Integrated Seb's comments in Boson 4 page

docs/reference/boson_4_chips.md

@@ -6,13 +6,13 @@ Boson 4 is part of the “Boson” series of chip designs, meant to demonstrate
 
 ![A Boson 4 chip](../media/backends/boson4.png)
 
-Just like its two predecessors, Boson 4 is a transmon-free design. This enables it to reach very long bit-flip lifetimes (up to 276 seconds measured in Alice & Bob’s lab).
+Just like its two predecessors, Boson 4 is a transmon-free design. This enables it to reach very long bit-flip lifetimes (up to 430 seconds).
 
 As its name suggests, Boson 4 follows three previous designs:
 
-- [Boson 1](https://www.nature.com/articles/s41567-020-0824-x) is the design which started Alice & Bob. It demonstrated the possibility to exponentially suppress bit-flips, while only linearly increasing phase-flips. Its bit-flip lifetime however saturated at 1 ms, due to the presence of a transmon in the experimental setup.
-- [Boson 2](https://arxiv.org/abs/2204.09128) showed it is possible to take bit-flip lifetime up to 100 seconds by removing the transmon from the setup. This design however lacked the possibility to measure phase-flips.
-- [Boson 3](https://arxiv.org/abs/2307.06617) improved upon Boson 2 by introducing a new readout protocol making it possible to do transmon-free measurements along the X and the Z axis. Its bit-flip lifetime reached over 10 seconds.
+- [Boson 1](https://www.nature.com/articles/s41567-020-0824-x) ([arXiv link](https://arxiv.org/abs/1907.11729)) is the design which started Alice & Bob. It demonstrated the possibility to exponentially suppress bit-flips, while only linearly increasing phase-flips. Its bit-flip lifetime however saturated at 1 ms, due to the presence of a transmon in the experimental setup.
+- [Boson 2](https://journals.aps.org/prxquantum/abstract/10.1103/PRXQuantum.4.020350) showed it is possible to take bit-flip lifetime up to 100 seconds by removing the transmon from the setup. This design however lacked the possibility to measure phase-flips.
+- [Boson 3](https://www.nature.com/articles/s41586-024-07294-3) ([arXiv link](https://arxiv.org/abs/2307.06617)) improved upon Boson 2 by introducing a new readout protocol making it possible to do transmon-free measurements along the X and the Z axis. Its bit-flip lifetime reached over 10 seconds.
 
 Boson 4 uses the same readout protocols as Boson 3, but manages to reach longer bit-flip lifetimes than any of its predecessors.
 
@@ -30,97 +30,102 @@ The figures below can all be reproduced using [this notebook](https://github.com
 
 ### Lifetime
 
-These figures represent the chip's bit-flip and phase-flip lifetime.
+These figures represent the qubit's bit-flip and phase-flip lifetime.
 
-When preparing the $\ket{0}$ state, the probability of a Z measurement yielding 0 after a delay of duration $t$ decays as $\exp(-t/T_{bf})$, where $T_{bf}$ is the bit-flip lifetime.
+When preparing the $\ket{0}$ state, the probability of the Pauli operator Z yielding 0 after a delay of duration $t$ decays as $\exp(-t/T_{Z})$, where $T_{Z}$ is the bit-flip lifetime.
 
-When preparing the $\ket{+}$ state, the probability of an X measurement yielding + after a delay of duration $t$ decays as $\exp(-t/T_{pf})$, where $T_{pf}$ is the phase-flip lifetime.
+When preparing the $\ket{+}$ state, the probability of the Pauli operator X yielding + after a delay of duration $t$ decays as $\exp(-t/T_{X})$, where $T_{X}$ is the phase-flip lifetime.
 
-|  | average_nb_photons = 4 | average_nb_photons = 16 |
+|  | average_nb_photons = 4 | average_nb_photons = 11 |
 | --- | --- | --- |
-| Bit-flip | 1 ms | > 100 seconds |
-| Phase-flip | 1 µs | 0.5 µs |
+| Bit-flip | $830 µs$ | $120 s$ |
+| Phase-flip | $0.95 µs$ | $0.3 µs$ |
 
-💡 **Note:** Measuring lifetimes over 100 seconds is challenging using repeated measurements and a chip shared between users:
+> 💡 **Note:** Measuring lifetimes over 100 seconds is challenging using repeated measurements and a chip shared between users:
 
