††thanks: These authors co-supervised the project††thanks: These authors co-
supervised the project

# Autoparametric resonance extending the bit-flip time of a cat qubit up to
0.3 s

A. Marquet  Alice & Bob, 53 Bd du Général Martial Valin, 75015 Paris, France
Ecole Normale Supérieure de Lyon, CNRS, Laboratoire de Physique, F-69342 Lyon,
France  A. Essig  Alice & Bob, 53 Bd du Général Martial Valin, 75015 Paris,
France  J. Cohen  Alice & Bob, 53 Bd du Général Martial Valin, 75015 Paris,
France  N. Cottet  Alice & Bob, 53 Bd du Général Martial Valin, 75015 Paris,
France  A. Murani  Alice & Bob, 53 Bd du Général Martial Valin, 75015 Paris,
France  E. Abertinale  Alice & Bob, 53 Bd du Général Martial Valin, 75015
Paris, France  S. Dupouy  Alice & Bob, 53 Bd du Général Martial Valin, 75015
Paris, France  Ecole Normale Supérieure de Lyon, CNRS, Laboratoire de
Physique, F-69342 Lyon, France  A. Bienfait  Ecole Normale Supérieure de Lyon,
CNRS, Laboratoire de Physique, F-69342 Lyon, France  T. Peronnin  Alice & Bob,
53 Bd du Général Martial Valin, 75015 Paris, France  S. Jezouin  Alice & Bob,
53 Bd du Général Martial Valin, 75015 Paris, France  R. Lescanne  Alice & Bob,
53 Bd du Général Martial Valin, 75015 Paris, France  B. Huard  Ecole Normale
Supérieure de Lyon, CNRS, Laboratoire de Physique, F-69342 Lyon, France

###### Abstract

Cat qubits, for which logical $|0\rangle$ and $|1\rangle$ are coherent states
$|\pm\alpha\rangle$ of a harmonic mode, offer a promising route towards
quantum error correction. Using dissipation to our advantage so that photon
pairs of the harmonic mode are exchanged with single photons of its
environment, it is possible to stabilize the logical states and exponentially
increase the bit-flip time of the cat qubit with the photon number
$|\alpha|^{2}$. Large two-photon dissipation rate $\kappa_{2}$ ensures fast
qubit manipulation and short error correction cycles, which are instrumental
to correct the remaining phase-flip errors in a repetition code of cat qubits.
Here we introduce and operate an autoparametric superconducting circuit that
couples a mode containing the cat qubit to a lossy mode whose frequency is set
at twice that of the cat mode. This passive coupling does not require a
parametric pump and reaches a rate $\kappa_{2}/2\pi\approx 2~{}\mathrm{MHz}$.
With such a strong two-photon dissipation, bit-flip errors of the
autoparametric cat qubit are prevented for a characteristic time up to 0.3 s
with only a mild impact on phase-flip errors. Besides, we illustrate how the
phase of a quantum superposition between $|\alpha\rangle$ and
$|-\alpha\rangle$ can be arbitrarily changed by driving the harmonic mode
while keeping the engineered dissipation active.

Quantum error correction is instrumental to building useful quantum
processors. It is based on the gathering of many physical quantum systems in
order to form protected logical qubits. The number of required physical
systems can be daunting but strategies involving harmonic oscillators instead
of qubits promise to reduce that number by a large factor [1, 2, 3, 4, 5, 6].

Figure 1: (a) A four wave mixing coupler such as a transmon or ATS swaps pairs
of photons of a memory mode at $\omega_{m}$ for single photons of a buffer
mode at $\omega_{b}$ at a rate $g_{2}$ owing to a pump at
$|\omega_{b}-2\omega_{m}|$. The buffer loss rate $\kappa_{b}$ thus leads to an
effective two-photon dissipation rate $\kappa_{2}$ that scales with pump
amplitude. Driving the buffer mode on resonance at a displacement rate
$\epsilon_{d}$ stabilizes a cat code $\\{|\pm\alpha\rangle\\}$. (b) A three-
wave mixing coupler passively performs the same photon exchange when
$\omega_{b}=2\omega_{m}$. (c) Scheme of the autoparametric cat. The three-wave
mixing coupler is a ring of three Josephson junctions threaded by a flux
$\phi_{\mathrm{ext}}=\varphi_{\mathrm{ext}}\varphi_{0}$, with
$\varphi_{0}=\hbar/2e$, Josephson energies $E_{W}/h\approx 115~{}\mathrm{GHz}$
and $E_{J}/h\approx 250~{}\mathrm{GHz}$. The buffer (green) and memory (blue)
mode geometries are represented as field vectors. (d) Optical image of the
device. False colors highlight the input buffer/flux line (green), the
tomography transmon and coupling capacitor to its readout resonator (red) as
well as the three-wave mixing coupler (purple) (see Sec. D). (e) Dots:
measured resonance frequency of the buffer $\omega_{b}$ (green) and twice the
measured resonance frequency of the memory $2\omega_{m}$ (blue) as a function
of the flux threading the ring. Dashed lines indicate the flux biases where
the circuit is operated. Solid lines: Best fit of the circuit parameters (see
Sec. B).

A prominent example is the cat qubit, whose computational states are coherent
states $|\pm\alpha\rangle$ of a harmonic oscillator, such as a superconducting
microwave cavity [7, 8, 9, 1]. These states can be stabilized through
measurement based feedback [10, 11, 12], Hamiltonian engineering [13, 14, 15,
16] or reservoir engineering [17, 18, 19, 20, 21, 22]. The latter strategy
prevents the state of that memory cavity from leaking out of the cat qubit
subspace by engineering its coupling to the environment with the key feature
that photon pairs are lost at a rate $\kappa_{2}$. The bit flip time
$T_{\mathrm{X}}$ then increases exponentially with the photon number
$|\alpha|^{2}$ at the modest expense of a linear deterioration of the phase
flip rate $\Gamma_{\mathrm{Z}}\propto|\alpha|^{2}$ [19]. Reaching large values
of two-photon rate $\kappa_{2}$ is critical in this strategy. First, to ensure
exponential bit-flip protection, it should overcome any parasitic processes
affecting the memory such as dephasing, thermal excitation, Kerr effect,
frequency shifts due to a thermally populated ancilla qubit [19, 23] or gate
drives. Second, its value sets a higher bound on cat qubit gate speed and
needs to be large compared to the residual single photon loss rate
$\kappa_{1}$, the main cause of phase-flip errors [24]. Besides, a repetition
code made of a chain of cat qubits can protect against phase-flip errors under
the condition $\kappa_{2}/\kappa_{1}\gtrsim 10^{2}$ [25, 26, 24]. Introducing
a new scheme called the autoparametric cat, our experiment reaches
$\kappa_{2}$ rates as high as $2\times 2\pi~{}\mathrm{MHz}$, about 150 times
larger than $\kappa_{1}$. As we increase the photon number, we observe an
improvement of the bit-flip time by more than a factor $25,000$ reaching
$0.36\pm 0.15~{}\mathrm{s}$ while the phase-flip rate degrades by less than a
factor of $6$. In addition, we demonstrate a Z gate of the cat code with a
fidelity of $96.5\pm 2\%$ in 28 ns.

To achieve two-photon coupling between the memory and its environment, an
intermediary mode with single photon coupling $\kappa_{b}$ to the environment
acts as a buffer and a two-to-one photon exchange Hamiltonian $\hat{H}=\hbar
g_{2}\hat{m}^{2}\hat{b}^{\dagger}+h.c.$ is activated with a rate $g_{2}$. The
operators $\hat{m}$ and $\hat{b}$ are the annihilation operators of the memory
and buffer mode respectively. In the limit where $g_{2}$ is small enough
compared to $\kappa_{b}$, the two-photon dissipation rate reads
$\kappa_{2}=4g_{2}^{2}/\kappa_{b}$. This interaction has been engineered in
the past by parametrically pumping a 4 wave mixing non-linearity (Fig. 1a),
such as a single junction or an asymmetrically threaded squid (ATS), at the
frequency matching condition $\omega_{p}=2\omega_{m}-\omega_{b}$. However,
although the two-to-one photon exchange rate $g_{2}$ scales with the pump
amplitude, increasingly large pump powers are known to affect coherence times
and activate higher non-linear terms in the Hamiltonian [18]. In practice, the
limited range of $\kappa_{2}$ rates that could be obtained with pumped
nonlinearities prevents surpassing the self-Kerr rate or the dispersive
coupling rate between transmon and memory [19, 18, 27]. In this work, we use 3
wave mixing instead (Fig. 1b), thus alleviating the need for a pump to mediate
the 2 photon interaction as in the 1989 proposal in Ref. [28]. The frequency
matching condition then becomes $2\omega_{m}=\omega_{b}$. This condition is
characteristic of autoparametric systems so that the buffer field passively
performs a parametric driving of the memory [29]. Remarkably, the resulting
exchange rate $g_{2}$ is much larger than what can be reached for 4 wave
mixing.

The mixing element of the autoparametric cat consists of two main Josephson
junctions with energy $E_{J}$ symmetrically arranged within a superconducting
loop that is threaded with an external magnetic flux $\phi_{\mathrm{ext}}$
(Fig. 1c). These two junctions in parallel configuration have a common mode
serving as a memory mode and a differential mode, associated with the flux
degree of freedom of the loop and serving as a buffer mode (see Sec. A). In
order to lower the relatively high frequency of the buffer mode and increase
its flux tunability, a third Josephson junction with energy $E_{W}$ is added
in the loop making its configuration similar to previously realized circuits
[30, 31, 32, 33, 34]. Besides, it endows the memory mode with a frequency
sweet spot provided $E_{W}<E_{J}/\sqrt{2}$. By symmetry, the memory mode does
not participate in this third junction which plays no role in the two-to-one
photon exchange Hamiltonian. To tune the mode frequencies and participation in
the mixing element, the superconducting loop is further integrated within a
linear microwave network that preserves the mode symmetries (Fig. 1c).
Finally, a single input line (green in Fig. 1d) couples to the circuit to
provide both fast flux bias and drive the buffer. The frequency tunability
makes it difficult to engineer a filter that protects the memory lifetime.
Instead, we leverage the symmetries of the circuit (Fig. 1d) and position the
input line such that it does not impact the memory quality factor while
preserving a strong coupling to the buffer (see Sec. C). We achieve a buffer
coupling rate $\kappa_{b}/2\pi\approx 40~{}\mathrm{MHz}$ and a much lower
memory loss rate $\kappa_{1}/2\pi\approx 14~{}\mathrm{kHz}$. Note that an RF-
SQUID or a SNAIL [35] could have been used as a 3-wave mixing element.
However, the existence of a sweet-spot in flux and the possibility to leverage
the circuit symmetries to preserve the memory quality factor made us favor
this design. Finally, a transmon qubit is inductively coupled to the memory
with $\chi/2\pi=170~{}\mathrm{kHz}$ to perform the Wigner tomography of the
memory mode [36, 37, 38].

The mixing element enforces a two-to-one photon exchange Hamiltonian with
strength $g_{2}$. Around the memory sweet spot, its value is well approximated
by (see Sec. B)

$g_{2}\approx\frac{E_{W}}{\hbar}\left(1-\frac{{\delta\varphi_{\mathrm{ext}}}^{2}}{2}\right)\varphi_{\mathrm{zpf},m}^{2}\varphi_{\mathrm{zpf},b}$
(1)

where
$\delta\varphi_{\mathrm{ext}}=(\phi_{\mathrm{ext}}-\phi_{\mathrm{ext}}^{(\mathrm{sweet})})/\varphi_{0}$
is the distance from the sweet spot and $\varphi_{\mathrm{zpf},m/b}$ are the
zero point fluctuation of the phase from the memory and buffer mode
respectively across either of the main junction. In practice, it is hard to
ensure the frequency matching condition precisely at the sweet spot. In the
experiment, $g_{2}$ is close to its maximum value at the flux
$\phi_{\mathrm{QEC}}$ for which the frequency of the buffer matches twice the
frequency of the memory
$\omega_{b}(\phi_{\mathrm{QEC}})=2\omega_{m}(\phi_{\mathrm{QEC}})\approx~{}2\pi\times
7.896\mathrm{GHz}$ (see Fig. 1e). Therefore the three-wave mixing term
performs the desired swap between pairs of memory photons and single buffer
photons at $\phi_{\mathrm{QEC}}$.

Figure 2: (a) Dots: Measured two-photon relaxation rate $\kappa_{2}$ as a
function of flux bias close to $\phi_{\mathrm{QEC}}$. Error bars represent
statistical uncertainties. Inset: pulse sequence used for the measurement. The
detuning between buffer frequency and twice the memory frequency is indicated
on the top axis. (b) Top: Measured Wigner functions of the memory after the
decay times indicated on the figure for $\left|\alpha\right|=2.5$ and at the
flux indicated by the star in (a). Bottom: results of the simulation using
$\kappa_{2}/2\pi=2.16~{}\mathrm{MHz}$.

The preparation of a cat state
$|C_{+}^{\alpha}\rangle\propto|\alpha\rangle+|-\alpha\rangle$ consists in
starting from the vacuum state in the memory at $\phi_{\mathrm{QEC}}$ and
turning on a drive with an amplitude $|\epsilon_{d}|=|\alpha|^{2}g_{2}$ (Fig.
1b) at twice the memory frequency $\omega_{d}=2\omega_{m}$ for a time of $300$
ns. It is shorter than the characteristic time $1/|\alpha|^{2}\kappa_{1}$ of a
photon loss yet long enough for the two-photon dissipation and drive to
stabilize the memory into the cat qubit manifold. From there, one may
determine the two-photon loss rate $\kappa_{2}$ by monitoring how the memory
decays once the drive has been turned off. Fitting the measured Wigner
function $W(\beta)$ at various time steps with the solution of a Lindblad
master equation where $\kappa_{2}$ is the single free parameter allows us to
determine $\kappa_{2}/2\pi\approx 2.16\pm 0.1~{}\mathrm{MHz}$ at
$\phi_{\mathrm{QEC}}$. The ratio $\kappa_{2}/\kappa_{1}\approx 1.5\times
10^{2}$ is much larger than in previous implementations of two-photon
dissipation using four-wave mixing [17, 18, 39, 23]. This procedure is
repeated for various flux biases around $\phi_{\mathrm{QEC}}$ in Fig. 2a,
which shows the range of detuning $\omega_{b}-2\omega_{m}$ over which the two-
photon loss rate decreases. The discrepancy between experiment and simulation
of the evolution of the Wigner functions (Fig. 2b), mainly visible at 8 ns,
can be attributed to the breakdown of the condition for adiabatic elimination
of the buffer mode (see Sec. G).

Note that quantum state tomography of the memory is performed using a cavity
displacement and the dispersive coupling to the transmon [36, 37, 38].
However, since both are inhibited by two-photon loss, Wigner tomography is
performed at a flux $\phi_{\mathrm{tomo}}$ (see Fig. 1e) such that the strong
detuning $\omega_{b}-2\omega_{m}$ disables two-photon dissipation. In
practice, the tomography thus requires to abruptly change the flux bias
between $\phi_{\mathrm{QEC}}$ and $\phi_{\mathrm{tomo}}$ (see Sec. E.2).
Besides, we use a tomography technique that strongly benefits from the two-
photon dissipation to avoid measurement errors associated with large photon
numbers [40].

Figure 3:  (a) Pulse sequence of the phase flip rate measurement. (b) Dots:
measured phase flip rate $\Gamma_{Z}$ (linear scale) of the cat code as a
function of photon number $|\alpha|^{2}$. All values are obtained by fitting
$W(0)$ to an exponential decay in time. Error bars represent statistical
uncertainties (see Sec. I.1). Line: expected rate $2|\alpha|^{2}\kappa_{1}$.
(c) Pulse sequence of the bit-flip time measurement. (d) Dots: measured bit
flip time (log scale) of the cat code as a function of photon number
$|\alpha|^{2}$. All values are obtained by fitting the difference
$W(\alpha)-W(-\alpha)$ to an exponential decay in time. Error bars represent
statistical uncertainties (see Sec. I.1). Solid black line: expected bit-flip
time with $\kappa_{\varphi}^{m}/2\pi=0.08~{}\mathrm{MHz}$ under the adiabatic
elimination of the buffer:
$e^{2|\alpha|^{2}}/(|\alpha|^{2}\kappa_{\varphi}^{m})$. Dashed blue line:
simulated bit-flip time with the same $\kappa_{\varphi}^{m}$, assuming a
detuning $2\omega_{m}-\omega_{b}=2\pi\times 3.5~{}\mathrm{MHz}$. Red dots:
predicted limitation of $T_{\mathrm{X}}$ imposed by the measured excitation of
higher states of the transmon (Fig. 25).

