglh',\n",
+       "showGraph('#graph-output-vUMr2sN0',\n",
        "  JSON.parse('{\"nodes\": [{\"name\": \"0\", \"x\": 50, \"y\": 100, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"1\", \"x\": 50, \"y\": 150, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"2\", \"x\": 50, \"y\": 200, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"3\", \"x\": 100, \"y\": 150, \"t\": 2, \"phase\": \"\\u03c0/2\", \"ground\": false, \"vdata\": []}, {\"name\": \"4\", \"x\": 100, \"y\": 100, \"t\": 2, \"phase\": \"\\u03c0/2\", \"ground\": false, \"vdata\": []}, {\"name\": \"5\", \"x\": 100, \"y\": 200, \"t\": 1, \"phase\": \"\\u03c0/2\", \"ground\": false, \"vdata\": []}, {\"name\": \"14\", \"x\": 150, \"y\": 150, \"t\": 1, \"phase\": \"5\\u03c0/4\", \"ground\": false, \"vdata\": []}, {\"name\": \"16\", \"x\": 200, \"y\": 100, \"t\": 2, \"phase\": \"7\\u03c0/4\", \"ground\": false, \"vdata\": []}, {\"name\": \"43\", \"x\": 250, \"y\": 100, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"54\", \"x\": 250, \"y\": 150, \"t\": 1, \"phase\": \"5\\u03c0/4\", \"ground\": false, \"vdata\": []}, {\"name\": \"58\", \"x\": 250, \"y\": 200, \"t\": 2, \"phase\": \"\\u03c0\", \"ground\": false, \"vdata\": []}, {\"name\": \"62\", \"x\": 300, \"y\": 100, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"63\", \"x\": 300, \"y\": 150, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"64\", \"x\": 300, \"y\": 200, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}], \"links\": [{\"source\": \"0\", \"target\": \"4\", \"t\": 2, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"1\", \"target\": \"3\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"2\", \"target\": \"5\", \"t\": 2, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"3\", \"target\": \"43\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"3\", \"target\": \"5\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"3\", \"target\": \"58\", \"t\": 2, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"3\", \"target\": \"54\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"3\", \"target\": \"14\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"4\", \"target\": \"43\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"4\", \"target\": \"58\", \"t\": 2, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"4\", \"target\": \"54\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"5\", \"target\": \"16\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"5\", \"target\": \"43\", \"t\": 2, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"5\", \"target\": \"58\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"5\", \"target\": \"54\", \"t\": 2, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"14\", \"target\": \"58\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"14\", \"target\": \"16\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"16\", \"target\": \"43\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"16\", \"target\": \"58\", \"t\": 2, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"43\", \"target\": \"54\", \"t\": 2, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"43\", \"target\": \"58\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"43\", \"target\": \"62\", \"t\": 2, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"54\", \"target\": \"58\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"54\", \"target\": \"63\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"58\", \"target\": \"64\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}], \"pauli_web\": []}'), 350, 250, 50,\n",
        "  10.0, false, true, '');\n",
        "</script>"
@@ -13164,22 +17006,14 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 15,
+   "execution_count": 18,
    "metadata": {},
    "outputs": [
     {
      "data": {
       "text/plain": [
-       "{(16, 5): 'Y',\n",
-       " (5, 16): 'Y',\n",
-       " (16, 43): 'Z',\n",
-       " (43, 16): 'Z',\n",
-       " (16, 58): 'Z',\n",
-       " (58, 16): 'X',\n",
-       " (16, 14): 'Y',\n",
-       " (14, 16): 'Y',\n",
-       " (3, 1): 'Z',\n",
-       " (1, 3): 'Z',\n",