-- Doing 1000 shots of a 100-second experiment takes almost 28 hours
-- Doing shorter experiments yields too few errors; this requires using more shots and does not make experiments significantly shorter
+> - Doing 1000 shots of a 100-second experiment takes almost 28 hours
+> - Doing shorter experiments yields too few errors; this requires using more shots and does not make experiments significantly shorter
 
-We are working on adding the “real-time trajectories” protocol described in [our latest Nature paper](https://www.nature.com/articles/s41586-024-07294-3) ([arXiv link](https://arxiv.org/pdf/2307.06617.pdf)), which enables shorter measurements.
+> We are working on adding the “real-time trajectories” protocol described in [our latest Nature paper](https://www.nature.com/articles/s41586-024-07294-3) ([arXiv link](https://arxiv.org/pdf/2307.06617.pdf)), which enables shorter measurements.
 
-💡 **Note:** if you're used to working with transmons, you know that state decay only happens if you start from the $\ket{1}$ state. With cat qubits, the $\ket{0}$ and $\ket{1}$ states are virtually interchangeable. Experimental differences might remain, but they're mostly due to sampling noise and calibration inaccuracies.
+> 💡 **Note:** if you're used to working with transmons, you know that state decay only happens if you start from the $\ket{1}$ state. With cat qubits, the $\ket{0}$ and $\ket{1}$ states are virtually interchangeable. Experimental differences might remain, but they're mostly due to sampling noise, calibration inaccuracies or [readout](#measure).
 
 ### SPAM errors
 
-These figures represent sequence error (# shots giving the expected result / # of shots).
+These figures represent sequence error (# shots giving the wrong result / # of shots).
 
-| Sequence | average_nb_photons = 4 | average_nb_photons = 16 |
+| Sequence | average_nb_photons = 4 | average_nb_photons = 11 |
 | --- | --- | --- |
-| P0 - Mz | 2 % | < 0.001 % |
-| P+ - Mx | 41 % | 47 % |
+| $P_0$ - $M_Z$ | $1.7\%$ | $7.6\times 10^{-6}$ |
+| $P_+$ - $M_X$ | $40 \%$ | $47 \%$ |
 
-💡 **Note:** As you notice, while this chip's bit-flip performance is stellar, the phase-flip performance is still somewhat underwhelming.
-
-Cat qubit architectures are less demanding regarding qubit quality (a repetition code has a higher threshold than a surface code), but phase-flip performance still needs to improve by 1 to 2 orders of magnitude for error correction to work reliably. We are focused on improving this, with several promising solutions being tested in our lab. Stay tuned!
-
-Or, if you think this is an interesting research topic, don't hesitate to [drop us a line](../contact_us.md), we are happy to explore potential collaborations.
-
-💡 **Note:** If you run your own experiments, you will notice that P1 - Mz does not give the same results as P0 - Mz. This is because while our [state preparation](#initialize0-or-1) is symmetrical, our [readout protocol](#measure) is not. We could have randomly mapped $\ket{0}$ to $\ket{+\alpha}$ or $\ket{-\alpha}$, which would have yielded symmetrical results on average, but we chose to stay closer to the behavior of the physical system.
+> 💡 **Note:** If you run your own experiments, you will notice that $P_1$ - $M_Z$ does not give the same results as $P_0$ - $M_Z$. This is because while our [state preparation](#initialize0-or-1) is symmetrical, our [readout protocol](#measure) is not.
 
 ### Z gate performance
 
 These figures represent the probability of getting a bit-flip or phase-flip during a Z-gate.
 
-|  | average_nb_photons = 4 | average_nb_photons = 16 |
+|  | average_nb_photons = 4 | average_nb_photons = 11 |
 | --- | --- | --- |
-| Bit-flip | 0.15 % | < 0.001 % |
-| Phase-flip | 20 % | 40 % |
+| Bit-flip | $5.0 \times 10^{-4}$ | $5.8 \times 10^{-8}$ |
+| Phase-flip | $22 \%$ | $27 \%$ |
+
+> 💡 **Note:** As you'll notice, while this chip's bit-flip performance is stellar, the phase-flip performance is still somewhat underwhelming.
+
+> A few things to keep in mind when reading these results:
+
+> - The readout protocol has not been optimized and can still be further improved.
+> - Better performance has already been achieved with different designs. For example, the [AutoCat design](https://journals.aps.org/prx/abstract/10.1103/PhysRevX.14.021019) features a 3.5 % error rate for the Z gate.
+> - Cat qubit architectures are less demanding regarding qubit quality (a repetition code has a higher threshold than a surface code).
+
+> Overall, phase-flip performance still needs to improve by 1 to 2 orders of magnitude for error correction to work reliably. We are focused on improving this, with several promising solutions being tested in our lab. Stay tuned!
+
+> Or, if you think this is an interesting research topic, don't hesitate to [drop us a line](../contact_us.md), we are happy to explore potential collaborations.
 