The phase-flip rate $\Gamma_{Z}$ of the cat code can be measured in a similar
manner as the two photon loss rate (Fig. 3a). Starting from the quantum
superposition $|C_{+}^{\alpha}\rangle$, the parity, hence the measured $W(0)$,
decays exponentially with time at a rate $\Gamma_{Z}$ while the drive
$\epsilon_{d}(\alpha)$ is kept on. In Fig. 3b are shown the observed
$\Gamma_{Z}$ rates as a function of photon number $\left|\alpha\right|^{2}$ in
the cat code. As expected, the phase-flip rate increases linearly as
$\Gamma_{Z}=2|\alpha|^{2}\kappa_{1}$ until it goes above 20 photons.

Figure 4: (a) Pulse sequence used to perform a $Z$ rotation in the cat qubit
$\\{|\alpha\rangle,|-\alpha\rangle\\}$ basis. (b) Dots: Measured oscillations
of $W(0)$ as a function of time $t$ using the pulse sequence of Fig. 3a where
the cat code is stabilized with a photon number $|\alpha|^{2}=9.3$. An
additional displacement drive at $\omega_{m}$ starts $240~{}\mathrm{ns}$ after
the buffer drive is turned on. Here, its amplitude $\epsilon_{Z}(t)$ is
Gaussian shaped with a mean amplitude
$\bar{\epsilon}_{Z}/2\pi=1.625~{}\mathrm{MHz}$ and its phase is chosen to
displace in the direction indicated by the arrow in c). Line: fit to
oscillations at a frequency $\Omega_{Z}/2\pi=19.8~{}\mathrm{MHz}$, which are
decaying at a rate $\kappa_{Z}/2\pi=0.62~{}\mathrm{MHz}$. (c) Measured Wigner
functions $W(\beta)$ after a $Z$ rotation of angle $\theta=2\pi$, $3\pi/2$,
and $\pi$ from top to bottom. (d) Dots: Inferred rotation frequency
$\Omega_{Z}$ as a function of cat code amplitude $\alpha$, and for various
mean drive amplitudes
$\bar{\epsilon}_{Z}/2\pi=0.32,0.625,0.965,1.295,1.625,1.955,2.285$, and
$2.66~{}\mathrm{MHz}$ from bright to dark orange. Lines: expected rotation
frequency $\Omega_{Z}=4\mathrm{Re}(\bar{\epsilon}_{Z}\alpha)$ around $Z$. (e)
Dots: Inferred decay rate $\kappa_{Z}$ as a function of $|\alpha|$ for the
same drive amplitudes. Lines: simulated decay rate with
$g_{2}/2\pi=6~{}\mathrm{MHz}$ as a fit parameter and the same detuning as in
Fig. 3d.

The bit-flip time $T_{\mathrm{X}}$ characterizes how fast the coherent state
$|\alpha\rangle$ decays to an equal mixture of $|-\alpha\rangle$ and
$|\alpha\rangle$. In order to measure it, the flux is first set to
$\phi_{\mathrm{tomo}}$ so that a memory drive can prepare the state
$|\alpha\rangle$. The flux is then turned back to $\phi_{\mathrm{QEC}}$ and
the buffer drive is immediately turned on with an amplitude
$\epsilon_{d}(\alpha)$ (Fig. 3c). We measure the Wigner functions
$W(\pm\alpha)$ for various waiting times and fit their difference
$W(\alpha)-W(-\alpha)\propto e^{-t/T_{\mathrm{X}}}$. The resulting bit-flip
time is shown in Fig. 3d and rises exponentially with photon number
$|\alpha|^{2}$ until about 12 photons. There, $T_{\mathrm{X}}$ grows by a
scaling factor of about $3.5$ per added photon, smaller than the limit of
$7.4$ predicted in case of pure dephasing alone (solid line) [24].

We explain this discrepancy by the breakdown of the approximation of adiabatic
elimination of the buffer. Indeed simulation of the evolution of the buffer-
memory bipartite system predicts smaller scaling factors (dashed line in Fig.
3d). Using the experimentally measured pure dephasing rate of the memory
$\kappa_{\varphi}^{m}/2\pi=0.08~{}\mathrm{MHz}$ (Fig. 24) is key to obtaining
a good match between simulations and experiment before saturation. Besides the
simulated bit-flip times strongly depend on the flux $\phi_{\mathrm{ext}}$ and
we set a detuning $2\omega_{m}-\omega_{b}=2\pi\times 3.5~{}\mathrm{MHz}$ in
order to better reproduce the measured bit-flip times. This corresponds to a
flux offset of $3~{}\times 10^{-4}\phi_{0}$, which could be attributed to flux
drifts during the month that separates the measurements of Fig. 2 and 3.

The bit-flip time saturates at $0.36\pm 0.15~{}\mathrm{s}$, reached for
$|\alpha|^{2}\approx 20$. Simulations reveal that while the transmon first
excited state no longer limits $T_{\mathrm{X}}$ [19], residual excitation of
transmon higher states sets a higher bound to the bit-flip time. Despite a
relatively small dispersive shift $\chi/2\pi=170~{}\mathrm{kHz}$, these higher
excited states of the transmon exit the dispersive regime yielding a large
frequency shift on the memory comparable to $\kappa_{2}|\alpha|^{2}$,
effectively turning off the cat qubit stabilization and inducing a bit-flip
error. We measure the higher excited states of the transmon while stabilizing
cat qubits of various mean photon number $|\alpha|^{2}$ (Fig. 25), and infer
the rate at which they get populated (red dots in Fig. 3d). The measured
saturation in bit-flip time seems to reach this upper bound, indicating that
the excitation of transmon higher states is the dominant limiting mechanism.
The intermediate regime $10<|\alpha|^{2}<15$ shows a sub-exponential increase
of $T_{\mathrm{X}}$ that could be attributed to self-Kerr effect and pure
dephasing of the buffer mode. In this region, we observe that the variance of
the two peaks at $\beta=\pm\alpha$ in $W(\beta)$ is larger than for a coherent
state. This is consistent with a thermalization of the memory by a thermal
occupation of the buffer mode owing to self-Kerr effect combined with pure
dephasing. Complementary measurements and simulations at various drive
frequencies corroborate this explanation (see Sec. I.6).

With such a large ratio $\kappa_{2}/\kappa_{1}$, it is possible to realize $Z$
gates in the cat code $\\{|\alpha\rangle,|-\alpha\rangle\\}$ using quantum
Zeno dynamics [41, 42, 43, 44, 45, 18]. Starting from $|C_{+}^{\alpha}\rangle$
with the buffer drive $\epsilon_{d}(\alpha)$ turned on, we continuously drive
the memory on resonance with an amplitude $\epsilon_{Z}$ [18] (Fig. 4a). The
phase of the latter is chosen so that, without two-photon loss, the memory
drive would induce a displacement (arrow in Fig. 4c) perpendicular to $\alpha$
in the memory phase space (see calibration procedure in Sec. H). The measured
Wigner functions of the memory are shown in Fig. 4c after three different
waiting times. Owing to quantum Zeno dynamics, the combined effects of two-
photon loss and buffer drive keep the memory state into the qubit space
generated by $\\{|\alpha\rangle,|-\alpha\rangle\\}$, but the memory drive
$\epsilon_{Z}$ rotates the phase $\theta$ of the quantum superposition
$(|\alpha\rangle+e^{i\theta}|-\alpha\rangle)/\sqrt{\mathcal{N}}$, which can be
seen as translated fringes in the Wigner function. The measured Wigner
function $W(0)$ as a function of time in Fig. 4b exhibits decaying
oscillations around the $Z$ axis of the cat qubit at a frequency $\Omega_{Z}$
and decay rate $\kappa_{Z}$.

The rotation frequency is expected to be given by
$\Omega_{Z}=4\mathrm{Re}(\epsilon_{Z}\alpha)$, which is precisely what we
observe in Fig. 4d. The decay rate $\kappa_{Z}$ has a more subtle dependence
on photon number $|\alpha|^{2}$ and memory drive $\epsilon_{Z}$ as seen in
Fig. 4e. In the ideal case and for constant $\epsilon_{Z}$, the rate is
expected to decay as
$\kappa_{Z}=2\kappa_{1}|\alpha|^{2}+\kappa_{b}\epsilon_{Z}^{2}/(2|\alpha|^{2}g_{2}^{2})$
[25]. The first term corresponds to phase flips at a rate $\Gamma_{Z}$ while
the second one corresponds to an induced leakage out of the confined cat qubit
Hilbert space when the drive amplitude $\epsilon_{Z}$ is too strong to be Zeno
blocked. This expression remains valid even outside of the adiabatic
elimination regime [46].

In practice, the experiment deviates from this simple picture owing to the
self-Kerr effect on the memory, slight detuning of the drive frequency, and
resonance frequency detuning induced by drifts in the flux bias. Complete
simulations of $\kappa_{Z}\left(\alpha\right)$ are sensitive to the two-to-one
photon coupling rate $g_{2}$ so that they are used to determine it. Adjusting
this parameter to $g_{2}/2\pi\approx 6\pm 0.5~{}\mathrm{MHz}$ leads to a good
match between measurement and simulations (see Fig. 4e). With such a coupling,
the adiabatic elimination of the buffer predicts a larger two-photon
dissipation rate $4g_{2}^{2}/\kappa_{b}\approx 3.6\times 2\pi~{}\mathrm{MHz}$
than what is measured in Fig. 2, which is expected since the condition
$8g_{2}|\alpha|<\kappa_{b}$ is not met for $\left|\alpha\right|^{2}\gtrsim 1$
[47]. Interestingly, despite the observed limitation on $T_{\mathrm{X}}$ in
the non adiabatic regime, it is still possible to improve gate speed and
fidelity by going to large values of $g_{2}|\alpha|$.

It is possible to infer the $Z$ gate fidelity from the measured evolution of
$W(\beta)$ during the drive. Owing to the noise bias of the cat code, the $Z$
gate is the least faithful when applied to Clifford states $|\pm X\rangle$ and
$|\pm Y\rangle$. In contrast, we measure that $|\pm Z\rangle$ states are
mapped onto themselves by the $Z$ gate with less than $3~{}10^{-6}$ error
probability under the driving conditions of Fig. 4b (see Sec. H.3). The gate
fidelity can be estimated to $F=1/2+\exp(-\pi\kappa_{z}/2\Omega_{z})/2$ (see
Sec. H.4). Using Gaussian pulses for the memory drive leads to $Z$ gate
fidelities $F=95\pm 2\%$ in 26 ns for $\left|\alpha\right|^{2}=9.3$ and for a
drive amplitude $\bar{\epsilon}_{Z}/2\pi=1.625~{}\mathrm{MHz}$. We further
improve the gate fidelity by using square pulses. Indeed, for a fixed average
drive amplitude $\bar{\epsilon}_{Z}$, the square pulse has the smallest
maximum amplitude, hence induces the least non-adiabatic errors. We then reach
$F=96.5\pm 2\%$ in 28 ns for the $Z$ gate.

In this work, we show that the autoparametric approach to two-photon exchange
can significantly enhance the two-photon dissipation rate $\kappa_{2}$,
exceeding the values obtained with prior parametric pumping strategies.
Remarkably, the autoparametric scheme does not seem to activate extra
relaxation processes as indicated by the linear increase of phase-flip rate up
to 20 average photons. Moreover, the associated photon exchange rate $g_{2}$
allows us to perform $Z$ gates with up to 96.5 % fidelity in 28 ns. While we
achieved a notable bit flip time $T_{\mathrm{X}}$ of up to 0.3 s, the pure
dephasing induced by flux noise currently limits the scaling rate with respect
to the photon number. This issue could be mitigated by operating closer to the
flux sweet spot through improved design choices. Additionally, removing the
transmon from the device should alleviate the limitation on the photon number
at which $T_{\mathrm{X}}$ reaches a saturation point. Consequently, the
readout of the memory could be conducted via the buffer mode, leveraging the
sizable two-to-one photon exchange rate [47]. Moreover, a limitation of the
current device originates from the pure dephasing of the buffer mode resulting
in thermal occupation of the buffer and in turn in an upper bound on
$T_{\mathrm{X}}$. Improving the insensitivity of the device to flux noise
would therefore be of prime importance. The figure-of-merit
$\kappa_{2}/\kappa_{1}=1.5\times 10^{2}$ exceeds the error correction
threshold for a repetition code based on cat qubits [24]. We foresee that
enhancing the memory lifetime by 1 or 2 orders of magnitude, similar to other
platforms, could dramatically bolster the scaling of error correction. In the
future, the large coupling rates of the autoparametric approach could be
leveraged under the frequency matching condition $\omega_{b}=4\omega_{m}$.
This could then stabilize cat qubits that are composed of superpositions of
the four coherent states $|\alpha\rangle$, $|i\alpha\rangle$,
$|-\alpha\rangle$, and $|-i\alpha\rangle$, whose autonomous stabilization is
still missing despite the strong interest of this approach for quantum error
correction [1].

###### Acknowledgements.

This research was supported by the QuantERA grant QuCos ANR-19-QUAN-0006, the
Plan France 2030 through the project ANR-22-PETQ-0003. We acknowledge IARPA
and Lincoln Labs for providing a Josephson Traveling-Wave Parametric
Amplifier. We thank the SPEC at CEA Saclay for providing fabrication
facilities. We thank Zaki Leghtas, Mazyar Mirrahimi, Ulysse Reglade and Marius
Villiers for inspiring discussions and feedback.

##  Appendix A Circuit design

Our autoparametric circuit can be understood as a limit case of a degenerate
parametric amplifier [48, 49, 50]. Such amplifiers are usually made of a
resonator $m$ connected to a pump mode $c$ via a nonlinear element such that
their interaction Hamiltonian reads $\hbar
g_{2}\hat{m}^{2}\hat{c}^{\dagger}+h.c.$.

When driving the pump mode $c$ at $2\omega_{m}$ with a large enough power such
that the number of photons in the pump mode exceeds the threshold
$n_{\mathrm{thr}}=\kappa_{m}^{2}/(4g_{2})^{2}$, parametric oscillation occurs.
The parametric oscillation threshold corresponds to the number of photons at
which the gain process compensates the losses of mode $m$. Above threshold,
the number of photons in $m$ stays finite owing to one of two possible
mechanisms: Kerr effect or pump depletion. The mode $m$ then emits radiation
with two possible phases [51, 52, 53]. These two phases correspond to two
coherent states $|\alpha\rangle$ and $|-\alpha\rangle$ of the mode $m$. This
phenomenon can be used to stabilize cat qubits.

The first limiting mechanism is the Kerr effect, which is a byproduct of the
nonlinearity used to generate the $g_{2}$ rate. As the number of photons
increases, the mode resonance frequency shifts up to a point where the pump at
$2\omega_{m}$ is so detuned that the gain and loss processes balance each
other. This Kerr limitation is the most standard one in Josephson circuits and
is at the origin of so called Kerr cats [13, 14, 15, 16]. The second limiting
mechanism is pump depletion, which is more common at optical wavelength. The
stiff pump approximation breaks down if the number of photons in the pump mode
is affected by how fast pump photons are consumed to generate photons in mode
$m$. The rate at which pump photons are regenerated by the drive is not fast
enough to produce pairs of photons in mode $m$.

The autoparametric design evades the Kerr limitation and operates in the pump
depletion limit, which engineers the reservoir of mode $m$ and not its
Hamiltonian. Instead of a far detuned pump mode, we use the buffer mode $b$
that resonates at the pump frequency $2\omega_{m}$. The high Q limit of mode
$a$ lowers the parametric oscillation threshold to $n_{\mathrm{thr}}\ll 1$
while the resonant driving of mode $b$ ensures that the regeneration rate of
pump photons in the buffer mode is minimal. Therefore, pump depletion is the
dominating mechanism for parametric oscillation stabilization. In the steady
state, the buffer is subject to two opposing driving forces. The buffer drive
and the action of the memory mode on the buffer via the 2-to-1 photon exchange
Hamiltonian compensate exactly. The buffer then reaches a steady state close
to the vacuum while the memory state converges to a superposition of
$|\alpha\rangle$ and $|-\alpha\rangle$.

This realization is at the origin of the autoparametric circuit design. We
started from a superconducting circuit widely used for degenerate parametric
amplification: a resonator comprising a DC flux biased SQUID that is flux
pumped at twice its frequency and acts as our memory mode [49]. Other non-
linear elements could be chosen from the variety of Josephson amplifiers that
have been designed over the past two decades. In this peculiar circuit, there
exists another mode associated to the flux degree of freedom which has a
differential symmetry with respect to the SQUID junctions but often ignored
for its very high frequency [34]. For our purpose, this mode, our buffer mode,
is brought down in frequency while preserving its symmetry by adding an
inductive element in the SQUID loop (the junction $E_{W}$) and splitting the
capacitance on either side of the SQUID. The symmetry is preserved and used
advantageously to be able to couple preferentially to the buffer mode without
needing frequency selective filtering to protect the memory lifetime. This
basis structure is then diluted with open and shorted stubs as described in
Fig. 1 to tune the modes $\varphi_{\mathrm{zpf}}$.

