4b. Algebra-CompositeHamiltonian.ipynb

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   "cell_type": "markdown",
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   "source": [
    "### Algebra: Composite Hamiltonian"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "1. https://numpy.org/doc/stable/reference/routines.linalg.html\n",
    "2. http://qutip.org/docs/latest/guide/dynamics/dynamics-master.html"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "from qutip import *\n",
    "import seaborn as sns\n",
    "import matplotlib.pyplot as plt\n",
    "%matplotlib inline\n",
    "sns.set()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Constructing composite Hamiltonians\n",
    "\n",
    "1. Two coupled qubits"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "Quantum object: dims = [[2], [2]], shape = (2, 2), type = oper, isherm = True\\begin{equation*}\\left(\\begin{array}{*{11}c}1.0 & 0.0\\\\0.0 & -1.0\\\\\\end{array}\\right)\\end{equation*}"
      ],
      "text/plain": [
       "Quantum object: dims = [[2], [2]], shape = (2, 2), type = oper, isherm = True\n",
       "Qobj data =\n",
       "[[ 1.  0.]\n",
       " [ 0. -1.]]"
      ]
     },
     "execution_count": 2,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sigmaz()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "Quantum object: dims = [[2], [2]], shape = (2, 2), type = oper, isherm = True\\begin{equation*}\\left(\\begin{array}{*{11}c}1.0 & 0.0\\\\0.0 & 1.0\\\\\\end{array}\\right)\\end{equation*}"
      ],
      "text/plain": [
       "Quantum object: dims = [[2], [2]], shape = (2, 2), type = oper, isherm = True\n",
       "Qobj data =\n",
       "[[1. 0.]\n",
       " [0. 1.]]"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "identity(2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "Quantum object: dims = [[2], [2]], shape = (2, 2), type = oper, isherm = True\\begin{equation*}\\left(\\begin{array}{*{11}c}0.0 & 1.0\\\\1.0 & 0.0\\\\\\end{array}\\right)\\end{equation*}"
      ],
      "text/plain": [
       "Quantum object: dims = [[2], [2]], shape = (2, 2), type = oper, isherm = True\n",
       "Qobj data =\n",
       "[[0. 1.]\n",
       " [1. 0.]]"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sigmax()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [],
   "source": [
    "H2 = tensor(sigmaz(), identity(2)) \\\n",
    "    + tensor(identity(2),sigmaz()) \\\n",
    "    + 0.05 * tensor(sigmax(), sigmax())"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "Quantum object: dims = [[2, 2], [2, 2]], shape = (4, 4), type = oper, isherm = True\\begin{equation*}\\left(\\begin{array}{*{11}c}2.0 & 0.0 & 0.0 & 0.050\\\\0.0 & 0.0 & 0.050 & 0.0\\\\0.0 & 0.050 & 0.0 & 0.0\\\\0.050 & 0.0 & 0.0 & -2.0\\\\\\end{array}\\right)\\end{equation*}"
      ],
      "text/plain": [
       "Quantum object: dims = [[2, 2], [2, 2]], shape = (4, 4), type = oper, isherm = True\n",
       "Qobj data =\n",
       "[[ 2.    0.    0.    0.05]\n",
       " [ 0.    0.    0.05  0.  ]\n",
       " [ 0.    0.05  0.    0.  ]\n",
       " [ 0.05  0.    0.   -2.  ]]"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "H2"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "2. Three Coupled Qubit"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [],
   "source": [
    "H3 = (tensor(sigmaz(), identity(2), identity(2)) +\n",
    "     tensor(identity(2), sigmaz(), identity(2)) +\n",
    "    tensor(identity(2), identity(2), sigmaz()) +\n",
    "     0.5 * tensor(sigmax(), sigmax(), identity(2)) +\n",
    "     0.25 * tensor(identity(2), sigmax(), sigmax())) "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "Quantum object: dims = [[2, 2, 2], [2, 2, 2]], shape = (8, 8), type = oper, isherm = True\\begin{equation*}\\left(\\begin{array}{*{11}c}3.0 & 0.0 & 0.0 & 0.250 & 0.0 & 0.0 & 0.500 & 0.0\\\\0.0 & 1.0 & 0.250 & 0.0 & 0.0 & 0.0 & 0.0 & 0.500\\\\0.0 & 0.250 & 1.0 & 0.0 & 0.500 & 0.0 & 0.0 & 0.0\\\\0.250 & 0.0 & 0.0 & -1.0 & 0.0 & 0.500 & 0.0 & 0.0\\\\0.0 & 0.0 & 0.500 & 0.0 & 1.0 & 0.0 & 0.0 & 0.250\\\\0.0 & 0.0 & 0.0 & 0.500 & 0.0 & -1.0 & 0.250 & 0.0\\\\0.500 & 0.0 & 0.0 & 0.0 & 0.0 & 0.250 & -1.0 & 0.0\\\\0.0 & 0.500 & 0.0 & 0.0 & 0.250 & 0.0 & 0.0 & -3.0\\\\\\end{array}\\right)\\end{equation*}"
      ],
      "text/plain": [
       "Quantum object: dims = [[2, 2, 2], [2, 2, 2]], shape = (8, 8), type = oper, isherm = True\n",
       "Qobj data =\n",
       "[[ 3.    0.    0.    0.25  0.    0.    0.5   0.  ]\n",
       " [ 0.    1.    0.25  0.    0.    0.    0.    0.5 ]\n",
       " [ 0.    0.25  1.    0.    0.5   0.    0.    0.  ]\n",
       " [ 0.25  0.    0.   -1.    0.    0.5   0.    0.  ]\n",
       " [ 0.    0.    0.5   0.    1.    0.    0.    0.25]\n",
       " [ 0.    0.    0.    0.5   0.   -1.    0.25  0.  ]\n",
       " [ 0.5   0.    0.    0.    0.    0.25 -1.    0.  ]\n",
       " [ 0.    0.5   0.    0.    0.25  0.    0.   -3.  ]]"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "H3"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Time Evolution"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "Quantum object: dims = [[4], [1]], shape = (4, 1), type = ket\\begin{equation*}\\left(\\begin{array}{*{11}c}1.0\\\\0.0\\\\0.0\\\\0.0\\\\\\end{array}\\right)\\end{equation*}"
      ],
      "text/plain": [
       "Quantum object: dims = [[4], [1]], shape = (4, 1), type = ket\n",
       "Qobj data =\n",
       "[[1.]\n",
       " [0.]\n",
       " [0.]\n",
       " [0.]]"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "psi0 = basis(4, 0)\n",
    "psi0"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [],
   "source": [
    "times = np.linspace(0.0, 10.0, 20)\n",
    "result = sesolve(H2, psi0, times, [sigmaz()])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Result object with sesolve data.\n",
       "--------------------------------\n",
       "expect = True\n",
       "num_expect = 1, num_collapse = 0"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "result"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[array([1.        , 0.99952844, 0.99953782, 0.99999981, 0.99951924,\n",
       "        0.99954738, 0.99999925, 0.99951025, 0.99955711, 0.99999832,\n",
       "        0.99950147, 0.999567  , 0.99999702, 0.99949292, 0.99957703,\n",
       "        0.99999534, 0.99948459, 0.99958719, 0.9999933 , 0.99947651])]"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "result.expect"
   ]
  },
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