+       "{(3, 1): 'Y',\n",
+       " (1, 3): 'Y',\n",
        " (3, 43): 'Z',\n",
        " (43, 3): 'Z',\n",
        " (3, 5): 'Y',\n",
@@ -13190,8 +17024,10 @@
        " (54, 3): 'Z',\n",
        " (3, 14): 'Y',\n",
        " (14, 3): 'Y',\n",
-       " (5, 2): 'X',\n",
-       " (2, 5): 'Z',\n",
+       " (5, 16): 'Y',\n",
+       " (16, 5): 'Y',\n",
+       " (5, 2): 'Y',\n",
+       " (2, 5): 'Y',\n",
        " (5, 43): 'X',\n",
        " (43, 5): 'Z',\n",
        " (5, 58): 'X',\n",
@@ -13199,10 +17035,16 @@
        " (5, 54): 'X',\n",
        " (54, 5): 'Z',\n",
        " (14, 58): 'X',\n",
-       " (58, 14): 'X'}"
+       " (58, 14): 'X',\n",
+       " (14, 16): 'Y',\n",
+       " (16, 14): 'Y',\n",
+       " (16, 43): 'Z',\n",
+       " (43, 16): 'Z',\n",
+       " (16, 58): 'Z',\n",
+       " (58, 16): 'X'}"
       ]
      },
-     "execution_count": 15,
+     "execution_count": 18,
      "metadata": {},
      "output_type": "execute_result"
     }
@@ -13214,13 +17056,13 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 16,
+   "execution_count": 19,
    "metadata": {},
    "outputs": [
     {
      "data": {
       "text/html": [
-       "<div style=\"overflow:auto; background-color: white\" id=\"graph-output-jPQD1p9T\"></div>\n",
+       "<div style=\"overflow:auto; background-color: white\" id=\"graph-output-FgJzKB5C\"></div>\n",
        "<script type=\"module\">\n",
        "var d3;\n",
        "if (d3 == null) { d3 = await import(\"https://cdn.skypack.dev/d3@5\"); }\n",
@@ -13583,8 +17425,8 @@
        "}\n",
        "\n",
        "\n",
-       "showGraph('#graph-output-jPQD1p9T',\n",
-       "  JSON.parse('{\"nodes\": [{\"name\": \"0\", \"x\": 50, \"y\": 100, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"1\", \"x\": 50, \"y\": 150, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"2\", \"x\": 50, \"y\": 200, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"3\", \"x\": 100, \"y\": 150, \"t\": 2, \"phase\": \"\\u03c0/2\", \"ground\": false, \"vdata\": []}, {\"name\": \"4\", \"x\": 100, \"y\": 100, \"t\": 2, \"phase\": \"\\u03c0/2\", \"ground\": false, \"vdata\": []}, {\"name\": \"5\", \"x\": 100, \"y\": 200, \"t\": 1, \"phase\": \"\\u03c0/2\", \"ground\": false, \"vdata\": []}, {\"name\": \"14\", \"x\": 150, \"y\": 150, \"t\": 1, \"phase\": \"5\\u03c0/4\", \"ground\": false, \"vdata\": []}, {\"name\": \"16\", \"x\": 200, \"y\": 100, \"t\": 2, \"phase\": \"7\\u03c0/4\", \"ground\": false, \"vdata\": []}, {\"name\": \"43\", \"x\": 250, \"y\": 100, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"54\", \"x\": 250, \"y\": 150, \"t\": 1, \"phase\": \"5\\u03c0/4\", \"ground\": false, \"vdata\": []}, {\"name\": \"58\", \"x\": 250, \"y\": 200, \"t\": 2, \"phase\": \"\\u03c0\", \"ground\": false, \"vdata\": []}, {\"name\": \"62\", \"x\": 300, \"y\": 100, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"63\", \"x\": 300, \"y\": 150, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"64\", \"x\": 300, \"y\": 200, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}], \"links\": [{\"source\": \"0\", \"target\": \"4\", \"t\": 2, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"1\", \"target\": \"3\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"2\", \"target\": \"5\", \"t\": 2, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"3\", \"target\": \"43\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"3\", \"target\": \"5\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"3\", \"target\": \"58\", \"t\": 2, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"3\", \"target\": \"54\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"3\", \"target\": \"14\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"4\", \"target\": \"43\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"4\", \"target\": \"58\", \"t\": 2, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"4\", \"target\": \"54\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"5\", \"target\": \"16\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"5\", \"target\": \"43\", \"t\": 2, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"5\", \"target\": \"58\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"5\", \"target\": \"54\", \"t\": 2, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"14\", \"target\": \"58\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"14\", \"target\": \"16\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"16\", \"target\": \"43\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"16\", \"target\": \"58\", \"t\": 2, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"43\", \"target\": \"54\", \"t\": 2, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"43\", \"target\": \"58\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"43\", \"target\": \"62\", \"t\": 2, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"54\", \"target\": \"58\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"54\", \"target\": \"63\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"58\", \"target\": \"64\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}], \"pauli_web\": [{\"source\": 16, \"target\": 5, \"t\": \"Y\"}, {\"source\": 5, \"target\": 16, \"t\": \"Y\"}, {\"source\": 16, \"target\": 43, \"t\": \"Z\"}, {\"source\": 43, \"target\": 16, \"t\": \"Z\"}, {\"source\": 16, \"target\": 58, \"t\": \"Z\"}, {\"source\": 58, \"target\": 16, \"t\": \"X\"}, {\"source\": 16, \"target\": 14, \"t\": \"Y\"}, {\"source\": 14, \"target\": 16, \"t\": \"Y\"}, {\"source\": 3, \"target\": 1, \"t\": \"Z\"}, {\"source\": 1, \"target\": 3, \"t\": \"Z\"}, {\"source\": 3, \"target\": 43, \"t\": \"Z\"}, {\"source\": 43, \"target\": 3, \"t\": \"Z\"}, {\"source\": 3, \"target\": 5, \"t\": \"Y\"}, {\"source\": 5, \"target\": 3, \"t\": \"Y\"}, {\"source\": 3, \"target\": 58, \"t\": \"Z\"}, {\"source\": 58, \"target\": 3, \"t\": \"X\"}, {\"source\": 3, \"target\": 54, \"t\": \"Z\"}, {\"source\": 54, \"target\": 3, \"t\": \"Z\"}, {\"source\": 3, \"target\": 14, \"t\": \"Y\"}, {\"source\": 14, \"target\": 3, \"t\": \"Y\"}, {\"source\": 5, \"target\": 2, \"t\": \"X\"}, {\"source\": 2, \"target\": 5, \"t\": \"Z\"}, {\"source\": 5, \"target\": 43, \"t\": \"X\"}, {\"source\": 43, \"target\": 5, \"t\": \"Z\"}, {\"source\": 5, \"target\": 58, \"t\": \"X\"}, {\"source\": 58, \"target\": 5, \"t\": \"X\"}, {\"source\": 5, \"target\": 54, \"t\": \"X\"}, {\"source\": 54, \"target\": 5, \"t\": \"Z\"}, {\"source\": 14, \"target\": 58, \"t\": \"X\"}, {\"source\": 58, \"target\": 14, \"t\": \"X\"}]}'), 350, 250, 50,\n",
+       "showGraph('#graph-output-FgJzKB5C',\n",
+       "  JSON.parse('{\"nodes\": [{\"name\": \"0\", \"x\": 50, \"y\": 100, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"1\", \"x\": 50, \"y\": 150, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"2\", \"x\": 50, \"y\": 200, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"3\", \"x\": 100, \"y\": 150, \"t\": 2, \"phase\": \"\\u03c0/2\", \"ground\": false, \"vdata\": []}, {\"name\": \"4\", \"x\": 100, \"y\": 100, \"t\": 2, \"phase\": \"\\u03c0/2\", \"ground\": false, \"vdata\": []}, {\"name\": \"5\", \"x\": 100, \"y\": 200, \"t\": 1, \"phase\": \"\\u03c0/2\", \"ground\": false, \"vdata\": []}, {\"name\": \"14\", \"x\": 150, \"y\": 150, \"t\": 1, \"phase\": \"5\\u03c0/4\", \"ground\": false, \"vdata\": []}, {\"name\": \"16\", \"x\": 200, \"y\": 100, \"t\": 2, \"phase\": \"7\\u03c0/4\", \"ground\": false, \"vdata\": []}, {\"name\": \"43\", \"x\": 250, \"y\": 100, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"54\", \"x\": 250, \"y\": 150, \"t\": 1, \"phase\": \"5\\u03c0/4\", \"ground\": false, \"vdata\": []}, {\"name\": \"58\", \"x\": 250, \"y\": 200, \"t\": 2, \"phase\": \"\\u03c0\", \"ground\": false, \"vdata\": []}, {\"name\": \"62\", \"x\": 300, \"y\": 100, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"63\", \"x\": 300, \"y\": 150, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"64\", \"x\": 300, \"y\": 200, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}], \"links\": [{\"source\": \"0\", \"target\": \"4\", \"t\": 2, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"1\", \"target\": \"3\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"2\", \"target\": \"5\", \"t\": 2, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"3\", \"target\": \"43\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"3\", \"target\": \"5\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"3\", \"target\": \"58\", \"t\": 2, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"3\", \"target\": \"54\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"3\", \"target\": \"14\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"4\", \"target\": \"43\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"4\", \"target\": \"58\", \"t\": 2, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"4\", \"target\": \"54\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"5\", \"target\": \"16\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"5\", \"target\": \"43\", \"t\": 2, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"5\", \"target\": \"58\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"5\", \"target\": \"54\", \"t\": 2, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"14\", \"target\": \"58\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"14\", \"target\": \"16\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"16\", \"target\": \"43\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"16\", \"target\": \"58\", \"t\": 2, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"43\", \"target\": \"54\", \"t\": 2, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"43\", \"target\": \"58\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"43\", \"target\": \"62\", \"t\": 2, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"54\", \"target\": \"58\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"54\", \"target\": \"63\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"58\", \"target\": \"64\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}], \"pauli_web\": [{\"source\": 3, \"target\": 1, \"t\": \"Y\"}, {\"source\": 1, \"target\": 3, \"t\": \"Y\"}, {\"source\": 3, \"target\": 43, \"t\": \"Z\"}, {\"source\": 43, \"target\": 3, \"t\": \"Z\"}, {\"source\": 3, \"target\": 5, \"t\": \"Y\"}, {\"source\": 5, \"target\": 3, \"t\": \"Y\"}, {\"source\": 3, \"target\": 58, \"t\": \"Z\"}, {\"source\": 58, \"target\": 3, \"t\": \"X\"}, {\"source\": 3, \"target\": 54, \"t\": \"Z\"}, {\"source\": 54, \"target\": 3, \"t\": \"Z\"}, {\"source\": 3, \"target\": 14, \"t\": \"Y\"}, {\"source\": 14, \"target\": 3, \"t\": \"Y\"}, {\"source\": 5, \"target\": 16, \"t\": \"Y\"}, {\"source\": 16, \"target\": 5, \"t\": \"Y\"}, {\"source\": 5, \"target\": 2, \"t\": \"Y\"}, {\"source\": 2, \"target\": 5, \"t\": \"Y\"}, {\"source\": 5, \"target\": 43, \"t\": \"X\"}, {\"source\": 43, \"target\": 5, \"t\": \"Z\"}, {\"source\": 5, \"target\": 58, \"t\": \"X\"}, {\"source\": 58, \"target\": 5, \"t\": \"X\"}, {\"source\": 5, \"target\": 54, \"t\": \"X\"}, {\"source\": 54, \"target\": 5, \"t\": \"Z\"}, {\"source\": 14, \"target\": 58, \"t\": \"X\"}, {\"source\": 58, \"target\": 14, \"t\": \"X\"}, {\"source\": 14, \"target\": 16, \"t\": \"Y\"}, {\"source\": 16, \"target\": 14, \"t\": \"Y\"}, {\"source\": 16, \"target\": 43, \"t\": \"Z\"}, {\"source\": 43, \"target\": 16, \"t\": \"Z\"}, {\"source\": 16, \"target\": 58, \"t\": \"Z\"}, {\"source\": 58, \"target\": 16, \"t\": \"X\"}]}'), 350, 250, 50,\n",
        "  10.0, false, true, '');\n",
        "</script>"
       ],
@@ -13613,13 +17455,13 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 17,
+   "execution_count": 20,
    "metadata": {},
    "outputs": [
     {
      "data": {
       "text/html": [
-       "<div style=\"overflow:auto; background-color: white\" id=\"graph-output-Q6J8ByNj\"></div>\n",
+       "<div style=\"overflow:auto; background-color: white\" id=\"graph-output-GPnBMd5D\"></div>\n",
        "<script type=\"module\">\n",
        "var d3;\n",
        "if (d3 == null) { d3 = await import(\"https://cdn.skypack.dev/d3@5\"); }\n",
@@ -13982,7 +17824,7 @@
        "}\n",
        "\n",
        "\n",
-       "showGraph('#graph-output-Q6J8ByNj',\n",
+       "showGraph('#graph-output-GPnBMd5D',\n",