Compared to previously pumped circuits comprising an ATS [19, 23], this design
does not ensure that the memory Kerr non-linearity vanishes at the working
point, and it is hard to ensure by fabrication that $\phi_{\mathrm{QEC}}$ is a
local extremum for the memory frequency so that flux noise affects the memory
pure dephasing rate. However, on the one hand, it has been shown that Kerr
effect is not detrimental to cat qubit stabilization as long as it is smaller
than $g_{2}$ [27]. This condition is here satisfied by choosing the modes
$\varphi_{\mathrm{zpf}}$ adequately. On the other hand, cat-qubit
stabilization is primarily designed to correct against dephasing [1]. The pure
dephasing of the memory mode is further mitigated by design. Since the memory
mode does not participate in the central inductive element of the SQUID, we
implement it using a single Josephson junction acting as a weak link in the
loop $E_{W}<E_{J}$ without increasing the memory self-Kerr rate. Using a
single Josephson junction instead of a more linear inductance dramatically
increases the buffer susceptibility to flux compared to memory, hence the
buffer is responsible for meeting the frequency matching condition
$2\omega_{m}=\omega_{b}$ and the memory can afford a much weaker dependence on
flux, hence on flux noise. Using a Josephson junction with a lower Josephson
energy $E_{W}$ than the SQUID junctions $E_{J}$ creates sweet spots in the
memory flux dispersion and the circuit parameters are designed such that the
frequency matching condition occurs close to it. We provide further details on
the sweet spot in the next section.

Figure 5: Simplified circuit diagram of the device. Only the modes of the ring
of junctions are considered for simplicity. For each junction
$\varphi_{x}=\tilde{\varphi}_{x}+\bar{\varphi}_{x}$, respectively the
oscillating part and the equilibrium part.

##  Appendix B Hamiltonian derivation

This section derives the Hamiltonian of the simplified version of the circuit
represented in Fig. 5. Since this circuit operates in the regime of small zero
point fluctuations of the phase across the Josephson junctions, we will first
compute the effective inductance of each junction with the flux bias, then
find the eigenmodes of the linear part of the Hamiltonian and finally compute
the nonlinearities and discuss the impact of junction asymmetry.

Figure 6: Flux sweet spots of the mixing element. (a) Sum $\bar{\varphi}$ of
the phase differences across the three junctions as a function of the phase
difference $\bar{\varphi}_{W}$ across the weak junction, calculated using Eq.
(3) for $\beta_{J}=E_{W}/E_{J}=0.46$ as in the experiment (blue),
$\beta_{J}<1/\sqrt{2}$ (orange) and $\beta_{J}>1/\sqrt{2}$ (green). A flux
bias imposes $\varphi_{\mathrm{ext}}=\bar{\varphi}$ so that these curves show
which values of $\bar{\varphi}_{W}$ are possible. (b) Phase differences
$\bar{\varphi}_{J}$ (solid-dotted) and $\bar{\varphi}_{W}$ (dashed-dotted) as
a function of $\varphi_{\mathrm{ext}}$. Same color code for the values of
$\beta_{J}$ as in (a). Dotted lines correspond to extensions of the solutions
that are stable but not the lowest energy configuration. (a/b) The black
vertical/horizontal line corresponds to $\bar{\varphi}_{W}=\pi/2$. In (a) it
crosses $\bar{\varphi}$ at $\varphi_{\mathrm{ext}}^{\mathrm{(sweet)}}$ (thin
horizontal colored lines). In (b) it crosses $\bar{\varphi}_{W}$ where
$\bar{\varphi}_{J}$ has a sweet spot at the same values of
$\varphi_{\mathrm{ext}}^{\mathrm{(sweet)}}$ (thin vertical colored lines). The
blue and orange closed circles correspond to sweet spots in the lowest energy
configuration. The open green circle emphasizes that while the sweet spot
exists, it is not the lowest energy configuration. At this value of
$\varphi_{\mathrm{ext}}$ (vertical green line) another configuration is
favored (green closed circle) which is not a flux sweet spot.

###  B.1 Equilibrium phase configuration

First, we compute the effective inductance of the central mixing element. This
element comprises 2 identical main junctions with Josephson energy $E_{J}$ and
one weaker junction with energy $E_{W}=\beta_{J}E_{J}$ with $\beta_{J}<1$.
Following the procedure detailed in [54], we compute the equilibrium phase
drop across each junction $\bar{\varphi}_{1}$ and $\bar{\varphi}_{2}$ and
$\bar{\varphi}_{W}$. Note that we decompose each phase drop $\varphi_{x}$ as a
sum of a constant part $\bar{\varphi}_{x}$ and a dynamical part
$\tilde{\varphi}_{x}$ with zero mean. Current conservation inside the loop
imposes that $\bar{\varphi}_{1}=-\bar{\varphi}_{2}$ so that we denote
$\bar{\varphi}_{J}=\bar{\varphi}_{1}$ the main junction phase drop. Current
conservation further dictates that

$\bar{\varphi}_{J}=\arcsin(\beta_{J}\sin{\bar{\varphi}_{W}}).$ (2)

The phase drop around the loop
$\bar{\varphi}=\bar{\varphi}_{W}+\bar{\varphi}_{1}-\bar{\varphi}_{2}$ then
reads

$\bar{\varphi}=\bar{\varphi}_{W}+2\arcsin(\beta_{J}\sin{\bar{\varphi}_{W}}).$
(3)

Besides the superconducting loop is threaded by the external flux
$\phi_{\mathrm{ext}}=\varphi_{\mathrm{ext}}\varphi_{0}$, which leads to the
constraint $\varphi_{1}+\varphi_{W}-\varphi_{2}=\varphi_{\mathrm{ext}}$. The
constraint translates into

$\bar{\varphi}=\varphi_{\mathrm{ext}}\textrm{ and
}\tilde{\varphi}_{1}+\tilde{\varphi}_{W}-\tilde{\varphi}_{2}=0\,.$ (4)

Finding the configuration of phase drops at equilibrium thus consists in first
determining $\bar{\varphi}_{W}(\varphi_{\mathrm{ext}})$ by solving the
equation $\varphi_{\mathrm{ext}}=\bar{\varphi}(\bar{\varphi}_{W})$ of which a
graphical representation is shown in Fig. 6a. Then, one gets
$\bar{\varphi}_{J}$ from Eq. (2) and the effective inductive energies of each
junction $\bar{E}_{J}=E_{J}\cos(\bar{\varphi}_{J})$,
$\bar{E}_{W}=E_{W}\cos(\bar{\varphi}_{W})$. The effective inductances follow
as $\bar{L}_{J}=\varphi_{0}^{2}/\bar{E}_{J}$ and
$\bar{L}_{W}=\varphi_{0}^{2}/\bar{E}_{W}$. In case where $\beta_{J}>1/2$, the
function $\bar{\varphi}_{W}(\varphi_{\mathrm{ext}})$ is multi-valued [54] as
shown in Fig. 6b. In order to avoid hysteresis effects or instabilities, we
chose $\beta_{J}<1/2$ which is equivalent to $E_{W}<E_{J}/2$. We can check in
Fig. 6a that $\bar{\varphi}(\bar{\varphi}_{W})$ is indeed monotonous in that
case.

For $\beta_{J}=0.46$ as in our device, at $\phi_{\mathrm{QEC}}=0.311\phi_{0}$
that is slightly offset from the sweet spot ($0.40\phi_{0}$), we find
$\bar{\varphi}_{J}=0.42$ and $\bar{\varphi}_{W}=1.11$ leading to
$\bar{E}_{J}/h=228~{}\mathrm{GHz}$ and $\bar{E}_{W}/h=51~{}\mathrm{GHz}$. At
$\phi_{\mathrm{tomo}}=0.168\phi_{0}$, we find $\bar{\varphi}_{J}=0.25$ and
$\bar{\varphi}_{W}=0.56$ leading to $\bar{E}_{J}/h=242~{}\mathrm{GHz}$ and
$\bar{E}_{W}/h=97~{}\mathrm{GHz}$.

We further study the mixing element to determine the value of the flux sweet
spot to evade the memory from flux noise. It is defined as the flux
$\varphi_{0}\varphi_{\mathrm{ext}}^{\mathrm{(sweet)}}$ such that

$\left.\frac{d\omega_{m}}{d\varphi_{\mathrm{ext}}}\right|_{\varphi_{\mathrm{ext}}^{\mathrm{(sweet)}}}=0.$
(5)

We focus on the case where
$\varphi_{\mathrm{ext}}^{\mathrm{(sweet)}}\in[0,\pi]$ because of the $2\pi$
periodicity of the solutions and by symmetry around
$\varphi_{\mathrm{ext}}=0$. Moreover, we dismiss the trivial solution
$\varphi_{\mathrm{ext}}=0$ since we require that $\bar{\varphi}_{J}\neq 0$ at
the sweet spot to preserve a non zero 3-wave mixing interaction rate. The
symmetry of the circuit of Fig. 5 implies that the memory mode, which is the
common mode, does not participate in the weak junction. Hence the flux
dependence of $\omega_{m}$ originates from
$\bar{E}_{J}(\varphi_{\mathrm{ext}})=E_{J}\cos(\bar{\varphi}_{J})$. The flux
sweet spot is therefore a local extremum of
$\bar{\varphi}_{J}(\varphi_{\mathrm{ext}})$. From Eq. (2),
$\varphi_{\mathrm{ext}}^{\mathrm{(sweet)}}$ is therefore close to
$\bar{\varphi}_{W}=\pi/2$ since $\sin{\varphi_{W}}$ is maximal there. Finally,
using the condition that $\varphi_{\mathrm{ext}}<\pi$ imposes an upper bound
on $\beta_{J}$. Indeed, for $\bar{\varphi}_{W}=\pi/2$, Eq. (3) comes down to
$\pi/2+2\arcsin{\beta_{J}}<\pi$ and thus $\beta_{J}<1/\sqrt{2}$.

Provided $E_{W}<E_{J}/\sqrt{2}$, a sweet spot will thus be visible at
$\varphi_{\mathrm{ext}}^{\mathrm{(sweet)}}=\pi/2+2\arcsin\beta_{J}$ (blue and
orange closed circle in Fig. 6b). Close to the sweet spot, at first order,
$\bar{\varphi}_{J}$ is constant and
$\bar{\varphi}_{W}=\pi/2+\delta\varphi_{\mathrm{ext}}$ where
$\delta\varphi_{\mathrm{ext}}=\varphi_{\mathrm{ext}}-\varphi_{\mathrm{ext}}^{(\mathrm{sweet})}$
is the flux offset from the sweet spot. At second order, we then have

$\displaystyle\bar{\varphi}_{J}$
$\displaystyle=\arcsin(\beta_{J}\sin(\pi/2+\delta\varphi_{\mathrm{ext}}))$ (6)
$\displaystyle=\arcsin(\frac{E_{W}}{E_{J}}(1-\frac{\delta\varphi_{\mathrm{ext}}^{2}}{2}))\,.$

Note that the single minimum condition $\beta_{J}<1/2$ that is verified in the
experiment is well in this limit. The regime $1/2<\beta_{J}<1/\sqrt{2}$
comprises a sweet spot in the lowest energy flux configuration (orange closed
circle in Fig. 6b) but the existence of another higher energy minimum might be
detrimental. Finally for $\beta_{J}>1/\sqrt{2}$, there exists a sweet spot
when but it is not the lowest energy phase configuration (green open circle in
Fig. 6b).

###  B.2 Eigenmodes

Now that the equilibrium phase differences and the effective inductive
energies $\bar{E}_{J}$ and $\bar{E}_{W}$ are determined, we can determine the
eigenmodes of the system and compute the zero point fluctuation of the phase
across each dipole of the linear equivalent circuit.

The potential energy of the linear system modeling the circuit reads
$U_{\mathrm{lin}}=\bar{E}_{J}\tilde{\varphi}_{1}^{2}/2+\bar{E}_{J}\tilde{\varphi}_{2}^{2}/2+\bar{E}_{W}\tilde{\varphi}_{W}^{2}/2$.
Hence, incorporating the constraints of Eq. (4), we find

$\displaystyle
U_{\mathrm{lin}}\left(\tilde{\varphi}_{1},\tilde{\varphi}_{2}\right)$
$\displaystyle=\frac{\bar{E}_{J}}{2}\tilde{\varphi}_{1}^{2}+\frac{\bar{E}_{J}}{2}\tilde{\varphi}_{2}^{2}+\frac{\bar{E}_{W}}{2}(\tilde{\varphi}_{1}-\tilde{\varphi}_{2})^{2}$
(7) $\displaystyle
T\left(\dot{\tilde{\varphi}}_{1},\dot{\tilde{\varphi}}_{2}\right)$
$\displaystyle=\frac{\hbar^{2}}{16E_{C}}\dot{\tilde{\varphi}}_{1}^{2}+\frac{\hbar^{2}}{16E_{C}}\dot{\tilde{\varphi}}_{2}^{2}$

where T is the kinetic energy of the system and $E_{C}$ is the charging energy
of each capacitor. This system can be readily diagonalized by performing the
change of variable

$\left\\{\begin{aligned}
\varphi_{m}=\frac{\tilde{\varphi}_{1}+\tilde{\varphi}_{2}}{2}\\\
\varphi_{b}=\frac{\tilde{\varphi}_{1}-\tilde{\varphi}_{2}}{2},\end{aligned}\right.$

where $\varphi_{m}$ and $\varphi_{b}$ correspond to the common and
differential modes of the circuit, respectively denoted as memory and buffer
modes. The potential and kinetic energies of the circuit then read

$\displaystyle U_{\mathrm{lin}}\left(\varphi_{m},\varphi_{b}\right)$
$\displaystyle=\frac{E_{L,m}}{2}\varphi_{m}^{2}+\frac{E_{L,b}}{2}\varphi_{b}^{2}$
$\displaystyle T\left(\dot{\varphi}_{m},\dot{\varphi}_{b}\right)$
$\displaystyle=\frac{\hbar^{2}}{8E_{C}}\dot{\varphi}_{m}^{2}+\frac{\hbar^{2}}{8E_{C}}\dot{\varphi}_{b}^{2}.$

with $E_{L,m}=2\bar{E}_{J}$ and $E_{L,m}=2\bar{E}_{J}+4\bar{E}_{W}$. The mode
frequencies and zero-point fluctuations of the buffer and memory mode are
given by

$\begin{matrix}\omega_{b}=\sqrt{4E_{C}E_{L,b}}&\varphi_{\mathrm{zpf},b}=\left(E_{C}/E_{L,b}\right)^{1/4}\\\
\omega_{m}=\sqrt{4E_{C}E_{L,m}}&\varphi_{\mathrm{zpf},m}=\left(E_{C}/E_{L,m}\right)^{1/4}\end{matrix}$

so that, in second quantization, the linear part of the Hamiltonian is

$\displaystyle\hat{H}_{\mathrm{lin}}/\hbar=\omega_{m}\hat{m}^{\dagger}\hat{m}+\omega_{b}\hat{b}^{\dagger}\hat{b}$
(8)

with annihilation operators defined by their relation to the phase differences

$\displaystyle\hat{\varphi}_{m}$
$\displaystyle=\varphi_{\mathrm{zpf},m}\left(\hat{m}+\hat{m}^{\dagger}\right)$
(9) $\displaystyle\hat{\varphi}_{b}$
$\displaystyle=\varphi_{\mathrm{zpf},b}\left(\hat{b}+\hat{b}^{\dagger}\right).$

The frequencies of both the memory and buffer modes depend on
$\phi_{\mathrm{ext}}$ through the dependency of $\bar{\varphi}_{J}$ and
$\bar{\varphi}_{W}$. In the actual circuit, stub resonators are connected to
the ring of junctions so that the mode frequencies and zero point fluctuations
are modified. $\phi_{\mathrm{ext}}$ is then chosen so that the condition
$\omega_{b}=2\omega_{m}$ is matched, defining the value of
$\phi_{\mathrm{QEC}}$ as shown in Fig. 1. Numerically, fitting the frequency
dispersion versus flux with the full model gives us
$\varphi_{\mathrm{zpf},m}=0.0305$ and $\varphi_{\mathrm{zpf},b}=0.0648$ at
$\phi_{\mathrm{QEC}}$.