 ### Chip parameters
 
 These parameters were measured in Alice & Bob’s lab and cannot be reproduced using Felis.
 
 | Metric | Measured value |
 | --- | --- |
-| f_a | 1.079 GHz |
-| f_b | 7.898 GHz |
-| κ_1/2π | 2.26 kHz |
-| κ_1_eff/2π | 22.7 kHz |
-| κ_b/2π | 22 MHz |
-| κ_2/2π | 250 kHz |
-| g_2/2π | 1.2 MHz |
-| K/2π | -12 kHz |
-| κ_φ/2π | ... kHz |
-| n_th | 2 |
-| n_th_buffer | not measured |
+| $f_a$ | 1.079 GHz |
+| $f_b$ | 7.898 GHz |
+| $\kappa_1/2\pi$ | 2.26 kHz |
+| $\kappa_{1_{eff}}/2\pi$ | 19.9 kHz |
+| $\kappa_b/2\pi$ | 22 MHz |
+| $\kappa_2/2\pi$ | 250 kHz |
+| $g_2/2\pi$ | 1.2 MHz |
+| $K/2\pi$ | -12 kHz |
+| $\kappa_φ/2\pi$ | < 10 kHz |
+| $n_th$ | 2 |
 
 ## Gate implementation details
 
 There are three types of gates available: 
 
-- "preparation gates" (P0, P1, P+, P-), used to initialize the state of the cat on some remarkable spots on the Bloch sphere;
-- "one-qubit operations" (Z, I), used to transform one state into an other one
-- "measurement gates" (MX, MZ), used to measure the state of the qubit.
+- "preparation gates" ($P_0$, $P_1$, $P_+$, $P_-$), used to initialize the state of the cat on some remarkable spots on the Bloch sphere;
+- "one-qubit operations" ($Z$, $I$), used to transform one state into an other one
+- "measurement gates" ($M_X$, $M_Z$), used to measure the state of the qubit.
 
 ![Bloch sphere for cat qubits](../media/reference/bloch_cats.png)
 
 *Bloch sphere for cat qubits. The Z basis states is constituted from the stable coherent state, while the X axis basis is constituted from the even and odd cat states.*
 
 For cat qubits, the basis states along the Z axis and the X axis are not equivalent. Indeed, the states along Z are stable (long lived) since they are protected by the two-photon dissipation process, while the states along the X axis are not protected: their lifetime is given by the lifetime of the memory under pump.
 
-Therefore, we can regroup the available gates into two families: the Z family associated to the the Z basis (P0, P1, I, MZ), and the X family associated to the X basis (P+, P-, Z, I, MX).
+Therefore, we can regroup the available gates into two families: the Z family associated to the the Z basis ($P_0$, $P_1$, $I$, $M_Z$), and the X family associated to the X basis ($P_+$, $P_-$, $Z$, $I$, $M_X$).
 
 Furthermore, each of these gates should be ''bias preserving'', which means that they preserve the high stability of the Z axis. In the following, we present each of these gates according to the Z and X family classification.  
 
-### initialize('0' or '1')
+### `initialize('0' or '1')`
 
-The preparation of the states $\left|0\right\rangle$ and $\left|1\right\rangle$ is done by displacing the memory from the vacuum state to the given locations $\alpha$ and $-\alpha$ respectively in phase space. This displacement is achieved by driving the memory resonantly while the two photon stabilization is turned off, for a given fixed duration and complex amplitude that has already been calibrated (figure).
+The preparation of the states $\left|0\right\rangle$ and $\left|1\right\rangle$ is done by displacing the memory from the vacuum state to the given locations $\alpha$ and $-\alpha$ respectively in phase space. This displacement is achieved by driving the memory resonantly while the two photon stabilization is turned off, for a given fixed duration and complex amplitude.
 