        "  JSON.parse('{\"nodes\": [{\"name\": \"0\", \"x\": 50, \"y\": 100, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"1\", \"x\": 100, \"y\": 100, \"t\": 1, \"phase\": \"\\u03c0/4\", \"ground\": false, \"vdata\": []}, {\"name\": \"2\", \"x\": 150, \"y\": 100, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}], \"links\": [{\"source\": \"0\", \"target\": \"1\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"1\", \"target\": \"2\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}], \"pauli_web\": []}'), 200, 150, 50,\n",
        "  10.0, false, false, '');\n",
        "</script>"
@@ -13997,7 +17839,7 @@
     {
      "data": {
       "text/html": [
-       "<div style=\"overflow:auto; background-color: white\" id=\"graph-output-Ci5Knzfw\"></div>\n",
+       "<div style=\"overflow:auto; background-color: white\" id=\"graph-output-7bG2EwPF\"></div>\n",
        "<script type=\"module\">\n",
        "var d3;\n",
        "if (d3 == null) { d3 = await import(\"https://cdn.skypack.dev/d3@5\"); }\n",
@@ -14360,8 +18202,8 @@
        "}\n",
        "\n",
        "\n",
-       "showGraph('#graph-output-Ci5Knzfw',\n",
-       "  JSON.parse('{\"nodes\": [{\"name\": \"0\", \"x\": 50, \"y\": 100, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"1\", \"x\": 100, \"y\": 100, \"t\": 1, \"phase\": \"\\u03c0/4\", \"ground\": false, \"vdata\": []}, {\"name\": \"2\", \"x\": 150, \"y\": 100, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}], \"links\": [{\"source\": \"0\", \"target\": \"1\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"1\", \"target\": \"2\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}], \"pauli_web\": []}'), 200, 150, 50,\n",
+       "showGraph('#graph-output-7bG2EwPF',\n",
+       "  JSON.parse('{\"nodes\": [{\"name\": \"0\", \"x\": 50, \"y\": 100, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"1\", \"x\": 100, \"y\": 100, \"t\": 1, \"phase\": \"\\u03c0/4\", \"ground\": false, \"vdata\": []}, {\"name\": \"2\", \"x\": 150, \"y\": 100, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}], \"links\": [{\"source\": \"0\", \"target\": \"1\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"1\", \"target\": \"2\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}], \"pauli_web\": [{\"source\": 0, \"target\": 1, \"t\": \"Z\"}, {\"source\": 1, \"target\": 0, \"t\": \"Z\"}]}'), 200, 150, 50,\n",
        "  10.0, false, true, '');\n",
        "</script>"
       ],
@@ -14385,7 +18227,7 @@
     "order, zwebs, xwebs = compute_pauli_webs(g)\n",
     "\n",
     "# highlight the web associated to the T spider\n",
-    "zx.draw(g, labels=True, pauli_web=None)"
+    "zx.draw(g, labels=True, pauli_web=zwebs[1])"
    ]
   },
   {
@@ -14397,13 +18239,13 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 18,
+   "execution_count": 21,
    "metadata": {},
    "outputs": [
     {
      "data": {
       "text/html": [
-       "<div style=\"overflow:auto; background-color: white\" id=\"graph-output-gP8xA43C\"></div>\n",
+       "<div style=\"overflow:auto; background-color: white\" id=\"graph-output-l12LCTYB\"></div>\n",
        "<script type=\"module\">\n",
        "var d3;\n",
        "if (d3 == null) { d3 = await import(\"https://cdn.skypack.dev/d3@5\"); }\n",
@@ -14766,7 +18608,7 @@
        "}\n",
        "\n",
        "\n",
-       "showGraph('#graph-output-gP8xA43C',\n",
+       "showGraph('#graph-output-l12LCTYB',\n",
        "  JSON.parse('{\"nodes\": [{\"name\": \"0\", \"x\": 50, \"y\": 100, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"1\", \"x\": 50, \"y\": 150, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"2\", \"x\": 50, \"y\": 200, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"3\", \"x\": 100, \"y\": 150, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"4\", \"x\": 100, \"y\": 100, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"5\", \"x\": 150, \"y\": 200, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"6\", \"x\": 150, \"y\": 150, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"7\", \"x\": 200, \"y\": 200, \"t\": 1, \"phase\": \"\\u03c0/4\", \"ground\": false, \"vdata\": []}, {\"name\": \"8\", \"x\": 