###  B.3 Nonlinearities

Around the DC solution of the system, the full potential

$\displaystyle
U(\tilde{\varphi}_{1},\tilde{\varphi}_{2})=-E_{J}\cos(\tilde{\varphi}_{1}+\bar{\varphi}_{J})-E_{J}\cos(\tilde{\varphi}_{2}-\bar{\varphi}_{J})$
(10) $\displaystyle-
E_{W}\cos(\tilde{\varphi}_{2}-\tilde{\varphi}_{1}+\bar{\varphi}_{W})$

can be used to compute the nonlinearities. Using the former change of variable
we get

$\displaystyle U\left(\varphi_{m},\varphi_{b}\right)=$
$\displaystyle-2E_{J}\cos{\left(\varphi_{m}\right)}\cos{\left(\varphi_{b}+\bar{\varphi}_{J}\right)}$
(11) $\displaystyle-E_{W}\cos{\left(2\varphi_{b}-\bar{\varphi}_{W}\right)}$

The potential energy is represented in Fig. 7 for the circuit parameters
$E_{W}/h\approx 115~{}\mathrm{GHz}$ and $E_{J}/h\approx 250~{}\mathrm{GHz}$
for several values of the external flux bias $\phi_{\mathrm{ext}}$. As
expected from the change of variable, the global minimum is located at
$(\varphi_{m},\varphi_{b})=(0,0)$ for any flux bias $\varphi_{\mathrm{ext}}$.
Other minima exist owing to the periodicity of the Josephson potential but
there is a potential barrier of $2E_{J}\approx k_{B}\times 22~{}\mathrm{K}$ to
overcome in order to transit from one solution to the other as can be seen in
Fig. 7. Expanding the $\sin$ and $\cos$ function up to $4^{\mathrm{th}}$ order
in the phases then gives the Hamiltonian parameters.

Figure 7: Potential energy $U\left(\varphi_{m},\varphi_{b}\right)/E_{W}$ for
$E_{W}/h=115~{}\mathrm{GHz}$ and $E_{J}/h=250~{}\mathrm{GHz}$ at six different
flux biases. For each flux bias, an offset
$U_{\mathrm{min}}(\varphi_{\mathrm{ext}})$ is subtracted to Eq. (11) in order
to set the potential global minimum to $0$ and better highlight the height of
the potential barrier.

#####  $3^{\mathrm{rd}}$ order terms

We now consider working at $\phi_{\mathrm{QEC}}$ yielding the condition
$\omega_{b}=2\omega_{m}$, and restrict our analysis to terms surviving the
Rotating Wave Approximation. The only $3^{\mathrm{rd}}$ order terms to
consider is then

$\displaystyle\hat{H}_{3^{\mathrm{rd}}}/\hbar=E_{J}\sin\left(\bar{\varphi}_{J}\right)\varphi_{\mathrm{zpf},b}\,\varphi_{\mathrm{zpf},m}^{2}\left(\hat{b}\,\hat{m}^{\dagger\,2}+\hat{b}^{\dagger}\hat{m}^{2}\right).$

This Hamiltonian corresponds to the 2-photon exchange Hamiltonian, responsible
for converting single photons in the buffer into pairs of photons in the
memory mode and back. Plugging in the value of $\bar{\varphi}_{J}$ around the
sweet spot Eq. (6), we recover Eq. (1) $\hbar g_{2}\approx
E_{W}\left(1-\delta\varphi_{\mathrm{ext}}^{2}/2\right)\varphi_{\mathrm{zpf},m}^{2}\varphi_{\mathrm{zpf},b}$.
From the frequency dispersion versus flux we expect
$g_{2}/2\pi=6.2~{}\mathrm{MHz}$, very close to what is experimentally
extracted from Fig. 4.

#####  $4^{\mathrm{th}}$ order terms

Expanding even further the Hamiltonian to $4^{\mathrm{th}}$ order terms leads
to the expression of the memory and buffer self-Kerr rates, as well as the
cross-Kerr coupling between these 2 modes

$\displaystyle\hbar\chi_{m,m}=\bar{E}_{J}\varphi_{\mathrm{zpf},m}^{4}$ (12)
$\displaystyle\hbar\chi_{b,b}=\bar{E}_{J}\varphi_{\mathrm{zpf},b}^{4}+8\bar{E}_{W}\varphi_{\mathrm{zpf},b}^{4}$
$\displaystyle\hbar\chi_{m,b}=2\bar{E}_{J}\varphi_{\mathrm{zpf},m}^{2}\varphi_{\mathrm{zpf},b}^{2}.$

Note that it is the equivalent Josephson energies $(\bar{E}_{J},\bar{E}_{W})$
that appears and not the bare ones $(E_{J},E_{W})$. The corresponding
Hamiltonian reads

$\displaystyle\hat{H}_{4^{\mathrm{th}}}/\hbar=-\frac{\chi_{m,m}}{2}\hat{m}^{\dagger\,2}\hat{m}^{2}-\frac{\chi_{b,b}}{2}\hat{b}^{\dagger\,2}\hat{b}^{2}-\chi_{m,b}\,(\hat{m}^{\dagger}\hat{m})(\hat{b}^{\dagger}\hat{b}).$

The effect of these spurious terms is taken into account in the simulations
used in Figs. 3 and 4. The memory self Kerr rate is directly estimated at
$\phi_{\mathrm{tomo}}$, $\chi_{m,m}/2\pi\approx 220\,\mathrm{kHz}$ from the
dynamics of a coherent state. The parameters $\chi_{b,b}$ and $\chi_{m,b}$ are
not measured in this experiment and instead inferred from the other measured
variables using the expressions above. At $\phi_{\mathrm{QEC}}$, from the
frequency dispersion versus flux, we expect
$\chi_{m,m}/2\pi=200~{}\mathrm{kHz}$, $\chi_{b,b}/2\pi=11.3~{}\mathrm{MHz}$
and $\chi_{m,b}=1.8~{}\mathrm{MHz}$.

##  Appendix C Microwave Design

###  C.1 Input line

Figure 8: Optical image of the autoparametric circuit showing its symmetry
axis in blue, and the offset of the input line (green) with respect to the
symmetry axis in orange.

Because the memory mode (respectively buffer mode) is symmetric (respectively
anti-symmetric, see Fig. 1c) with respect to the circuit symmetry axis (blue
in Fig. 8), one can use the properties of transmission line modes of the input
to disable the coupling to the memory mode. Memory and buffer modes only
couple to propagating modes having the same symmetries.

Figure 9: Electromagnetic simulations (using Ansys HFSS) of the current field
of the memory mode around the input line for various offsets of the input line
with respect to the circuit symmetry axis. Colors indicate surface currents
according to the legend. For large offsets (-0.5 $\mathrm{\mu m}$ and 0.5
$\mathrm{\mu m}$), the current field is characteristic of a slot-line geometry
(opposite current in the two ground planes) indicating that the memory is
mostly coupled to the slot-line propagating mode. For zero offset, the current
field is characteristic of a co-planar geometry (identical current on the two
ground planes and opposite current in the central track) indicating that the
memory is mostly coupled to the co-planar waveguide propagating mode. For the
two optimal offsets (-0.175 $\mathrm{\mu m}$ and 0.425 $\mathrm{\mu m}$)
leading to the highest memory coupling quality factor, the current field is a
mix of slot-line and co-planar geometries.

A first input line design that can benefit from these symmetries is a slotline
made of a gap separating two ground planes [55]. This transmission line has a
single propagating mode, which is anti-symmetric since opposed currents flow
in the two ground planes. When the slotline is aligned with the circuit
symmetry axis, only the buffer mode is coupled to the transmission line while
the memory mode is protected by symmetry. If this input line design is optimal
from a point of view of memory filtering, it is not compatible with fast flux
bias as it cannot bring a DC current close to the Josephson junction loop.

In contrast, a co-planar waveguide (CPW) transmission line contains a center
track so that it can be used for fast flux biasing the loop. A CPW
transmission line contains two quasi-transverse electromagnetic propagating
modes [56]. One is symmetric with respect to the transmission line axis with
identical currents in the ground planes but an opposite one in the central
line. We call it the co-planar waveguide mode. The other is anti-symmetric
with respect of the transmission line axis with opposing currents in the two
ground planes and no current in the central line. We call it the slot line
mode. Importantly, the slot line mode is suppressed by increasing the density
of wirebonds that connect the two ground planes.

When the CPW transmission line is aligned on the circuit symmetry axis, the
buffer (respectively memory) mode is only coupled to the slot-line mode
(respectively co-planar waveguide mode) by symmetry. The mutual inductance of
2 pH that we design for flux biasing sets the geometry of the input line close
to the loop. However, the position and density of wirebonds can be chosen. In
practice, we tune the buffer coupling rate using wirebond positions as they
affect the slot-line mode. In contrast, lowering the memory coupling rate
requires another control parameter.

To this aim, we offset the CPW transmission line away from the circuit
symmetry axis (orange arrow in Fig. 8). The buffer mode stays dominantly
coupled to the slot-line mode whereas the memory mode is coupled to both slot-
line and co-planar waveguide modes. This can be seen in electromagnetic
simulations of Fig. 9 by looking at the currents of the memory mode around the
input line. The memory mode coupling to the co-planar waveguide (respectively
slot-line) mode decreases (respectively increases) with the shift length. This
can be observed by looking at the dominant geometry of the memory currents
around the input line (see Fig. 9).

Figure 10: Simulated memory (blue) and buffer (green) coupling quality factors
(Q) as a function of the input line offset with respect to the circuit
symmetry axis. The memory Q factor reaches maximums for two sweet spots
defined as offset lengths of 425 $\mathrm{\mu m}$ and -175 $\mathrm{\mu m}$.

The memory coupling rate to the transmission line is given by the sum of the
coupling rates to the co-planar waveguide and slot-line modes and can be
simulated. Fig. 10 shows the variation of the simulated memory coupling
quality factor $Q$ as a function of the input line offset. There are two
optimal offsets (a positive one $l_{+}=425\mathrm{\mu m}$ and a negative one
$l_{-}=-175\mathrm{\mu m}$) for which the total coupling rate to the slot-line
and co-planar waveguide modes is minimized leading to an enhancement of the Q
factor of the memory. The $l_{-}$ offset has a large enough mutual inductance
(about 2 pH) for fast flux biasing and it is the one we choose.

##  Appendix D Cabling

Figure 11: Schematic of the setup. Each electromagnetic mode in the circuit is
driven by an RF source detuned by the modulation frequency and whose color
matches that of the corresponding mode.

The transmon qubit, readout resonator, memory and buffer modes are driven by
pulses whose envelope is generated using an Arbitrary Waveform Generator
(AWG), an OPX by Quantum Machine in this experiment. These pulses are
respectively modulated at $\omega_{\mathrm{IF},q}/2\pi=100\,\text{MHz}$,
$\omega_{\mathrm{IF},r}/2\pi=75\,\text{MHz}$,
$\omega_{\mathrm{IF},m,\mathrm{tomo}}/2\pi=40\,\text{MHz}$ or
$\omega_{\mathrm{IF},m,\mathrm{QEC}}=\left(\omega_{m,\mathrm{tomo}}+\omega_{\mathrm{IF},m,\mathrm{tomo}}\right)-\,\omega_{b,\mathrm{QEC}}/2$,
and
$\omega_{\mathrm{IF},b,\mathrm{QEC}}=2\omega_{\mathrm{IF},m,\mathrm{QEC}}$.
$\omega_{\mathrm{IF},m,\mathrm{tomo}}$ and
$\omega_{\mathrm{IF},m,\mathrm{QEC}}$ are the modulation frequencies used to
respectively drive the memory at $\phi_{\mathrm{tomo}}$ or
$\phi_{\mathrm{QEC}}$, $\omega_{m,\mathrm{tomo}}$ and
$\omega_{b,\mathrm{QEC}}$ are the frequencies of the memory mode at
$\phi_{\mathrm{tomo}}$ and the buffer mode at $\phi_{\mathrm{QEC}}$. The above
condition on $\omega_{\mathrm{IF},m,\mathrm{QEC}}$ and
$\omega_{\mathrm{IF},b}$ ensures the phase stability of the encoded cat in the
frame rotating at the memory frequency.

These signals are up-converted using single sideband mixers for the transmon
qubit and readout resonator, and IQ mixers for the memory and buffer mode,
with Radio Frequency signals generated by a 4 channel Anapico APUASYN20. The
signals at frequencies $\omega_{q}$, $\omega_{r}$, and
$\omega_{m,\mathrm{tomo}}/\omega_{m,\mathrm{QEC}}$ are all combined and then
sent via the readout port of the device using a 6 GHz frequency diplexer. The
memory drive subsequently traverses the transmon qubit, readout resonator, and
its Purcell filter before reaching the memory cavity. Given this complex path,
employing a room-temperature amplifier becomes indispensable for achieving
displacements $\hat{D}(\beta)$ where $\beta\geq 2$.

The signal driving the buffer mode at $\omega_{b}$ is transmitted through the
alternative port of the device. It’s combined with a DC tone directly
generated by one DAC of the OPX using a 3 GHz frequency diplexer, facilitating
a swift transition from $\phi_{\mathrm{tomo}}$ to $\phi_{\mathrm{QEC}}$. We
attempted to drive the memory through this port to bypass the previously
described elements, but the protection from the symmetry of the non-linear
coupler (Fig. 1c) is excessively effective and prevented the achievement of
large enough displacements.

The two reflected signals from the buffer and readout modes merge at the
mixing chamber. The latter is first pre-amplified by a Travelling Wave
Parametric Amplifier (TWPA) [57]. Further amplification is performed by a High
Electron Mobility Transistor at the 4K stage, and then a room-temperature
amplifier. Subsequently, the signal is down-converted using an image reject
mixer, followed by filtering, amplification, and acquisition by an ADC of the
OPX. With its capacity for real-time digitization and demodulation, the OPX
allows for real-time feedback and implementation of the transmon qubit reset
at the beginning of each pulse sequence. The complete setup is depicted in
(Fig. 11).

##  Appendix E Wigner measurement of cat states

###  E.1 The use of a fast flux line

As explained in the main text, the preparation of the cat state
$|C_{+}^{\alpha}\rangle\propto|\alpha\rangle+|-\alpha\rangle$ is as simple as
starting from the memory vacuum state at $\phi_{\mathrm{QEC}}$ and turning on
a drive with the right amplitude $|\epsilon_{d}|=\alpha^{2}g_{2}$ at twice the
memory frequency $\omega_{d}=2\omega_{m,\mathrm{QEC}}$. This drive, resonant
with the buffer mode, injects photons with energy
$\hbar\omega_{b,\mathrm{QEC}}$ which are converted into pairs of photons in
the memory. By adiabatically eliminating the buffer, we obtain the desired
effective memory dynamics, characterized by the loss operator

$\hat{L}_{2}=\sqrt{\kappa_{2}}\left(\hat{m}^{2}-\alpha^{2}\right),$

with $\kappa_{2}=4g_{2}^{2}/\kappa_{b}$.

Our task is to measure the Wigner function $W$ of the encoded state. The
conventional technique for measuring $W\left(\beta\right)$ [36, 37, 38] begins
by displacing the memory by $\hat{D}\left(-\beta\right)$ before applying an
unconditional $\pi/2$ pulse on the transmon qubit. The system then remains
idle for a time period of $\pi/\chi\approx 2.8\mu\mathrm{s}$, during which the
qubit state gains information about the parity of the number of photons in the
memory. Subsequently, a second $\pi/2$ pulse is applied to map the memory
parity into the states $\ket{g}$ or $\ket{e}$ of the qubit.

However, this method would not be effective at $\phi_{\mathrm{QEC}}$ due to
the intense 2-photon dissipation. This would impede proper displacements
$\hat{D}\left(-\beta\right)$. Moreover, this dissipation broadens the memory
energy levels by $\kappa_{2}\gg\chi$, effectively neutralizing the dispersive
coupling between the memory and transmon qubit.

To overcome these challenges, Wigner tomography is performed at
$\phi_{\mathrm{tomo}}$ where $\omega_{b,\mathrm{tomo}}\neq
2\omega_{m,\mathrm{tomo}}$. At this flux, two-photon dissipation is inactive
because the 2-photon exchange Hamiltonian is not preserved in the rotating
wave approximation, enabling the usual Wigner tomography. To rapidly switch
between $\phi_{\mathrm{QEC}}$ and $\phi_{\mathrm{tomo}}$, we employ a fast
flux line that sets the desired magnetic flux in approximately
$20,\mathrm{ns}$.

While the memory dynamics at $\phi_{\mathrm{tomo}}$ are primarily dominated by
the self-Kerr rate $\chi_{m,m}/2\pi\approx 220\mathrm{kHz}$, which only
marginally impacts during the 20 ns it takes to switch the flux, it is crucial
to keep the drive $\epsilon_{d}\left(\alpha\right)$ on before shifting from
$\phi_{\mathrm{QEC}}$ to $\phi_{\mathrm{tomo}}$. The memory dynamics at
$\phi_{\mathrm{QEC}}$ are indeed dominated by the 2-photon dissipation with a
rate $\kappa_{2}$, which significantly impacts the system in 20 ns. To prevent
state distortion prior to the Wigner tomography, the drive
$\epsilon_{d}\left(\alpha\right)$ is thus maintained during the flux change.
This drive at $\omega_{b,\mathrm{QEC}}$ does not affect the memory at
$\phi_{\mathrm{tomo}}$, where the frequency matching condition is no longer
satisfied.