 ![Pulse sequence for initialize('0' or '1')](../media/reference/prepare0_1.png)
 
 *Pulse sequence for preparing $\left|0\right\rangle$ and $\left|1\right\rangle$ with a resonant drive on the memory.*
 
-### initialize('+')
+### `initialize('+')`
 
 Preparing the cat state $\left|+\right\rangle$ is done by ''inflation from vacuum.'' Starting from an empty memory, we turn on the two photon stabilization and the buffer drive, thereby leading to a stabilized cat state for timescales greater than $1 /\kappa_2$ : this process is called an ''inflation''. Since the parity of the state is preserved through this process, the stabilized cat state will be the even cat state $\left|+\right\rangle$. 
 
@@ -129,45 +134,42 @@ Preparing the cat state $\left|+\right\rangle$ is done by ''inflation from vacuu
 
 *Pulse sequence for preparing the even cat $\left|+\right\rangle$, and the corresponding Wigner figure*
 
-### initialize('-')
+### `initialize('-')`
 
 Preparing the odd cat state $\left|-\right\rangle$ is done by applying a gate $Z_{\pi} $ to the even cat state $\left|+\right\rangle$.
 ![Pulse sequence for preparing the odd cat](../media/reference/prepare-.png)
 
 *Pulse sequence for preparing $\ket{-}$*
 
-### measure
+### `measure(0)`
 
-Given a state $\left|\psi\right\rangle = a \left|0\right\rangle + b\left|1\right\rangle = a \left|\alpha\right\rangle + b\left|-\alpha\right\rangle + O\left(e^{-2|\alpha|^2}\right)$, the measurement of Z is done using the ''Cat Longitudinal Readout'' (CaLoR) protocol. It consists in turning off the 2 photon pump, then displacing the resulting state by $+\alpha$, and finally reading out the number of photons. 
+Given a state $\left|\psi\right\rangle = a \left|0\right\rangle + b\left|1\right\rangle = a \left|\alpha\right\rangle + b\left|-\alpha\right\rangle + O\left(e^{-2|\alpha|^2}\right)$, the measurement of Z is done using the ''Cat Longitudinal Readout'' (CaLoR) protocol. It consists in turning off the 2 photon pump, then displacing the resulting state by $+\alpha$, and finally reading out the number of photons. Read more about it in [our latest Nature paper](https://www.nature.com/articles/s41586-024-07294-3) ([arXiv link](https://arxiv.org/pdf/2307.06617.pdf)).
 
 The result of the displacement is a state $\left|\psi'\right\rangle \equiv D(\alpha)\left|\psi\right\rangle = a \left| 2\alpha \right\rangle + b\left|0\right\rangle$. Reading out the number of photons of $\left|\psi'\right\rangle$ yields the number $4 |\alpha|^2 |a|^2 $, which directly relates to $\left\langle \psi' |Z| \psi' \right \rangle = |a|^2 - |b|^2 = 2 |a|^2 - 1$ (recall that $|\alpha|^2 $ is a fixed quantity). Note that this measurement is destructive: it does not preserve the state, so the cat is dead after this readout!  
 
-![Pulse sequence for the MZ measurement gate](../media/reference/mz.png)
-
-*Pulse sequence for the MZ measurement gate*
+![Pulse sequence for the $M_Z$ measurement gate](../media/reference/mz.png)
 
-**Calibration:**
-In practice, we perform the complementary operation with the opposite displacement $D(-\alpha)$ to subtract eventual drifts in the measurement. Furthermore, since we are only concerned in the two pointer states $\left|0\right\rangle$ and $\left|1\right\rangle$, the distinction between the two can be enhanced using thresholding. Finally, the number of photons is measured through the ''longitudinal readout'' protocol (see [Réglade, Bocquet, ''et.al.'' '''Nature''' (2024)](https://www.nature.com/articles/s41586-024-07294-3) - or [arXiv](https://arxiv.org/abs/2307.06617) for more details).
+*Pulse sequence for the $M_Z$ measurement gate*
 