250, \"y\": 200, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"9\", \"x\": 250, \"y\": 150, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"10\", \"x\": 300, \"y\": 150, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"11\", \"x\": 300, \"y\": 100, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"12\", \"x\": 350, \"y\": 100, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"13\", \"x\": 350, \"y\": 150, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"14\", \"x\": 350, \"y\": 200, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}], \"links\": [{\"source\": \"0\", \"target\": \"4\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"1\", \"target\": \"3\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"2\", \"target\": \"5\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"3\", \"target\": \"4\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"3\", \"target\": \"6\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"4\", \"target\": \"11\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"5\", \"target\": \"6\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"5\", \"target\": \"7\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"6\", \"target\": \"9\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"7\", \"target\": \"8\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"8\", \"target\": \"9\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"8\", \"target\": \"14\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"9\", \"target\": \"10\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"10\", \"target\": \"11\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"10\", \"target\": \"13\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"11\", \"target\": \"12\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}], \"pauli_web\": []}'), 400, 250, 50,\n",
        "  10.0, false, false, '');\n",
        "</script>"
@@ -14781,7 +18623,7 @@
     {
      "data": {
       "text/html": [
-       "<div style=\"overflow:auto; background-color: white\" id=\"graph-output-40viJwea\"></div>\n",
+       "<div style=\"overflow:auto; background-color: white\" id=\"graph-output-3oru9Wy0\"></div>\n",
        "<script type=\"module\">\n",
        "var d3;\n",
        "if (d3 == null) { d3 = await import(\"https://cdn.skypack.dev/d3@5\"); }\n",
@@ -15144,8 +18986,8 @@
        "}\n",
        "\n",
        "\n",
-       "showGraph('#graph-output-40viJwea',\n",
-       "  JSON.parse('{\"nodes\": [{\"name\": \"0\", \"x\": 50, \"y\": 200, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"1\", \"x\": 50, \"y\": 250, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"2\", \"x\": 50, \"y\": 300, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"4\", \"x\": 100, \"y\": 200, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"12\", \"x\": 350, \"y\": 200, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"13\", \"x\": 350, \"y\": 250, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"14\", \"x\": 350, \"y\": 300, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"15\", \"x\": 200, \"y\": 100, \"t\": 1, \"phase\": \"\\u03c0/4\", \"ground\": false, \"vdata\": []}, {\"name\": \"16\", \"x\": 200, \"y\": 150, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"17\", \"x\": 110.00000000000001, \"y\": 300.0, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"18\", \"x\": 80.0, \"y\": 250.0, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}], \"links\": [{\"source\": \"0\", \"target\": \"4\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"1\", \"target\": \"18\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"2\", \"target\": \"17\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"4\", \"target\": \"12\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"4\", \"target\": \"16\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"13\", \"target\": \"18\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"14\", \"target\": \"17\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"15\", \"target\": \"16\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"16\", \"target\": \"17\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"16\", \"target\": \"18\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}], \"pauli_web\": [{\"source\": 