###  E.2 Phase correction of the stabilized cat

Owing to the change in memory frequency when the flux is switched between
$\phi_{\mathrm{QEC}}$ and $\phi_{\mathrm{tomo}}$, a carefully designed driving
sequence must be followed in order to track the reference frame of the cat
qubit. We set a local oscillator at $\omega_{\mathrm{LO},m}=2\pi\times
3.988481~{}\mathrm{GHz}$ and another one at twice this frequency
$\omega_{\mathrm{LO},b}=2\omega_{\mathrm{LO},m}$ (see Fig. 12). They are
generated using 2 channels of an Anapico APUASYN20 so that their phases are
locked.

Figure 12: Repartition of the frequencies of local oscillators generated by
the APUASYN20 synthetizer, intermediate frequencies generated by the OPX DACs,
and resonance frequencies of the device.

The memory displacement pulse applied for Wigner tomography at the flux
$\phi_{\mathrm{tomo}}$ is generated by mixing the local oscillator at
$\omega_{\mathrm{LO},m}$ with a pulse generated by the AWG at exactly
$\omega_{\mathrm{IF},m,\mathrm{tomo}}/2\pi=40~{}\mathrm{MHz}$. In contrast, at
the flux $\phi_{\mathrm{QEC}}$, the cat qubit is stabilized with a buffer
drive at a frequency $\omega_{b,\mathrm{QEC}}=2\omega_{m,\mathrm{QEC}}$ that
is not given by
$2(\omega_{\mathrm{LO},m}-\omega_{\mathrm{IF},m,\mathrm{tomo}})$ owing to the
frequency change of the memory between the two flux working points. The drive
at $\omega_{m,\mathrm{QEC}}$ is in fact generated by mixing the local
oscillator at $2\omega_{\mathrm{LO},m}$ with a pulse generated by the AWG at
$\omega_{\mathrm{IF},b,\mathrm{QEC}}=2(\omega_{\mathrm{LO},m}-\omega_{m,\mathrm{QEC}})\approx
2\pi\times 120~{}\mathrm{MHz}$.

Figure 13: Wigner Tomography of the cat state $\ket{C_{\alpha}^{+}}$ with
$\alpha=2$ for 3 stabilization times: 300, 304 and 308 ns. (a) Evolution of
the cat state when no compensation is applied. (b) Evolution of the cat when
the accumulated phase between the 2 local oscillators is taken into account
before memory displacement.

Owing to the detuning of about 20 MHz between
$\omega_{\mathrm{IF},b,\mathrm{QEC}}/2$ and
$\omega_{\mathrm{IF},m,\mathrm{tomo}}$, the stabilized coherent states
$|\pm\alpha\rangle$ of the cat qubit are offset by a time increasing phase in
the frame of the displacement pulses used for Wigner tomography. When trying
to perform the desired displacement
$\hat{D}(-\beta)=\hat{D}(-|\beta|e^{i\theta})$ for the Wigner tomography, this
accumulated phase induces a displacement with an angle

$\theta_{\mathrm{offset}}(t)=\theta+\left(\omega_{\mathrm{IF},b,\mathrm{QEC}}/2-\omega_{\mathrm{IF},m,\mathrm{tomo}}\right)t,$

where $t$ is the time spent at $\phi_{\mathrm{QEC}}$. This can be seen as cat
states whose direction in phase-space changes over time (Fig. 13b).

Taking this phase offset into account, we compensate the accumulated phase
directly on the AWG to keep the orientation of the cat qubit states constant
in phase space when reconstructing its Wigner Tomography (Fig. 13a). This is
of particular interest for the measurement of $T_{\mathrm{X}}$ where we need
to measure the evolution of $W\left(\pm\alpha\right)$, which can then be done
by measuring only 2 points of the Wigner function, greatly speeding up this
already time-consuming measurement.

##  Appendix F Different methods to calibrate memory displacements

The displacements $\hat{D}(\beta)$ applied on the memory during Wigner
tomography are performed by applying a drive at frequency
$\omega_{\mathrm{m,\,tomo}}$. We calibrate how the displacement amplitudes
$\beta$ depend on the voltage amplitude $V_{d}$ at the level of the DAC by 3
methods (Fig. 14). We then verify how good is the match between the
proportionality factor $\mu=\mathrm{d}\beta/\mathrm{d}V_{d}$ they provide.

Figure 14: (a) Ramsey interferometry. Dots: measured signal $S_{+}-S_{-}$
between 2 Ramsey-like experiments for various voltages $V_{d}=0$, $10$, $20$,
$30$, $40,~{}\mathrm{mV}$ from top to bottom. Lines: Fit of the measurements
to Eq. (13) leading to a photon number $\bar{n}=0.01$, $0.10$, $0.35$, $0.81$,
$1.45$. The residual thermal population is thus $n_{\mathrm{th}}=0.01$. (b)
Measured Wigner function of the memory in thermal equilibrium with its
environment. The conversion used between $V_{d}$ and $|\beta|$ to plot it is
made by a Gaussian fit of the measurement with the Wigner function of a
thermal state with $n_{\mathrm{th}}=0.01$ photons on average. (c) Dots: Cuts
of the Wigner tomography of a stabilized cat qubit along $\beta\in\mathbb{R}$
after $100~{}\mu\mathrm{s}$ of dephasing. Line: Theoretical prediction. (d)
Same plot along $\beta\in i\mathbb{R}$, $500~{}\mathrm{ns}$ after the buffer
drive is turned on.

###  F.1 Ramsey interferometry

The first calibration method relies on a Ramsey sequence [58]. Starting from
the memory in its vacuum state, a drive of amplitude $V_{d}$ is applied to
displace the memory to a coherent state $\ket{\beta}$. Accounting for the
residual thermal occupation of the memory mode, the mean number of photons is
$\bar{n}=\beta^{2}+n_{\mathrm{th}}$. The dispersively coupled qubit is then
prepared in an equal superposition of ground and excited state by applying an
unconditional $\pi/2$ pulse. After a varying time $t$, the superposition
accumulates a phase $\chi\hat{m}^{\dagger}\hat{m}t$ that depends on the memory
photon number $\hat{m}^{\dagger}\hat{m}$. A second unconditional $\pm\pi/2$
pulse is then applied on the qubit, which is then measured to give 2 average
signals $S_{\pm}$. The difference between these 2 signals then evolves as [58]

$S_{+}-S_{-}=\cos\left(\bar{n}\sin\left(\chi
t\right)\right)e^{\bar{n}\left(\cos\left(\chi t\right)-1\right)-t/T_{2}}.$
(13)

From this measurement, we can extract the cross-Kerr coupling rate
$\chi/2\pi=170\mathrm{kHz}$ between memory and transmon qubit. We also obtain
the thermal population $n_{\mathrm{th}}=0.011\pm 0.002$ and $\mu=31.33\pm
0.85~{}\mathrm{V}^{-1}$.

###  F.2 Thermal state tomography

Another calibration method is to perform a Wigner tomography of the memory
thermal state, using the independently measured average occupation
$n_{\mathrm{th}}=0.01$. The density matrix can be written as a Boltzmann
distribution
$\hat{\rho}_{\mathrm{th}}=\sum_{n}\frac{n_{\mathrm{th}}^{n}}{\left(1+n_{\mathrm{th}}\right)^{n}}\ket{n}\bra{n}$
and the Wigner function associated

$W_{\mathrm{th}}\left(\beta\right)=\sum_{n}\frac{n_{\mathrm{th}}^{n}}{\left(1+n_{\mathrm{th}}\right)^{n}}W_{n}\left(\beta\right),$
(14)

where
$W_{n}\left(\beta\right)=\left(-1\right)^{n}\frac{2}{\pi}e^{-2\left|\beta\right|^{2}}L_{n}\left(4\left|\beta\right|^{2}\right)$
is the Wigner function of Fock state $\ket{n}$. Laguerre polynomials obey the
following rule

$\sum_{n}t^{n}L_{n}\left(4\left|\beta\right|^{2}\right)=\frac{1}{1-t}e^{4t\left|\beta\right|^{2}/(1-t)}.$

Therefore, we obtain

$\displaystyle W_{\mathrm{th}}\left(\beta\right)$
$\displaystyle=\frac{2}{\pi}e^{-2\left|\beta\right|^{2}}\sum_{n}\left(\frac{-n_{\mathrm{th}}}{1+n_{\mathrm{th}}}\right)^{n}L_{n}\left(4\left|\beta\right|^{2}\right)$
(15)
$\displaystyle=\frac{2\left(1+n_{\mathrm{th}}\right)}{\pi\left(1+2n_{\mathrm{th}}\right)}e^{-2\left|\beta\right|^{2}/\left(1+2n_{\mathrm{th}}\right)}.$

Using a conversion factor $\mu=31.31\pm 0.14~{}\mathrm{V}^{-1}$ rescales the
displacement amplitudes from voltages $V_{d}$ into complex amplitudes $\beta$
for the measured Wigner function in (Fig. 14b), in such a way that the
standard deviation $\sigma$, of this Gaussian distribution is
$\sigma=\sqrt{1+2n_{\mathrm{th}}}/2$, with $n_{\mathrm{th}}=0.01$.

###  F.3 Measurement of cat states fringes

Our last method to calibrate the conversion factor is based on the Wigner
tomography of a cat state [59]. The particular features of the cat Wigner
function allow to directly estimate $\mu$, assuming the distortion due to
memory self-Kerr or thermal population is negligible. The Wigner function of
an even cat state of size $\alpha$, $\ket{C_{\alpha}^{+}}$, can be written as

$\displaystyle W_{\alpha}^{+}\left(\beta\right)=\frac{1}{\pi}$
$\displaystyle\left(e^{-2\left|\alpha-\beta\right|^{2}}+e^{-2\left|\alpha+\beta\right|^{2}}\right.$
(16)
$\displaystyle+\left.2\cos\left(4\mathrm{Im}\left(\alpha^{*}\beta\right)\right)e^{-2\left|\beta\right|^{2}}\right).$

Introducing $\delta V_{\alpha}$ and $\delta V_{\mathrm{I}}$ the drive voltages
corresponding to respectively a displacement of $2\alpha$ (distance between
the two Gaussian distributions in Fig. 14c) and the periodicity of the fringes
$\pi/2\alpha$ (seen in Fig. 14d), this yields

$\left\\{\begin{aligned} 2\alpha&=\mu\,\delta V_{\alpha}\\\
\frac{\pi}{2\alpha}&=\mu\,\delta V_{\mathrm{I}}\\\ \end{aligned}\right.$

Therefore $\mu=\sqrt{\pi/\left(\delta V_{\alpha}\delta
V_{\mathrm{I}}\right)}$. The values of $\delta V_{\alpha}$ and $\delta
V_{\mathrm{I}}$ are measured via cuts of the Wigner function in the direction
and orthogonal to the cat state (Fig. 14c,d). The conversion factor obtained
via this method is $\mu=31.41\pm 0.04~{}\mathrm{V}^{-1}$, once again
compatible with the previous calibrations. With the conversion factor being
calibrated, this method can actually be used in order to estimate the cat size
by simply looking at the fringes periodicity, as has been done in [59].

###  F.4 Calibration

To conclude this section, the three methods are compatible with $\mu=31.4\pm
0.1~{}\mathrm{V}^{-1}$. We use this value of $\mu=31.4~{}\mathrm{V}^{-1}$ for
the whole article.

##  Appendix G Measurement of $\kappa_{2}$ and $\kappa_{1}$

###  G.1 Determination of $\kappa_{2}$ using engineered relaxation of cat
qubits

In order to measure the rate $\kappa_{2}$, we first prepare
$\ket{C_{\alpha}^{+}}$ or $\ket{C_{\alpha}^{-}}$ by driving the buffer with a
drive $\epsilon_{d}\left(\alpha\right)$ at $\phi_{\mathrm{QEC}}$. Turning off
$\epsilon_{d}\left(\alpha\right)$ while remaining at $\phi_{\mathrm{QEC}}$
then ensures the memory loses pairs of photons to the environment at the rate
$\kappa_{2}$. $\ket{C_{\alpha}^{\pm}}$ then converges to a state in the
manifold $\\{\ket{\alpha},\ket{-\alpha}\\}$ with the same parity as the
initial state. An example of such an evolution starting from
$\ket{C_{\alpha}^{+}}$, $\alpha=2.5$, is shown at a few decay times in Fig. 2.
The complete list of measured decay times for this evolution is t = 0, 4, 8,
12, 20, 28, 40, 60, 100, 160, 240 and 320 ns.

In order to extract the rate $\kappa_{2}$ from these dynamics, the initial
density matrix describing the memory is approximated by

$\rho=p\ket{C_{\alpha}^{+}}\bra{C_{\alpha}^{+}}+(1-p)\ket{C_{\alpha}^{-}}\bra{C_{\alpha}^{-}}.$
(17)

$\alpha$ is extracted by fitting the initial measured Wigner tomography, while
$p$ is deduced from the value of $W(0)$ which fully characterizes the parity
of the state. The obtained description of the initial density matrix is
however only an approximation as it does not take into account possible
leakage out of the code space due to the memory self Kerr effect, dispersive
coupling to transmon and buffer modes, or potential heating effect. From this
initial state (17), the evolution of the memory state is then simulated using
the Hamiltonian and loss operators

$\displaystyle\hat{H}/\hbar=-\frac{\chi_{m,m}}{2}\hat{m}^{\dagger
2}\hat{m}^{2}$ (18) $\displaystyle\hat{L}_{1}=\sqrt{\kappa_{1}}\hat{m},\
\hat{L}_{2}=\sqrt{\kappa_{2}}\hat{m}^{2}.$

The single photon loss rate $\kappa_{1}/2\pi\sim 14~{}\mathrm{kHz}$ is
extracted from the decay of the single photon state
$\ket{1}\rightarrow\ket{0}$ (see section G.2). Using the memory self-Kerr rate
of $2\pi\times 220~{}\mathrm{kHz}$ measured at $\phi_{\mathrm{tomo}}$, we use
its predicted flux dependence in Eq. (12) to estimate that the self-Kerr rate
at $\phi_{\mathrm{QEC}}$ is $\chi_{m,m}/2\pi\sim 206~{}\mathrm{kHz}$.
Minimizing the difference between measured and simulated Wigner functions at
all times $t$

$\sum_{\mathrm{t}}\
\int_{\mathbb{C}}\,\left|W_{\mathrm{exp}}\left(\beta,t\right)-W_{\mathrm{sim}}\left(\beta,t\right)\right|\mathrm{d}\beta$
(19)

then allows us to fit the value of $\kappa_{2}$ that best reproduces the
memory dynamics.

The uncertainty shown in Fig. 2a is then calculated using the result of the
minimization method $\Delta\kappa_{2}\leq\sqrt{\mathrm{tol}\,\times\,H^{-1}}$,
with tol being the tolerance given for the convergence of the algorithm and
$H^{-1}$ the inverse of the Hessian matrix. It should be noted that, due to
the condition for the adiabatic elimination of the buffer

$8g_{2}\alpha<\kappa_{b}$ (20)

not being verified for $\alpha\gtrsim 1$, we observe a deviation between
experimental data in Fig. 2b and the evolution predicted by this simple model.
Actually, the buffer mode gets populated during two-photon dissipation. In
turn, the memory sees an effective drive originating from this buffer
population, inducing small deformations of the Wigner function. Particularly
visible at 8 ns where the buffer is close to being maximally populated, this
effect vanishes at 40 ns after the memory loses enough photons for the system
to re-enter the adiabatic regime. This effect can be taken into account in the
simulation by including the buffer dynamics without adiabatic elimination
(Fig. 15). In practice, we use the same model as in Sec. H.2 apart from the
detunings $\Delta_{m}=1~{}\mathrm{MHz}$ and $\Delta_{b}=0$. However this
bipartite evolution does not provide an effective value of $\kappa_{2}$ acting
on the memory mode, hence the benefit to stick with the simpler model.

Figure 15: Evolution of the Wigner functions of the memory starting close to a
cat state $\ket{C_{\alpha}^{+}}$ and under the effect of two-photon
dissipation at $\phi_{\mathrm{QEC}}$ without driving the buffer. First and
third lines: measured Wigner functions at various times indicated on the
figure. Second and fourth lines: simulated Wigner functions of the memory
without the adiabatic elimination of the buffer, and with a two-photon
coupling rate $g_{2}/2\pi=6~{}\mathrm{MHz}$ (Fig. 3e)

We have tried other methods to estimate $\kappa_{2}$, in particular by
extracting $\left<\hat{m}^{\dagger}\hat{m}\right>$ from the measured Wigner
function

$\bar{n}=\int_{\mathbb{C}}W_{\mathrm{exp}}\left(\beta\right)\beta^{2}\mathrm{d}\beta-0.5$
(21)

and compare it with the theoretical expression given in [60]. However,
reconstructing $\bar{n}$ with this method has proven quite challenging due to
the measurement noise of the Wigner tomography, which would have made it
necessary to use Maximum Likelihood Estimation (MLE) [61] in order to
circumvent this issue.