-### measure_x
+### `measure_x(0)`
 
-Parity measurement is done by mapping the parity basis to the logical basis via a $Y_{\frac{\pi}{2}} = Z_{\frac{\pi}{2}} X_{\frac{\pi}{2}} $ gate $\{\left|+\right\rangle, \left|-\right\rangle\}  \rightarrow \{\left|0\right\rangle, \left|1\right\rangle\}$ . In practice this is realized through the ''Holonomic sequence'' which consists of a $Z_{\frac{\pi}{2}} $gate followed by a deflate and orthogonal inflate.
+Parity measurement is done by mapping the parity basis to the logical basis via a $Y_{\frac{\pi}{2}} = X_{\frac{\pi}{2}} Z_{\frac{\pi}{2}} $ gate $\{\left|+\right\rangle, \left|-\right\rangle\}  \rightarrow \{\left|0\right\rangle, \left|1\right\rangle\}$ . In practice this is realized through the ''Holonomic sequence'' which consists of a $Z_{\frac{\pi}{2}} $gate followed by a deflate and orthogonal inflate. Read more about this sequence in [our latest Nature paper](https://www.nature.com/articles/s41586-024-07294-3) ([arXiv link](https://arxiv.org/pdf/2307.06617.pdf)).
 
-![Pulse sequence for the MX measurement gate](../media/reference/mx.png)
+![Pulse sequence for the $M_X$ measurement gate](../media/reference/mx.png)
 ![Wigner 1](../media/reference/mx1.gif) ![Wigner 2](../media/reference/mx2.gif)
 
-*Pulse sequence and Wigner figures for the MX measurement gate*
+*Pulse sequence and Wigner figures for the $M_X$ measurement gate*
 
-### z
+### `z(0)`
 
-The Zeno gate allows to change the phase $\phi$ of the logical states superposition $\frac{\left|0\right\rangle + e^{i\phi} \left|1\right\rangle}{2}$, and thus for example change the parity of a cat state $Z_{\frac{\pi}{2}} \left|+\right\rangle = \left|-\right\rangle$. Zeno gate is realized by driving the memory resonantly with a phase orthogonal to the stabilization axis, while stabilizing the memory. The resonant memory drive causes the fringes to roll with a frequency $\Omega_z= 4\alpha\epsilon_z$ proportional to the cat size $\alpha$ and the Zeno drive amplitude $\epsilon_z$.
+The Zeno gate allows to change the phase $\phi$ of the logical states superposition $\frac{\left|0\right\rangle + e^{i\phi} \left|1\right\rangle}{2}$, and thus for example change the parity of a cat state, since $Z_{\pi} \left|+\right\rangle = \left|-\right\rangle$. Zeno gate is realized by driving the memory resonantly with a phase orthogonal to the stabilization axis, while stabilizing the memory. The resonant memory drive causes the fringes to roll with a frequency $\Omega_z= 4\alpha\epsilon_z$ proportional to the cat size $\alpha$ and the Zeno drive amplitude $\epsilon_z$.
 
 ![Pulse sequence for the Z gate](../media/reference/z.png)
 ![Wigner for the Z gate](../media/reference/z.gif)
 
 *Pulse sequence and Wigner figure for Zeno pulse while stabilizing cat qubits*
 
-### reset
+### `reset(0)`
 
 Resetting the memory is done by activating a so called SWAP interaction with the buffer $H_{\rm SWAP} = g_1 a^\dagger b + g_1^*a^\dagger b$. Since the buffer is cold and relaxes quickly to the environment, the memory will be endowed with a strong 1 photon loss mechanism, leading it to empty to the vacuum state.
 

docs/reference/papers.md

@@ -21,7 +21,7 @@ While the bit-flip lifetime of the chip in Raphaël Lescanne's seminal experimen
 
 The chip in this paper is now called "Boson 2".
 
-- arXiv: [https://arxiv.org/abs/2204.09128](https://arxiv.org/abs/2204.09128)
+- Physical Review X Quantum: [https://journals.aps.org/prxquantum/abstract/10.1103/PRXQuantum.4.020350](https://journals.aps.org/prxquantum/abstract/10.1103/PRXQuantum.4.020350)
 
 ## Quantum control of a cat-qubit with bit-flip times exceeding ten seconds
 
@@ -34,7 +34,7 @@ The chip in this paper is now called "Boson 3".
 
 ## Autoparametric Resonance Extending the Bit-Flip Time of a Cat Qubit up to 0.3 s
 
-In parallel, we also explore better transmon-based designs, such as AutoCat which reached a 0.3 s bit-flip lifetime.
+In parallel, we also explore better transmon-based designs, such as AutoCat which reached a 0.3 s bit-flip lifetime and a better phase-flip performance (error rate as low as 3.5 % for the Z gate).
 
 - Physical Review X: [https://journals.aps.org/prx/abstract/10.1103/PhysRevX.14.021019](https://journals.aps.org/prx/abstract/10.1103/PhysRevX.14.021019)