16, \"target\": 15, \"t\": \"Z\"}, {\"source\": 15, \"target\": 16, \"t\": \"Z\"}, {\"source\": 16, \"target\": 17, \"t\": \"Z\"}, {\"source\": 17, \"target\": 16, \"t\": \"Z\"}, {\"source\": 16, \"target\": 18, \"t\": \"Z\"}, {\"source\": 18, \"target\": 16, \"t\": \"Z\"}, {\"source\": 16, \"target\": 4, \"t\": \"Z\"}, {\"source\": 4, \"target\": 16, \"t\": \"Z\"}, {\"source\": 4, \"target\": 0, \"t\": \"Z\"}, {\"source\": 0, \"target\": 4, \"t\": \"Z\"}, {\"source\": 18, \"target\": 1, \"t\": \"Z\"}, {\"source\": 1, \"target\": 18, \"t\": \"Z\"}, {\"source\": 17, \"target\": 2, \"t\": \"Z\"}, {\"source\": 2, \"target\": 17, \"t\": \"Z\"}]}'), 400, 350, 50,\n",
+       "showGraph('#graph-output-3oru9Wy0',\n",
+       "  JSON.parse('{\"nodes\": [{\"name\": \"0\", \"x\": 50, \"y\": 200, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"1\", \"x\": 50, \"y\": 250, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"2\", \"x\": 50, \"y\": 300, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"4\", \"x\": 100, \"y\": 200, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"12\", \"x\": 350, \"y\": 200, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"13\", \"x\": 350, \"y\": 250, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"14\", \"x\": 350, \"y\": 300, \"t\": 0, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"15\", \"x\": 200, \"y\": 100, \"t\": 1, \"phase\": \"\\u03c0/4\", \"ground\": false, \"vdata\": []}, {\"name\": \"16\", \"x\": 200, \"y\": 150, \"t\": 2, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"17\", \"x\": 110.00000000000001, \"y\": 300.0, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}, {\"name\": \"18\", \"x\": 80.0, \"y\": 250.0, \"t\": 1, \"phase\": \"\", \"ground\": false, \"vdata\": []}], \"links\": [{\"source\": \"0\", \"target\": \"4\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"1\", \"target\": \"18\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"2\", \"target\": \"17\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"4\", \"target\": \"12\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"4\", \"target\": \"16\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"13\", \"target\": \"18\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"14\", \"target\": \"17\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"15\", \"target\": \"16\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"16\", \"target\": \"17\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}, {\"source\": \"16\", \"target\": \"18\", \"t\": 1, \"index\": 0, \"num_parallel\": 1}], \"pauli_web\": [{\"source\": 16, \"target\": 15, \"t\": \"Z\"}, {\"source\": 15, \"target\": 16, \"t\": \"Z\"}, {\"source\": 16, \"target\": 17, \"t\": \"Z\"}, {\"source\": 17, \"target\": 16, \"t\": \"Z\"}, {\"source\": 16, \"target\": 18, \"t\": \"Z\"}, {\"source\": 18, \"target\": 16, \"t\": \"Z\"}, {\"source\": 16, \"target\": 4, \"t\": \"Z\"}, {\"source\": 4, \"target\": 16, \"t\": \"Z\"}, {\"source\": 0, \"target\": 4, \"t\": \"Z\"}, {\"source\": 4, \"target\": 0, \"t\": \"Z\"}, {\"source\": 1, \"target\": 18, \"t\": \"Z\"}, {\"source\": 18, \"target\": 1, \"t\": \"Z\"}, {\"source\": 2, \"target\": 17, \"t\": \"Z\"}, {\"source\": 17, \"target\": 2, \"t\": \"Z\"}]}'), 400, 350, 50,\n",
        "  10.0, false, true, '');\n",
        "</script>"
       ],
@@ -15184,6 +19026,13 @@
     "zx.draw(g, labels=True, pauli_web=zwebs[15])"
    ]
   },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": []
+  },
   {
    "cell_type": "code",
    "execution_count": null,
@@ -15194,7 +19043,7 @@
  ],
  "metadata": {
   "kernelspec": {
-   "display_name": "venv",
+   "display_name": "Python 3 (ipykernel)",
    "language": "python",
    "name": "python3"
   },
@@ -15208,7 +19057,7 @@
    "name": "python",
    "nbconvert_exporter": "python",
    "pygments_lexer": "ipython3",
-   "version": "3.12.3"
+   "version": "3.10.12"
   }