###  G.2 Determination of $\kappa_{1}$ using the relaxation of a single photon

The measurement of $\kappa_{1}$ is done by observing the decay from Fock state
$\ket{1}$ to the vacuum. If we prepare the state $\ket{C_{\alpha}^{-}}$ in the
memory, and let it evolve under the action of two photon dissipation at rate
$\kappa_{2}\gg\kappa_{1}$ (Fig. 16a), the parity of the memory is preserved so
that the memory state ends up in the subspace generated by
$\\{\ket{0},\ket{1}\\}$ with the same parity as $\ket{C_{\alpha}^{-}}$: that
is the Fock state $\ket{1}$. All that is necessary to measure $\kappa_{1}$ is
then to monitor the memory parity $\pi W(0)/2$ as it evolves towards $1$, here
corresponding to the vacuum.

In order to prepare $\ket{C_{\alpha}^{-}}$, we prepare $\ket{C_{\alpha}^{+}}$
as in Fig. 2 and then apply a $Z$ gate. Even if the preparation and gate are
not optimized, as in this measurement, the decay rate can still be extracted
with excellent accuracy as it only affects the initial value of $W(0)$ during
the decay from $\ket{1}$ to $\ket{0}$ (Fig. 16b).

Figure 16: (a) Measured Wigner functions of the memory starting close to
$\ket{C_{\alpha}^{-}}$ after the decay times indicated on the figure. (b)
Dots: Measured evolution of $W(0)$ as a function of the time $t$ spent after
the memory has been prepared close to $\ket{C_{\alpha}^{-}}$. Note the much
longer timescale for this single photon decay compared to the sub $\mu$s time
needed to prepare Fock state $|1\rangle$ in (a). Dashed line: Fit of the
exponential relaxation to vacuum.

The evolution of $W(0)$ in (Fig. 16b) is fitted by an exponential relaxation
at a rate $\kappa_{1}/2\pi=14~{}\mathrm{kHz}$. Repeating this measurement over
the course of months, we found that it is not stable. The rate
$\kappa_{1}/2\pi$ typically varies by $\pm 2~{}\mathrm{kHz}$ around this
average value.

##  Appendix H Calibration of the $Z$ gate

The quantum operation shown in Fig. 4 consists of a rotation around the $z$
axis of the cat qubit Bloch sphere (Fig. 17a). This is performed by driving
the memory with a drive at frequency $\omega_{m}$, effectively implementing a
displacement whose Hamiltonian reads

$\hat{H}_{z}/\hbar=-i\epsilon_{Z}e^{i\theta_{z}}\hat{m}+h.c.,$

where $\theta_{z}$ and $\epsilon_{Z}$ are the drive phase and amplitude.
Applying this drive while simultaneously driving the buffer at
$\phi_{\mathrm{QEC}}$ (Fig. 17b) implements a quantum Zeno dynamics of the
cavity which is restricted to states in the cat qubit subspace. The effect of
$\hat{H}_{z}$ on the Wigner function of the memory then simply consists in
shifting the phase of the interference fringes as seen in Fig. 17c,
effectively inducing the desired dynamics on the logical qubit.

Figure 17: (a) Bloch sphere of the cat qubit, whose computational basis is
${\ket{\alpha},\ket{-\alpha}}$. The effect of a $Z$ rotation is illustrated by
the blue arrow. (b) Pulse sequence for $Z$ rotation characterization. (c)
Effect of a drive $-i\epsilon_{Z}e^{i\theta_{z}}\hat{m}+h.c.$ acting on the
state $\ket{C_{\alpha}^{+}}$ of the memory.

###  H.1 Calibration of the drive phase

Figure 18: (a) Measured Wigner function $W(\beta)$ of the memory as a function
of $\beta\in i\mathbb{R}$ and time $t$. The displacement drive parameters are
$\bar{\epsilon}_{Z}/2\pi=1.25\,\mathrm{MHz}$ and $\theta_{z}=\pi/2$ (b)
Measured oscillation frequency $\Omega_{z}$ around the $z$ axis of the Bloch
sphere as a function of $\theta_{z}$.

The first calibration needed for this scheme is to set the value of
$\theta_{z}$ to $\pi$. This maximizes the gate speed for a given drive
amplitude $\epsilon_{Z}$, improving the gate fidelity by decreasing the time
during which single photon dissipation affects the memory.

This optimization is done by sweeping the phase of the memory drive and doing
a vertical cut of the memory Wigner tomography. Looking at how fast the Wigner
function fringes shift over time allows to extract the oscillation rate
$\Omega_{z}$. Note that measuring $W\left(0\right)$ alone would leave the sign
of $\Omega_{z}$ undetermined, which is why we measure a vertical cut of the
Wigner function (Fig. 18a).

Doing this measurement for different value of $\theta_{z}$, (Fig. 18b) shows
an evolution
$\Omega_{z}\left(\theta_{z}\right)\propto\cos\left(\theta_{z}\right)$. This is
expected as only the vertical component of the drive
$\mathrm{Re}\left(\epsilon_{z}e^{i\theta_{z}}\right)$ effectively displaces
the fringes of the cat Wigner function (Fig. 16c). The horizontal component
$\mathrm{Im}\left(\epsilon_{z}e^{i\theta_{z}}\right)$ is disabled by the 2
photon dissipation and does not affect the system, which can be seen as a
cancellation of $\Omega_{z}$ for $\theta_{z}=\pm\pi/2$.

###  H.2 Comparison of $\kappa_{\mathrm{Z}}$ with the theoretical model

The decay rate $\kappa_{\mathrm{Z}}$ of the oscillations around $Z$ (shown in
Fig. 4) is obtained by simulating the master equation

$\displaystyle\frac{d\hat{\rho}}{dt}=$
$\displaystyle-\frac{i}{\hbar}[\hat{H},\hat{\rho}]$ (22)
$\displaystyle+\mathcal{D}[\sqrt{\kappa_{1}}\hat{m}](\hat{\rho})+\mathcal{D}[\sqrt{\kappa_{\varphi}^{m}}\hat{m}^{\dagger}\hat{m}](\hat{\rho})$
$\displaystyle+\mathcal{D}[\sqrt{\kappa_{b}}\hat{b}](\hat{\rho})+\mathcal{D}[\sqrt{\kappa_{\varphi}^{b}}\hat{b}^{\dagger}\hat{b}](\hat{\rho})$

where
$\mathcal{D}[\hat{L}]\rho=\hat{L}\rho\hat{L}^{\dagger}-\hat{L}^{\dagger}\hat{L}\rho/2-\rho\hat{L}^{\dagger}\hat{L}/2$
is the Lindblad superoperator. The four last terms respectively model the
single photon dissipation of the memory, the pure dephasing of the memory, the
single photon dissipation of the buffer, and the pure dephasing of the buffer.
The effective Hamiltonian of the system takes the form

$\displaystyle\frac{\hat{H}}{\hbar}=$
$\displaystyle\Delta_{m}\hat{m}^{\dagger}\hat{m}+\Delta_{b}\hat{b}^{\dagger}\hat{b}$
$\displaystyle-\frac{\chi_{mm}}{2}\hat{m}^{\dagger
2}\hat{m}^{2}-\frac{\chi_{bb}}{2}\hat{b}^{\dagger
2}\hat{b}^{2}-\chi_{mb}\hat{m}^{\dagger}\hat{m}~{}\hat{b}^{\dagger}\hat{b}$
$\displaystyle+g_{2}(\hat{m}^{2}-\alpha^{2})\hat{b}^{\dagger}+h.c$
$\displaystyle+i\epsilon_{Z}e^{-i\theta_{z}}\hat{m}^{\dagger}-i\epsilon_{Z}e^{i\theta_{z}}\hat{m},$

where $\Delta_{b}=\omega_{b}-\omega_{d}$ is the detuning of the drive with
respect to the buffer frequency, and
$\Delta_{m}=\omega_{m}-\omega_{b}/2+\Delta_{b}/2$. Here, the Gaussian drive
envelope reads

$\displaystyle\epsilon_{Z}(t)=\bar{\epsilon}_{Z}\frac{6}{\sqrt{2\pi}}\exp{-\frac{(t-T/2)^{2}}{2w^{2}}},$

where the time window of the pulse is set to $w=T/6$, where $T$ is the total
time of the pulse and $\bar{\epsilon}_{Z}$ corresponds to the average drive
amplitude. The case of a square pulse is easily extended by choosing
$\epsilon_{Z}(t)=\bar{\epsilon}_{Z}$ over the same time window. As for the
experiment, we fit the decaying oscillations of the photon number parity to
extract the rotation frequencies $\Omega_{Z}$ and decay rate $\kappa_{Z}$
corresponding to each drive amplitude $\epsilon_{Z}$.

We numerically observe that the value of $\kappa_{\mathrm{Z}}$ is mostly
sensitive to the loss rate $\kappa_{1}$, the ratio $4g_{2}^{2}/\kappa_{b}$,
the self-Kerr rate $\chi_{mm}$ and the effective detuning $\Delta_{m}$. We
have independently measured $\chi_{mm}/2\pi=0.22~{}\mathrm{MHz}$,
$\kappa_{1}/2\pi=14~{}\mathrm{kHz}$, and $\kappa_{b}/2\pi=40~{}\mathrm{MHz}$.
This leaves us with two fit parameters, $g_{2}$ and $\Delta_{m}$. Let us
estimate the uncertainty on $\Delta_{m}$. As we have measured
$\Delta_{b}/2\pi=-3~{}\mathrm{MHz}$ (see Fig. 24), this leads to
$\Delta_{m}/2\pi=\omega_{m}-\omega_{b}/2-1.5~{}\mathrm{MHz}$. The resonance
condition $2\omega_{m}=\omega_{b}$ being sensitive to the external flux
threading the loop, we estimate that
$|2\omega_{m}-\omega_{b}|<5~{}\mathrm{MHz}$ from the width of the peak
$\kappa_{2}(\varphi_{\mathrm{ext}}$ in Fig. 2a, leading to
$\Delta_{m}/2\pi=-1.5\pm 2.5~{}\mathrm{MHz}$. From the fit of the frequencies
of the system and the formula derived in Sec. B, we estimate
$\chi_{mb}^{\mathrm{(model)}}/2\pi\approx 1.6~{}\mathrm{MHz}$,
$\chi_{bb}^{\mathrm{(model)}}/2\pi\approx 10~{}\mathrm{MHz}$ and
$g_{2}^{\mathrm{(model)}}/2\pi\approx 6.5~{}\mathrm{MHz}$. Interestingly, the
decay rate $\kappa_{Z}$ strongly depends on $g_{2}$ and we use it to extract
this parameter experimentally. For perfect frequency matching
$2\omega_{m}=\omega_{b}$, the best fit to the simulation is obtained for
$g_{2}/2\pi=6\mathrm{MHz}$ (solid line in Fig. 19).

To illustrate the sensitivity of the simulations to the value of $g_{2}$, we
also compute $\kappa_{Z}(\alpha)$ for $g_{2}/2\pi=5.5\mathrm{MHz}$ and
$g_{2}/2\pi=6.5\mathrm{MHz}$. The clear deviations in Fig. 19a show that under
the assumption that $2\omega_{m}=\omega_{b}$, $g_{2}$ can be determined with a
much better precision than $2\pi\times 0.5~{}\mathrm{MHz}$ from the measured
decay rates $\kappa_{Z}$.

Figure 19: (a) Triangles: measured decay rate $\kappa_{Z}$ of the oscillations
around $Z$ as a function of $|\alpha|^{2}$ for four drive amplitudes
$\epsilon_{Z}$ corresponding to distinct colors as in Fig. 4d. Lines:
simulated decay rates $\kappa_{Z}$ using Eq. (22) with three values of the
rate $g_{2}/2\pi$ indicated as an inset. (b) Triangles: same measurement as
above. Lines: simulated $\kappa_{Z}$ for three values of the detuning between
the buffer and the memory
$(2\omega_{m}-\omega_{b})/2\pi=-5,~{}0,~{}5~{}\mathrm{MHz}$ covering its the
uncertainty range, and the corresponding optimal values of
$g_{2}/2\pi=6.5,~{}6.0,~{}6.0~{}\mathrm{MHz}$.

The uncertainty on $g_{2}$ is actually dominated by the values it can take
over the range of conceivable detunings $\Delta_{m}$. In order to get a higher
bound on this uncertainty, we choose three values for the resonance condition,
$(2\omega_{m}-\omega_{b})/2\pi=-5,0,5~{}\mathrm{MHz}$, and search the rate
$g_{2}$ that best reproduce the experiment. The fitted $g_{2}$ rates are
between $2\pi\times 6.0~{}\mathrm{MHz}$, and $2\pi\times 6.5~{}\mathrm{MHz}$
(Fig. 19b). We therefore claim that $g_{2}/2\pi=6\pm 0.5\mathrm{MHz}$.

Note that in the simulations of the bit-flip time (Fig. 3) and in the
simulations of the $Z$ gate (Fig. 4), we have used the value
$g_{2}/2\pi=6~{}\mathrm{MHz}$ which best fits the gate rates and decay rates
for the detuning $(2\omega_{m}-\omega_{b})/2\pi=3.5~{}\mathrm{MHz}$.

###  H.3 Bias preserving nature of the $Z$ gate

In order to preserve the benefit offered by bit-flip protection in cat qubits,
it is crucial for logical gates to be bias-preserving [62], meaning they do
not convert phase-flip errors into bit-flip errors.

In order to verify the bias-preserving nature of the $Z$ gate, we measure the
dependence of $T_{\mathrm{X}}$ when continuously driving the memory with a
varying drive amplitude $\epsilon_{Z}$. Similarly to the measurement presented
in Fig. 3, the flux is first set to $\phi_{\mathrm{tomo}}$ and the memory
displaced by $\hat{D}\left(\alpha\right)$ in order to prepare the desired
state $\ket{\alpha}$. The flux is then changed to $\phi_{\mathrm{QEC}}$ and 2
drives are sent to the buffer and memory modes, with an amplitude of
$\epsilon_{d}\left(\alpha\right)$ and $\epsilon_{Z}$. The role of the drive
acting on the buffer is to stabilize the cat qubit, preventing bit-flip errors
from happening, while the drive acting on the memory performs the desired $Z$
gate. The Wigner function $W(\pm\alpha)$ is finally measured for various
waiting times, and their difference $W(\alpha)-W(-\alpha)\propto
e^{-t/T_{\mathrm{X}}}$ is fitted to extract $T_{\mathrm{X}}$ (Fig. 20a).

Figure 20: (a) Pulse sequence for the $T_{\mathrm{X}}$ measurement, while
continuously applying the $Z$ gate on the memory with an average drive
amplitude $\bar{\epsilon}_{Z}$. (b) Dots: Measured $T_{\mathrm{X}}$ as a
function of the average memory drive amplitude $\bar{\epsilon}_{Z}$ for
different cat qubit sizes $\alpha=2$, $2.4$, and $2.7$ (yellow to brown).
Lines: measured $T_{\mathrm{X}}$ at $\bar{\epsilon}_{Z}=0$.

The measured dependence of $T_{\mathrm{X}}$ on $\bar{\epsilon}_{Z}$ is shown
in Fig. 20b for different amplitudes $\alpha$. Despite a rather large
measurement uncertainty, no visible decrease of $T_{\mathrm{X}}$ can be
observed for
$\left|\bar{\epsilon}_{Z}/2\pi\right|<6\,\left(\mathrm{MHz}\right)$, after
which the displacement becomes strong enough to overcome the stabilization
provided by the 2 photon dissipation. This induces an increased number of bit-
flip errors, leading to a decrease of $T_{\mathrm{X}}$.

The $Z$ rotation presented in Fig. 4a is measured with
$\bar{\epsilon}_{Z}/2\pi=1.625\,\mathrm{MHz}$. It can thus be assumed the
drive did not induce additional bit-flip errors. Comparing the measured
$T_{\mathrm{X}}\sim 10~{}\mathrm{ms}$ for $\alpha^{2}=9.3$ with the
$Z\left(\pi\right)$ gate duration, we can estimate that bit-flip errors alone
would then limit the gate fidelity to about $99.9997\%$. The obtained gate
fidelity of $95\%$ thus primarily originates from phase flip errors.

