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

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 in 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].A prominent example is the cat qubit, whose computational states are coherent states | ± α⟩ of a harmonic oscillator, such as a superconducting microwave cavity [1,[7][8][9].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 κ 2 .The bit flip time T X then increases exponentially with the photon number |α| 2 at the modest expense of a linear deterioration of the phase flip rate Γ Z ∝ |α| 2 [19].Reaching large values of two-photon rate κ 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 κ 1 , the main cause of phase-flip errors [24].The remaining phase-flip errors could then be corrected using a repetition code made of a chain of cat qubits under the condition κ 2 /κ 1 ≳ 10 2 [24][25][26].Introducing a new non-linear design called the autoparametric cat, our experiment benefits from a stronger 3-wave mixing interaction compared * These authors co-supervised the project to the 4-wave mixing interaction of previous schemes.We demonstrate κ 2 rates as high as 2 × 2π MHz, about 150 times larger than κ 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 ± 0.15 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 ± 2% in 28 ns.
To achieve two-photon coupling between the memory and its environment, an intermediary mode with single photon coupling κ b to the environment acts as a buffer and a two-to-one photon exchange Hamiltonian Ĥ = ℏg 2 m2b † + h.c. is activated with a rate g 2 .The operators m and b are the annihilation operators of the memory and buffer mode respectively.In the limit where g 2 is small enough compared to κ b , the two-photon dissipation rate reads κ 2 = 4g 2  2 /κ 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 ω p = 2ω m − ω 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 κ 2 rates that could be obtained with pumped nonlinearities prevents surpassing the self-Kerr rate or the dispersive coupling rate between transmon and memory [18,19,27].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 2ω m = ω 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 FIG. 1.(a) A four-wave mixing coupler such as a transmon or ATS swaps pairs of photons of a memory mode at ωm for single photons of a buffer mode at ω b at a rate g2 owing to a pump at |ω b −2ωm|.The buffer loss rate κ b thus leads to an effective two-photon dissipation rate κ2 that scales with pump amplitude.Driving the buffer mode on resonance at a displacement rate ϵ d stabilizes a cat code {| ± α⟩}.(b) A three-wave mixing coupler passively performs the same photon exchange when ω b = 2ωm.(c) Scheme of the autoparametric cat.The threewave mixing coupler is a ring of three Josephson junctions threaded by a flux ϕext = φextφ0, with φ0 = ℏ/2e, Josephson energies EW /h ≈ 115 GHz and EJ /h ≈ 250 GHz.The buffer (green) and memory (blue) mode geometries are represented as field vectors.(d) Optical and e-beam images of a clone of the device.False colors highlight the input buffer/flux line (green), the tomography transmon and its readout resonator and Purcell filter (red) as well as the three-wave mixing coupler (purple) (see Sec. D).An array of holes in the tantalum (to prevent the apparition of vortices) appear as a white rectangle in the larger image.(e) Dots: measured resonance frequency of the buffer ω b (green) and twice the measured resonance frequency of the memory 2ωm (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).rate g 2 is much larger than what can be reached for 4 wave mixing (see comparison in section J).
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 ϕ 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 / √ 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).A single input line (green in Fig. 1d) then couples to the circuit in order 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 κ b /2π ≈ 40 MHz and a much lower memory loss rate κ 1 /2π ≈ 14 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 (see section C 2) to the memory with χ/2π = 170 kHz to perform the Wigner tomography of the memory mode [36][37][38].Note that a work conducted in parallel to ours demonstrates that Wigner tomography can be performed without the transmon [39].