  },
  "nbformat": 4,
diff --git a/pyzx/pauliweb.py b/pyzx/pauliweb.py
index 3b4b3529..1d85c6b5 100644
--- a/pyzx/pauliweb.py
+++ b/pyzx/pauliweb.py
@@ -116,10 +116,10 @@ def transpose_corrections(c: Dict[VT, Set[VT]]) -> Dict[VT, Set[VT]]:
             ct[v].add(k)
     return ct
 
-def compute_pauli_webs(g: BaseGraph[VT,ET]) -> Tuple[Dict[VT, int], Dict[VT, PauliWeb[VT,ET]], Dict[VT, PauliWeb[VT,ET]]]:
+def compute_pauli_webs(g: BaseGraph[VT,ET], backwards:bool=True) -> Tuple[Dict[VT, int], Dict[VT, PauliWeb[VT,ET]], Dict[VT, PauliWeb[VT,ET]]]:
     g1 = g.clone()
-    out_edge: Dict[VT, Tuple[VT,VT]] = dict()
-    interior_edge: Dict[VT, Tuple[VT,VT]] = dict()
+    color_edge: Dict[VT, Tuple[VT,VT]] = dict()
+    boundary: Dict[VT, VT] = dict()
 
     for e in g.edges():
         s, t = g.edge_st(e)
@@ -133,12 +133,11 @@ def compute_pauli_webs(g: BaseGraph[VT,ET]) -> Tuple[Dict[VT, int], Dict[VT, Pau
             g1.remove_edge(e)
             g1.add_edge((s, v), EdgeType.SIMPLE)
             g1.add_edge((v, t), et)
-            interior_edge[v] = (s,t)
+            color_edge[v] = (s,t)
         
-
-    for o in g1.outputs():
-        e = g1.incident_edges(o)[0]
-        v = next(iter(g1.neighbors(o)))
+    for b in g1.inputs() + g1.outputs():
+        e = g1.incident_edges(b)[0]
+        v = next(iter(g1.neighbors(b)))
         vt = g1.type(v)
         et = g1.edge_type(e)
         g1.remove_edge(e)
@@ -151,11 +150,12 @@ def compute_pauli_webs(g: BaseGraph[VT,ET]) -> Tuple[Dict[VT, int], Dict[VT, Pau
             v1 = g1.add_vertex(VertexType.X)
         g1.add_edge((v, v0), et)
         g1.add_edge((v0, v1), EdgeType.SIMPLE)
-        g1.add_edge((v1, o), EdgeType.SIMPLE)
-        out_edge[v0] = (o, v)
-        out_edge[v1] = (o, v)
+        g1.add_edge((v1, b), EdgeType.SIMPLE)
+        color_edge[v0] = (b, v)
+        boundary[v0] = b
+        boundary[v1] = b
 
-    gf = gflow(g1, focus=True, pauli=True)
+    gf = gflow(g1, focus=True, reverse=backwards, pauli=True)
     if not gf:
         raise ValueError("Graph must have gFlow")
     order, corr = gf
@@ -163,9 +163,9 @@ def compute_pauli_webs(g: BaseGraph[VT,ET]) -> Tuple[Dict[VT, int], Dict[VT, Pau
     zwebs: Dict[VT, PauliWeb[VT,ET]] = dict()
     xwebs: Dict[VT, PauliWeb[VT,ET]] = dict()
     vset = g.vertex_set()
-    corr_t = transpose_corrections(corr)
+    # corr_t = transpose_corrections(corr)
 
-    for v,c in corr_t.items():
+    for v,c in corr.items():
         pw = PauliWeb(g)
         for v1 in c:
             if v1 in vset:
@@ -175,34 +175,16 @@ def compute_pauli_webs(g: BaseGraph[VT,ET]) -> Tuple[Dict[VT, int], Dict[VT, Pau
                         pw.add_edge((v1, v2), p)
                     else:
                         pw.add_half_edge((v1, v2), p)
-            elif v1 in out_edge:
-                pw.add_edge(out_edge[v1], 'Z' if g1.type(v1) == VertexType.X else 'X')
-            elif v1 in interior_edge:
-                pw.add_edge(interior_edge[v1], 'Z' if g1.type(v1) == VertexType.X else 'X')
-        if v in out_edge:
-            ref = out_edge[v][0]
+            elif v1 in color_edge:
+                pw.add_edge(color_edge[v1], 'Z' if g1.type(v1) == VertexType.X else 'X')
+        if v in boundary:
+            ref = boundary[v]
         else:
             ref = v
 
-        pw.spread_to_boundary(inputs=True, outputs=False)
+        # pw.spread_to_boundary(inputs=True, outputs=False)
         if g1.type(v) == VertexType.Z: zwebs[ref] = pw
         elif g1.type(v) == VertexType.X: xwebs[ref] = pw
 
-    for i in g.inputs():
-        v = next(iter(g.neighbors(i)))
-        pw = PauliWeb(g)
-        if g.type(v) == VertexType.Z:
-            pw.add_edge((v, i), 'Z')
-            zwebs[v] = pw
-        elif g.type(v) == VertexType.X:
-            pw.add_edge((v, i), 'X')
-            xwebs[v] = pw
-        elif g.type(v) == VertexType.BOUNDARY:
-            pw.add_edge((v, i), 'X')
-            xwebs[v] = pw
-            pw1 = PauliWeb(g)
-            pw1.add_edge((v, i), 'Z')
-            zwebs[v] = pw1
-
 
     return (order, zwebs, xwebs)