###  H.4 Fidelity of the $Z$ gate

To evaluate the fidelity of the $Z$ gate, we use the evolution of $W(\beta)$
shown in Fig. 3c and directly estimate the evolution of the logical
$\left<\hat{\sigma}_{x,L}\right>$, $\left<\hat{\sigma}_{y,L}\right>$ and
$\left<\hat{\sigma}_{z,L}\right>$ from the Wigner functions. These operators
are defined in the cat encoding as

$\displaystyle\left<\hat{\sigma}_{x,L}\right>=\ket{C_{\alpha}^{+}}\bra{C_{\alpha}^{+}}-\ket{C_{\alpha}^{-}}\bra{C_{\alpha}^{-}}$
$\displaystyle\left<\hat{\sigma}_{y,L}\right>=\ket{C_{\alpha}^{+i}}\bra{C_{\alpha}^{+i}}-\ket{C_{\alpha}^{-i}}\bra{C_{\alpha}^{-i}}$
$\displaystyle\left<\hat{\sigma}_{z,L}\right>=\ket{\alpha}\bra{\alpha}-\ket{-\alpha}\bra{-\alpha},$

with $\ket{C_{\alpha}^{\pm}}=\frac{\ket{\alpha}\pm\ket{-\alpha}}{\sqrt{2}}$
and $\ket{C_{\alpha}^{\pm i}}=\frac{\ket{\alpha}\pm
i\ket{-\alpha}}{\sqrt{2}}$. The measured evolution of the three logical Bloch
vector coordinates is shown in Fig. 21a. Interestingly, the evolution of
$\left<\hat{\sigma}_{z,L}\right>(t)$ during the gate shows no visible
evolution (Fig. 21b), which is expected from its bias preserving property Fig.
20b.

Figure 21: (a) Trajectory of the cat qubit during the $Z$ gate estimated from
the Wigner functions of Fig. 3c. (b) Dots: Mean value of $\hat{\sigma}_{z,L}$
as a function of the gate time. Line: Linear fit of
$\left<\hat{\sigma}_{z,L}\right>(t)$. (c) Dots: Mean value of
$\hat{\sigma}_{x,L}$ as a function of the gate time. Line: fit used in Fig. 3b
to oscillations at a frequency $\Omega_{Z}/2\pi=19.8~{}\mathrm{MHz}$, decaying
at a rate $\kappa_{Z}/2\pi=0.62~{}\mathrm{MHz}$. A scaling factor of $\pi/2$
is applied to respect the relation
$W(0)=2\left<\hat{\sigma}_{x,L}\right>/\pi$. (c) Dots: Mean value of
$\hat{\sigma}_{y,L}$ as a function of the gate time. Line: fit used in (c)
dephased by $\pi/2$.

Using the formalism of the standard process matrix $\chi$ [63], the impact of
the $Z$ gate on an initial density matrix $\hat{\rho}$ is modeled as

$\mathcal{E}_{Z}\left(\hat{\rho}\right)=\sum_{m,n}\chi_{mn}\hat{E}_{m}\hat{\rho}\hat{E}_{n}^{\dagger}.$
(23)

where the Kraus operator are chosen to be
$\left\\{\hat{E}_{m}\right\\}_{m}\in\left\\{\mathds{1},\hat{\sigma}_{x,L},\hat{\sigma}_{y,L},\hat{\sigma}_{z,L}\right\\}$.
Furthermore, owing to the demonstrated negligible bit-flips occurring during
the $Z$ gate, we assume no term causing a bit-flip type of error appears in
Eq. (23) and only consider the simpler error model that provides a lower bound
on the fidelity by neglecting the $\chi_{z,1}$ and $\chi_{1,z}$ terms

$\mathcal{E}_{Z}\left(\hat{\rho}\right)=\left(1-\varepsilon\right)\hat{\sigma}_{z,L}\hat{\rho}\hat{\sigma}_{z,L}+\varepsilon\hat{\rho}.$
(24)

The parameter $\chi_{z,z}=(1-\varepsilon)$ is then the gate fidelity owing to
the definition $F=\mathrm{Tr}\left(\chi~{}\chi_{\mathrm{opt}}\right)$. Indeed,
the matrix $\chi_{\mathrm{opt}}$ is the $\chi$ matrix describing an ideal $Z$
gate, with all its terms null except for $\chi_{z,z}=1$. Using the general
form of a density matrix describing a qubit from its Bloch vector
$(x~{}y~{}z)^{T}$

$\hat{\rho}=\frac{1}{2}\left(\mathds{1}+x\hat{\sigma}_{x,L}+y\hat{\sigma}_{y,L}+z\hat{\sigma}_{z,L}\right),$

the density matrix after application of the Kraus map reads

$\displaystyle\mathcal{E}_{Z}\left(\hat{\rho}\right)=\frac{1}{2}$
$\displaystyle\left(\mathds{1}-x\left(1-2\varepsilon\right)\hat{\sigma}_{x,L}\right.$
$\displaystyle\left.-y\left(1-2\varepsilon\right)\hat{\sigma}_{y,L}+\left<\hat{\sigma}_{z,L}\right>\hat{\sigma}_{z,L}\right).$

The parameters $\varepsilon$ can then be estimated with the evolution of
$\left<\hat{\sigma}_{x,L}\right>(t)$ and $\left<\hat{\sigma}_{y,L}\right>(t)$.
We first check that both the evolution of
$\left<\hat{\sigma}_{x,L}\right>=W(0)\pi/2$ and
$\left<\hat{\sigma}_{y,L}\right>$ matches the fit of $W(0)$ used in Fig. 3b
and Fig. 3c, up to a scaling factor and a dephasing. Finally, we compute
$F=1/2+e^{-\pi\kappa_{z}/\Omega_{z}}/2=95\pm 2\%$ for the 26 ns long Gaussian
pulse with $\left|\alpha\right|^{2}=9.3$ and for a drive amplitude
$\bar{\epsilon}_{Z}/2\pi=1.625~{}\mathrm{MHz}$. The same expression for the
measured decaying oscillations of parity in the case of square pulses leads to
a slightly better fidelity of $F=96.5\pm 2\%$ in 28 ns. Note that the
infidelity in the preparation of $\ket{C_{\alpha}^{+}}$ is due to a
preparation time of $500$ ns starting from $\ket{0}$, similar to the phase-
flip time of the cat qubit $1/\Gamma_{\mathrm{Z}}\approx 500$ ns for
$\left|\alpha\right|^{2}=9.3$. This time should be optimized in future
measurements.

##  Appendix I Complementary data and analysis about bit-flip and phase-flip
rates

###  I.1 Measurement of $T_{\mathrm{X}}$, $\Gamma_{\mathrm{Z}}$

Figure 3 shows the measured $T_{\mathrm{x}}$ and $\Gamma_{\mathrm{Z}}$ for
various cat sizes $\alpha$. An example of a phase-flip rate and bit-flip time
measurement is shown in (Fig. 23) for $\alpha\approx 2.6$.

Figure 22: Determination of $T_{\mathrm{x}}$ and $\Gamma_{\mathrm{Z}}$ for
$\alpha^{2}\approx 6.5$. (a) Dots: measured $W(0)$ as a function of cat
stabilization time $t$. Solid line: fit with an exponential decay
$e^{-\Gamma_{\mathrm{z}}t}$. Insert: associated pulse sequence. (b) Dots:
measured $W(-\alpha)$, $W(\alpha)$ and $W(\alpha)-W(-\alpha)$ as a function of
cat stabilization time $t$. Solid line: fit with an exponential decay of the
measured data, the bit-flip time is deduced from the fit of
$W(\alpha)-W(-\alpha)$. Insert: associated pulse sequence.

The phase-flip rate corresponds to the loss of coherence of the cat qubit,
through which any superposition of $\ket{\pm\alpha}$ decays to a mixture of
these 2 coherent states. In order to probe it, a cat state
$\ket{C_{\alpha}^{+}}$ is first prepared in the memory by applying a drive
$\epsilon_{d}(\alpha)$ at $\phi_{\mathrm{QEC}}$, starting from an empty
cavity. The cat qubit coherence towards
$\left(\ket{\alpha}\bra{\alpha}+\ket{-\alpha}\bra{-\alpha}\right)/2$ is then
monitored by simply measuring $W(0)$. Fitting this evolution with an
exponential decay at a rate $\Gamma_{\mathrm{Z}}$ gives the value of the
phase-flip rate (Fig. 22a).

The bit-flip time $T_{\mathrm{X}}$ characterizes the typical time it takes for
the populations in $\ket{\alpha}$ and $\ket{-\alpha}$ to equilibrate. We
measure this value for various amplitudes $\alpha$ by first displacing the
memory at $\phi_{\mathrm{tomo}}$ to prepare $\ket{\alpha}$, before applying a
drive $\epsilon_{d}(\alpha)$ at $\phi_{\mathrm{QE}}$. The state $\ket{\alpha}$
is then protected by Zeno dynamics. We monitor the values of $W(\alpha)$ and
$W(-\alpha)$ over time and fit $W(\alpha)-W(-\alpha)$ with an exponential
decay at a rate $1/T_{\mathrm{X}}$ which gives the value of the bit-flip time
(Fig. 22b).

###  I.2 Dependence of $T_{\mathrm{X}}$ and $\Gamma_{\mathrm{Z}}$ on photon
number for various buffer drive frequencies

The cat stabilization works by driving the buffer on resonance at
$\phi_{\mathrm{QEC}}$. What happens if we drive it off resonant?

Figure 23a shows the measured $W(0)$ as a function of flux and drive frequency
$\omega_{d}$ after 5$\mu$s of stabilization. The red regions where
$W(0)\approx 2/\pi$ correspond to a memory unaffected by the drive so that it
is in the vacuum state. In contrast, a white region where $W(0)\ll 1$
corresponds to regions where a mixture of coherent states has formed in the
memory. The figure reminds an avoided level crossing and it is actually
autoparametric version of that between $\omega_{b}(\phi_{\mathrm{ext}})$ and
$2\omega_{m}(\phi_{\mathrm{ext}})$.

Figure 23: (a) Anticrossing of the autoparametric memory-buffer system.
Measured $W(0)$ after applying a buffer drive at $\omega_{d}$ for $5\mu$s
while at $\phi_{\mathrm{ext}}$. (b) Dots: measured $T_{\mathrm{x}}$ as a
function of $\alpha^{2}$ for 4 different drive frequencies. The color matches
with the dots shown in (a). (c) Dots: measured $\Gamma_{\mathrm{Z}}$ as a
function of $\alpha^{2}$ for 4 different drive frequencies. Dashed line:
Linear fit of the measured data with
$\Gamma_{\mathrm{Z}}=2\left|\alpha\right|^{2}/T_{\mathrm{1,eff}}$. where the
fit parameter $T_{\mathrm{1,eff}}$ is shown as an inset.

The dependence of $T_{\mathrm{x}}$ and $\Gamma_{\mathrm{Z}}$ on mean photon
number $\alpha^{2}$ is measured for 4 buffer drive frequencies at
$\phi_{\mathrm{QEC}}$ (dots in Fig. 23a). A similar behavior for
$T_{\mathrm{x}}\left(\alpha^{2}\right)$ can be seen for the 4 different drive
frequencies, with an initial exponential increase before reaching a maximum
for some optimal cat size (Fig. 23b). This optimal bit-flip time strongly
depends on the chosen drive frequency, with the curve shown in Fig. 3d
corresponding to the frequency that gives us the largest measured bit-flip
time $T_{\mathrm{X}}$.

Note that the 1D-cuts of the Wigner functions $W(\beta)$ from which are
extracted the bit-flip times (not shown here), exhibit a broadening of the
Gaussian distribution around $\pm\alpha$ with increasing $\alpha$ and with
increasing drive detuning. This can result from a distortion of the cat qubit
manifold and/or a thermalization of the memory. As discussed in Sec. I.6,
based on reasonable assumptions, we show numerical evidence that both the
smaller bit-flip times at other detuning and the broadening of the Wigner
distribution can originate from the dephasing and the self-Kerr effect of the
buffer mode.

The dependence of $\Gamma_{\mathrm{Z}}\left(\alpha^{2}\right)$ is measured for
the same four drive frequencies (Fig. 23c). The same initial behavior can be
seen for all 4 curves with an initial linear increase of the phase-flip rate.
The slope of the linear increase is given by
$\Gamma_{\mathrm{Z}}=2\left|\alpha\right|^{2}/T_{\mathrm{1,eff}}$, with
$T_{\mathrm{1,eff}}$ the effective memory lifetime. This effective lifetime
matches the memory lifetime measured by looking at the decay from $\ket{1}$ to
$\ket{0}$ at the optimal drive frequency (see section G.2), with a drastic
deterioration as the detuning of the drive frequency increases.

###  I.3 Dephasing rate measurement

Figure 24: Ramsey-like experiment on the $\left\\{\ket{0},\ket{1}\right\\}$
memory manifold. Dots: measured Wigner tomography $W(\pm\alpha)$. Dashed line:
Fit used to extract the dephasing rate $\kappa_{\varphi}^{m}$.

The memory pure dephasing rate $\kappa_{\varphi}^{m}$, used for numerical
simulations of the bit-flip time evolution, is measured with a Ramsey-like
experiment. A state close to $\left(\ket{0}+\ket{1}\right)/2$ is prepared in
the memory by first displacing the memory to the coherent state
$\ket{\alpha}$, with $\alpha=2.1$, and letting it decay under the loss
operator $\hat{L}_{2}=\sqrt{\kappa_{2}}\hat{m}^{2}$. By Zeno effect, the
2-photon loss constraints the memory dynamics to the
$\left\\{\ket{0},\ket{1}\right\\}$ manifold, hence acting as a qubit whose
basis states are $0$ and $1$ photon in the memory. This state is then left
idle for a time $t$, during which it rotates around the $Z$ axis of the Bloch
sphere at the detuning $\Delta$ between the memory drive frequency and
$\omega_{\mathrm{m,QEC}}$, in the frame of the drive frequency. The memory
state evolves under the Hamiltonian and loss operator

$\displaystyle\hat{H}/\hbar=-\frac{\Delta}{2}\left(\ket{1}\bra{1}-\ket{0}\bra{0}\right)$
(25)
$\displaystyle\hat{L}_{1}=\sqrt{\kappa_{1}}\ket{0}\bra{1},\,\hat{L}_{\varphi}=\sqrt{\frac{\kappa_{\varphi}^{m}}{2}}\left(\ket{1}\bra{1}-\ket{0}\bra{0}\right).$

For readout, the obtained state is then mapped into the
$\left\\{\ket{\alpha},\ket{-\alpha}\right\\}$ manifold by driving the buffer
mode with a drive $\epsilon_{d}\left(\alpha\right)$, mapping the eigenvectors
of $\left(\ket{1}\bra{0}+\ket{0}\bra{1}\right)$ to the 2 coherent states. The
Wigner function $W(\pm\alpha)$ (Fig. 24) is then measured, and the data fitted
to extract the detuning $\Delta$, and the dephasing rate
$\kappa_{\varphi}^{m}/2\pi\approx 0.08\mathrm{MHz}$.

###  I.4 Population of the higher excited states of the transmon

The transmon used for Wigner tomography, dispersively coupled to the memory
mode, has been shown to be one of the main factors limiting the bit-flip time
$T_{\mathrm{X}}$ at large photon numbers. In this experiment, the 2 photon
dissipation rate $\kappa_{2}$ is much greater than the dispersive shift
$\kappa_{2}\gg\chi$. This ensures that the population in the qubit first
excited state only has a negligible detrimental impact on the cat qubit
stabilization, and does not introduce additional bit-flip errors [19, 23].

Figure 25: (a) Histogram of $10^{7}$ measurements of the readout quadratures
when the transmon and memory are in thermal equilibrium with the cold
environment. (b) Same histogram when a cat qubit space is stabilized for
$100~{}\mu\mathrm{s}$ with a mean photon number
$\left|\alpha\right|^{2}\approx 30$. (c) Dots: measured occupation of the
transmon states as a function of the stabilized mean number of photons in the
memory. Excitation numbers are indicated by colors of increasing brightness.
Black dotted line: sum of the populations in states higher than $|3\rangle$.

However, simulations show that populations of higher excited states of the
transmon have an impact on the bit-flip time. The transmon populations are
probed by measuring the transmon state, while a cat qubit is stabilized in the
memory mode. The drive frequency of the readout resonator is chosen in order
to resolve the transmon states up to its $5^{\mathrm{th}}$ excited state. In
contrast, the readout frequency used everywhere else in this work was
optimized to distinguish between the transmon ground and first excited state.
A cutoff is then calibrated to separate the states
$\ket{0},\ket{1},\ket{2},\ket{3}$, and $\ket{4}$ from the others, allowing to
measure the transmon population for each amplitude $\alpha$ of the cat qubit
(Fig. 25b).

As can be seen in Fig. 25c, the transmon populations in states of higher
energy than $\ket{3}$ get excited for $\left|\alpha\right|^{2}>10$. This
increase in transmon higher excitations is clearly correlated with the
occupation of the memory mode. Besides it is not a Boltzmann distribution.