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) where δφ ext = (ϕ ext −ϕ )/φ 0 is the distance from the sweet spot and φ zpf,m/b are the zero point fluctuations 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 ϕ QEC for which the frequency of the buffer matches twice the frequency of the memory ω b (ϕ QEC ) = 2ω m (ϕ QEC ) ≈ 2π × 7.896GHz (see Fig. 1e).Therefore the three-wave mixing term performs the de-sired swap between pairs of memory photons and single buffer photons at ϕ QEC .The preparation of a cat state |C α + ⟩ ∝ |α⟩ + | − α⟩ consists in starting from the vacuum state in the memory at ϕ QEC and turning on a drive with an amplitude |ϵ d | = |α| 2 g 2 (Fig. 1b) at twice the memory frequency ω d = 2ω m for a time of 300 ns.It is shorter than the characteristic time 1/|α| 2 κ 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 κ 2 by monitoring how the memory decays once the drive has been turned off.Fitting the measured Wigner function W (β) at various time steps with the solution of a Lindblad master equation where κ 2 is the single free parameter allows us to determine κ 2 /2π ≈ 2.16±0.1 MHz at ϕ QEC .The ratio κ 2 /κ 1 ≈ 1.5 × 10 2 is much larger than in previous implementations of two-photon dissipation using four-wave mixing [17,18,23,40].This procedure is repeated for various flux biases around ϕ QEC in Fig. 2a, which shows the range of detuning ω b −2ω 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).Indeed this approximation breaks down when the frequency 8g 2 |α| at which single buffer photons are swapped with pairs of memory photons gets larger than the rate κ b at which they leak out from the buffer.
Note that Wigner tomography of W (β) is performed using a displacement of the memory mode by D(−β), followed by a parity measurement that exploits the dispersively coupled qubit in a Ramsey-like sequence Ref. [36][37][38].However, since both displacements and the dispersive coupling are inhibited by two-photon loss, these operations are performed at a flux ϕ tomo (see Fig. 1e) such that the strong detuning ω b − 2ω m disables twophoton dissipation, yet without too much nonlinearity that would distort the tomography.In practice, the tomography thus requires to abruptly change the flux bias between ϕ QEC and ϕ tomo (see Sec. E 2).Besides, we improve the performance of this tomography by initializing the parity measurement with an idle time of 300 ns at ϕ QEC so that all pairs of photons are dumped into the environment.The parity measurement then boils down to determining whether or not a single photon remains in the memory [25,41].Parity is preserved during this operation owing to the large ratio κ 2 /κ 1 .
The phase-flip rate Γ 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 α + ⟩, the parity, hence the measured W (0), decays exponentially with time at a rate Γ Z while the drive ϵ d (α) is kept on.In Fig. 3b are shown the observed Γ Z rates as a function of photon number |α| 2 in the cat code.As expected, the phase-flip rate increases linearly as Γ Z = 2|α| 2 κ 1 until it goes above 20 photons.
The bit-flip time T X characterizes how fast the coherent state |α⟩ decays to an equal mixture of | − α⟩ and |α⟩.
In order to measure it, the flux is first set to ϕ tomo so that a memory drive can prepare the state |α⟩.The flux is then turned back to ϕ QEC and the buffer drive is immediately turned on with an amplitude ϵ d (α) (Fig. 3c).We measure the Wigner functions W (±α) for various waiting times and fit their difference W (α) − W (−α) ∝ e −t/TX .The resulting bit-flip time is shown in Fig. 3d and rises exponentially with photon number |α| 2 until about 12 photons.There, T 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.We performed simulations of the evolution of the buffermemory bipartite system that indeed predict smaller scaling factors (dashed line in Fig. 3d and section I).Some parameters we use in the master equation (H1) are experimentally measured such as the pure dephasing rate of the memory κ m φ /2π = 0.16 MHz (Fig. 25), the self-Kerr rate of the memory χ m,m /2π = 0.22 MHz, and the measured detuning ∆ m /2π = 3 MHz between half the drive frequency and ω m .Other parameters are inferred from the circuit model and the pure dephasing rate of the buffer κ b φ /2π is assumed to be limited