We attribute the increase of the occupation of transmon states above $\ket{3}$
to a resonance between the memory and the higher levels of the transmon. The
negative anharmonicity $\omega_{12}-\omega_{01}=-2\pi\times
181~{}\mathrm{MHz}$ of the transmon results in a transition frequency between
the 6-th and 7-th excited states being close to that of the memory. Note that
such a resonance usually occurs when the resonator frequency is placed below
that of the transmon. This phenomenon, which also happens with a simple
Duffing oscillator, was recently investigated theoretically in Ref. [64] and
experimentally in Ref. [65]. As the number of photons in the resonator
increases, the states below and above the 6-th and 7-th states of the transmon
also hybridize. In the steady state, we expect these hybridized states to be
equally populated [66, 64], and this population to increase with the number of
photons in the resonator. This qualitative signature of a growing number of
hybridized states is observed in Fig. 25c. Note that this is in contrast with
an overall increase in the temperature, where one would expect the hierarchy
of populations to roughly follow the Boltzmann distribution. Below, we study
how a finite population of these hybridized states sets an upper bound on the
bit-flip time $T_{\mathrm{X}}$.

###  I.5 Impact of high excited states of the transmon on bit-flip time

Populating the higher excited states of the transmon can in turn result in a
shift of the memory frequency. If this shift exceeds the tolerance of the
stabilization scheme, even a small population could limit a bit-flip time
which is as high as hundreds of milliseconds. Below, we evaluate the magnitude
of the frequency shifts that can be reached when the memory field drives the
transmon.

In our system, the transmon is inductively coupled to the memory and
capacitively coupled to a readout resonator. For simplicity, we do not
consider the Purcell filter of the readout resonator. The Hamiltonian of the
system we consider reads

$\displaystyle\hat{H}=$ $\displaystyle
4E_{C}(\hat{n}_{t}-n_{g})^{2}-E_{J}\cos{\hat{\theta}_{t}}+\hbar\omega_{m}\hat{m}^{\dagger}\hat{m}+\hbar\omega_{c}\hat{c}^{\dagger}\hat{c}$
$\displaystyle+\hbar
g_{mt}\sin{\hat{\theta}_{t}}(\hat{m}+\hat{m}^{\dagger})-i\hbar
g_{ct}n_{t}(\hat{c}-\hat{c}^{\dagger}),$ (26)

where $\hat{n}_{t}$ and $\hat{\theta}_{t}$ are the transmon charge and phase
operators, $n_{g}$ is an offset charge, $\hat{c}$ is the annihilation operator
of the readout resonator, $\omega_{c}$ the frequency of the readout resonator,
$g_{mt}/2\pi$ and $g_{ct}/2\pi$ are the coupling rates between the memory and
the transmon, and between the readout resonator and the transmon. We find the
values of charging energy $E_{C}/h=169.4~{}\mathrm{MHz}$ and the Josephson
energy $E_{J}/h=22.85~{}\mathrm{GHz}$, as well as that of the coupling rates
$g_{ct}/2\pi=67~{}\mathrm{MHz}$ and $g_{mt}/2\pi=225~{}\mathrm{MHz}$, by
fitting the measured low energy spectrum of the system which includes
frequencies, anharmonicity and dispersive shifts of the system. Although we
are concerned with the interaction of the memory and transmon, we included the
readout resonator to correctly fit the spectrum of the system.

Figure 26: Dots: computed memory frequency shift as a function of mean photon
number $\bar{n}$ in the memory for various transmon states. The vertical lines
are due to state mistracking. Inset: zoom on the small dispersive shifts for
low transmon excitation.

By diagonalizing the Hamiltonian in Eq. (I.5), one can obtain the values of
the frequency shifts on the cavity as a function of the transmon state and the
number of photons in the resonator (see Fig. 26). In the inset, one recognizes
the dispersive shift $\chi/2\pi=0.170~{}\mathrm{MHz}$ when the transmon is in
its first excited state $|1\rangle$, which is used for the Wigner tomography.
While populating one of the first 3 excited states of the transmon cause small
enough frequency shifts of the memory (inset) so that they are handled by the
stabilization scheme, populating higher excited states can result in frequency
shifts as large as $30~{}\mathrm{MHz}$. Such large frequency shifts are due to
the non-perturbative hybridization of the transmon states, thus exiting the
dispersive regime of the coupling between transmon and memory.

For small frequency shifts, the state remains confined to a manifold spanned
by two coherent states, although the cat size might slightly vary. Indeed,
using a semi-classical analysis (Eq. (S22) of [23]), in the limit where
$\kappa_{1}\ll|\Delta_{m}|$ and $\kappa_{b}\gg|\Delta_{b}|$, the photon number
reads
$|\alpha_{\Delta}|^{2}=|\alpha_{0}|^{2}-\frac{|\Delta_{m}|\kappa_{b}}{4|g_{2}|^{2}}$,
where $|\alpha_{0}|^{2}$ is the photon number at zero detuning at the same
drive amplitude. This equation has a solution if
$|\Delta_{m}|<4|g_{2}|^{2}|\alpha_{0}|^{2}/\kappa_{b}$. Using the measured and
extracted values of $\kappa_{b}/2\pi=40~{}\mathrm{MHz}$ and
$g_{2}/2\pi=6~{}\mathrm{MHz}$ , this leads to
$|\Delta_{m}|<3.6|\alpha_{0}|^{2}~{}\mathrm{MHz}$, that is the minimal
condition for a 2D-manifold to be stabilized. The impact of the detuning
$\Delta_{m}$ on the bit-flip time is illustrated in Fig. 27 , where the bit-
flip time is plotted for various detunings as a function of the drive
amplitude. The drive amplitude is converted to the photon number corresponding
to the detuning $\Delta_{m}/2\pi=4$ MHz.

Figure 27: Effect of memory detuning $\Delta_{m}$ on the bit-flip time
$T_{X}$. The bit-flip time is plotted as a function of the drive amplitude for
several values of $\Delta_{m}$. The drive amplitude is converted to the
corresponding photon number at $\Delta_{m}=4~{}\mathrm{MHz}$, which
approximately corresponds to the detunings set for Fig. 3b. The black crosses
indicate the experimentally measured bit-flip times shown in Fig. 3d.

The detuning associated with the first and second excited states (below 20
photons) is below $1~{}\mathrm{MHz}$, making the condition
$|\Delta_{m}|<3.6|\alpha_{0}|^{2}~{}\mathrm{MHz}$ largely satisfied. This is
numerically verified in Fig. 27. In this case, the bit flip rate takes the
form
$T_{\mathrm{X}}^{-1}=\sum_{i,|\Delta_{i}|<\Delta_{\mathrm{max}}}p_{i}T_{\mathrm{X}}^{-1}(\Delta_{i})$.
The small detunings associated with the first and second excited states make
the weighted contribution $p_{i}T_{\mathrm{X}}^{-1}(\Delta_{i}),~{}i=1,2$
negligible compared to $p_{0}T_{\mathrm{X}}^{-1}(\Delta_{0})$.

However, detunings as large as $30~{}\mathrm{MHz}$ become difficult for the
dissipation scheme to compensate. For $|\Delta_{m}|=30~{}\mathrm{MHz}$, the
minimal condition to generate a cat state in the cavity is
$|\alpha_{0}|^{2}>8.5$ photons. Moreover, even if the memory becomes
populated, this large detuning comes with a large displacement on the buffer
mode, $\beta\sim\Delta_{m}/2g_{2}=2.5$ (see next section). Combined with its
dephasing noise and large Kerr nonlinearity, the resulting bit-flip time of
such a cat qubit is low. This is illustrated in Fig. 27, where the bit-flip
time corresponding to a detuning of $9~{}\mathrm{MHz}$ does not improve over
the bare memory lifetime until $|\alpha_{0}|^{2}=12.5$. For simplicity, we
assume that a bit-flip occurs every time the system is subject to such a large
detuning for the range of photons considered here, which represents the worst-
case scenario.

From the analysis of the transmon, it is likely that populating the layer of
hybridized states will result in a large detuning and therefore a bit-flip.
This allows us to derive a simple upper bound on the induced bit-flip time
$T_{\mathrm{X}}$ from the measured values of the state population: it is given
by the inverse rate at which this layer of states is populated
$\gamma_{\mathrm{hyb}}$. Let us call $p_{\mathrm{hyb}}$ the population of the
hybridized states, represented as a black dotted line in Fig. 25, and $p_{1}$
the population of the first excited state. Note that the state $\ket{2}$
merges with the rising plateau of states around $|\alpha|^{2}=20$ photons. The
rate at which the hybridized layer gets populated thus reads
$\gamma_{\mathrm{hyb}}=\gamma_{\mathrm{hyb\rightarrow 1}}p_{\mathrm{hyb}}$.
Using the measured $\gamma_{\mathrm{1\rightarrow 0}}^{-1}=18~{}\mu\mathrm{s}$
and assuming $\gamma_{\mathrm{hyb\rightarrow
1}}\approx\gamma_{\mathrm{2\rightarrow 1}}=2\gamma_{\mathrm{1\rightarrow 0}}$,
we obtain the red dotted line in Fig. 3b.

###  I.6 Main limitation of the bit-flip time

####  I.6.1 Impact of memory dephasing on bit-flip time $T_{\mathrm{X}}$

We expect pure dephasing on the memory
($T_{\varphi}^{m}=1/\kappa_{\varphi}^{m}=2~{}\mu\mathrm{s}$, see Fig. 24) to
impact the bit-flip time $T_{\mathrm{X}}$. In the adiabatic regime
$8|\alpha|g_{2}\ll\kappa_{b}$, the bit-flip time scales as
$T_{\varphi}^{m}|\alpha|^{-2}\exp(2|\alpha|^{2})$ [1]. However, in our case,
the adiabaticity condition is not respected for $|\alpha|\gtrsim 1$. While the
memory dephasing noise limits the scaling of $T_{X}$ with the photon number,
we observe numerically that it is not strong enough to lead to the observed
saturation.

####  I.6.2 Impact of buffer photons on bit-flip time $T_{\mathrm{X}}$

During the stabilization, the driven buffer ideally stays in the vacuum state.
A finite buffer amplitude $\beta=\langle\hat{b}\rangle_{\mathrm{ss}}\neq 0$ in
the steady state can result in unwanted sources of bit flips. First, dephasing
noise on the buffer can be upconverted to thermal photons via the drive
photons. These buffer thermal photons are converted to memory thermal photons
by the two-photon exchange term, thus creating a heating term of the form
$\hat{m}^{\dagger 2}$. Provided the large flux dispersion of the buffer
frequency, we expect the dephasing rate $\kappa_{\varphi}^{b}$ to be large. If
one assumes that the dephasing rate on the memory is mostly due to flux noise,
this gives

$\frac{\kappa_{\varphi}^{b}}{\kappa_{\varphi}^{m}}=\frac{\left|\frac{\partial\omega_{b}}{\partial\phi_{\mathrm{ext}}}(\phi_{\mathrm{QEC}})\right|}{\left|\frac{\partial\omega_{m}}{\partial\phi_{\mathrm{ext}}}(\phi_{\mathrm{QEC}})\right|}=60.$

The expected thermal population is given by
$n_{\mathrm{th}}^{b}=\kappa_{\varphi}^{b}|\beta|^{2}/\kappa_{b}$.

Figure 28: (a) Simulated bit-flip times for various values of the drive
detuning $\Delta_{b}=\omega_{b}-\omega_{d}$, corresponding to the values of
Fig. 23b. The dephasing rate is set to
$\kappa_{\varphi}^{b}=180\kappa_{\varphi}^{m}$, and the memory detuning from
the buffer is $2\omega_{m}-\omega_{b}=2\pi\times 3.5$ MHz. As a reference, we
indicate the measured optimal bit-flip times (black crosses) and the bound
independently set by the transmon (red dots). (b) Corresponding amplitudes of
the buffer mode in the steady state as a function of average photon number
$|\alpha|^{2}$ in the memory for the same detunings. The imaginary part is
zero. (c) Simulated probability distributions of the quadrature defined by the
direction of $\alpha$ in the memory phase space
$\mathrm{P}(\mathrm{Re(\beta)})=|\langle\mathrm{Re(\beta)}|\psi\rangle|^{2}$,
plotted for several values of $|\alpha|^{2}$ (arbitrary units). The detuning
is set to $\Delta_{b}/2\pi=-8.5~{}\mathrm{MHz}$. As the photon number
increases, the distribution broadens.

Second, the large expected self-Kerr rate of the buffer $\chi_{bb}/2\pi\approx
10~{}\mathrm{MHz}$ can result in a small squeezing on the buffer when the
latter is displaced. The single photon loss channel on a squeezed buffer will
yield an effective non-zero thermal occupation given by
$n_{\mathrm{th}}^{b}=\sinh^{2}(r)$, where $r$ is the effective squeezing
parameter. Besides, the squeezing on the buffer results in an anti-squeezing
term on the memory, which is likely to decrease the bit-flip time.

Let us estimate the displacement on the buffer. In the interaction picture,
the master equation on the memory mode Eq. (22) reads

$\displaystyle\frac{d\hat{m}}{dt}=$
$\displaystyle-i\Delta_{m}\hat{m}-i2\chi_{mm}\hat{m}^{\dagger}\hat{m}^{2}-i\chi_{mb}\hat{m}\hat{b}^{\dagger}\hat{b}$
$\displaystyle-\frac{\kappa_{\varphi}^{m}+\kappa_{1}}{2}\hat{m}-i2g_{2}\hat{m}^{\dagger}\hat{b}.$

Assuming a steady-state solution of the form
$\rho=\ket{\alpha}\bra{\alpha}\otimes\ket{\lambda}\bra{\lambda}$, with
$\alpha\neq 0$, and taking the trace of the above equation, we obtain

$\displaystyle\lambda=e^{i2\theta_{m}}\frac{-\Delta_{m}-2\chi_{mm}|\alpha|^{2}-\chi_{mb}|\lambda|^{2}+i(\kappa_{\varphi}^{m}+\kappa_{1})/2}{2g_{2}},$

where $\theta_{m}=\arg(\alpha)$.

In the limit
$\kappa_{\varphi}^{m},\kappa_{1}\ll\left|\Delta_{m}+2\chi_{mm}|\alpha|^{2}+\chi_{mb}|\lambda|^{2}\right|$,
the amplitude $\lambda$ becomes

$\displaystyle\lambda=-e^{i2\theta_{m}}\frac{\Delta_{m}+2\chi_{mm}|\alpha|^{2}+\chi_{mb}|\lambda|^{2}}{2g_{2}}.$

The displacement amplitude on the buffer depends on the effective memory
detuning, which in turn depends on the cat size due to the self-Kerr effect.
The amplitude of the buffer field in the steady state thus depends on the
effective detuning of the memory, and consequently the detuning of the buffer
drive $\Delta_{b}=\omega_{b}-\omega_{d}$ and the photon number in the memory.

Before addressing the impact of a populated buffer on the bit-flip time, let
us first outline the experimental data that cannot be explained solely by the
heated transmon. As a reminder, the transmon excited state population sets an
upper bound around $1~{}\mathrm{s}$ for the bit-flip time. However,
measurements taken at various drive detunings reveal earlier saturations of
the bit-flip time, almost two orders of magnitude below $0.3~{}\mathrm{s}$
(see Fig. 23b). This cannot be explained by the $1~{}\mathrm{s}$ bound imposed
by the transmon. Besides, the measured 1D-cut of the Wigner functions along
the cat axis displays a broadening of the coherent states along this axis (not
shown here). This distortion of the coherent states cannot be explained by the
weak occupancy of the excited transmon states (below $1~{}\%$ for the first
excited state and below $0.1~{}\%$ for the states above). It rather results
from a local distortion of the code space and/or a thermalization of the
memory.

Interestingly, we find that both of these experimental observations can be
corroborated by the large expected self-Kerr rate of the buffer and its pure
dephasing. Fig. 28a shows the bit-flip times extracted from the simulation of
Eq. (22), at the values of the drive detunings $\Delta_{d}$ corresponding to
the experimental values of Fig. 23b. We observe a qualitatively similar
behavior of the bit-flip times, in particular for the curves corresponding to
larger detunings. In order to see a notable similar effect, the dephasing rate
is here set to $\kappa_{\varphi}^{b}=180\kappa_{\varphi}^{m}$ which is three
times larger than what can be inferred from the flux dependence of the
transition frequencies. We note that this parameter is not directly measured
in the experiment so a larger buffer dephasing rate than predicted is
conceivable.

In Fig. 28b are shown the corresponding amplitudes of the buffer mode. As
expected, the amplitude correlates with the detuning and the maximal bit-flip
time of each curve. The linear increase or decrease with $|\alpha|^{2}$ is due
to the self-Kerr effect on the memory and to the cross-Kerr effect between
buffer and memory, which modify the effective detuning on the memory as
$|\alpha|^{2}$ and $|\beta|^{2}$ grow. The increasing amplitude combined with
the self-Kerr effect and dephasing rate of the buffer mode leads to
thermalization of the memory and eventually limits the bit-flip time.

Finally, Fig. 28c shows the probability distribution of the memory state along
the cat axis for $\Delta_{b}/2\pi=-8.5~{}\mathrm{MHz}$. The distribution
broadens as $|\alpha|$ increases due to buffer thermal photons, leading to a
saturation of the bit-flip time for this detuning.

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