by flux noise like κ m φ /2π (see section I).Besides the simulated bit-flip times strongly depend on the flux ϕ ext and we set a detuning 2ω m − ω b = 2π × 3.5 MHz in order to better reproduce the measured bit-flip times.This corresponds to a flux offset of 3 × 10 −4 ϕ 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 ± 0.15 s, reached for |α| 2 ≈ 20.We identify three possible mechanisms that present comparable contributions to the bit-flip rate at large photon numbers.First, varying the dephasing rates κ m φ in simulations lead to an apparent saturation of T X (see Fig. 29).A better tuning of the circuit parameters would cancel κ m φ by matching the flux at which the autoparametric condition occurs with the memory sweet spot in Fig. 1e.A second candidate for the bit-flip limitation is the thermalization of the buffer mode, which may come from various origins (see section I 6).In turn,  3a where the cat code is stabilized with a photon number |α| 2 = 9.3.An additional displacement drive at ωm starts 240 ns after the buffer drive is turned on.Here, its amplitude ϵZ (t) is Gaussian shaped with a mean amplitude εZ /2π = 1.625 MHz and its phase is chosen to displace in the direction indicated by the arrow in c).Line: fit to oscillations at a frequency ΩZ /2π = 19.8MHz, which are decaying at a rate κZ /2π = 0.62 MHz.(c) Measured Wigner functions W (β) after a Z rotation of angle θ = 2π, 3π/2, and π from top to bottom.(d) Dots: Inferred rotation frequency ΩZ as a function of cat code amplitude α, and for various mean drive amplitudes εZ /2π = 0.32, 0.625, 0.965, 1.295, 1.625, 1.955, 2.285, and 2.66 MHz from bright to dark orange.Lines: expected rotation frequency ΩZ = 4Re(εZ α) around Z. (e) Dots: Inferred decay rate κZ as a function of |α| for the same drive amplitudes.Lines: simulated decay rate with g2/2π = 6 MHz as a fit parameter and the same detuning as in Fig. 3d.
owing to the breakdown of adiabatic elimination, it limits the bit-flip time.A mitigation strategy would consist in adapting the drive frequency for each size α of the cat qubit to keep ∆ m = 0. Finally, simulations reveal that while the transmon first excited state no longer limits T X [19], residual excitations of the transmon's higher states sets a bound below which the bit flip time cannot be affected by the transmon.Despite a relatively small dispersive shift χ/2π = 170 kHz, these higher excited states of the transmon exit the dispersive regime yielding a large frequency shift on the memory comparable to κ 2 |α| 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 |α| 2 (Fig. 26), 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 bound, indicating that the excitation of transmon higher states may be a limiting mechanism.This limitation could be avoided by removing the transmon qubit altogether [39].
With such a large ratio κ 2 /κ 1 , it is possible to realize Z gates in the cat code {|α⟩, | − α⟩} using quantum Zeno dynamics [18,[42][43][44][45][46].Starting from |C α + ⟩ with the buffer drive ϵ d (α) turned on, we continuously drive the memory on resonance with an amplitude ϵ 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 α 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 {|α⟩, | − α⟩}, but the memory drive ϵ Z rotates the phase θ of the quantum superposition (|α⟩ + e iθ | − α⟩)/ √ 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 Ω Z and decay rate κ Z .
The rotation frequency is expected to be given by Ω Z = 4Re(ϵ Z α), which is precisely what we observe in Fig. 4d.The decay rate κ Z has a more subtle dependence on photon number |α| 2 and memory drive ϵ Z as seen in Fig. 4e.In the ideal case and for constant ϵ Z , the rate is expected to decay as κ Z = 2κ 1 |α| 2 + κ b ϵ 2 Z /(2|α| 2 g 2 2 ) [25].The first term corresponds to phase flips at a rate Γ Z while the second one corresponds to an induced leakage out of the confined cat qubit Hilbert space when the drive amplitude ϵ Z is too strong to be Zeno blocked.This expression remains valid even outside of the adiabatic elimination regime [47].
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 κ Z (α) 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 /2π ≈ 6±0.5 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 /κ b ≈ 3.6 × 2π MHz than what is measured in Fig. 2, which is expected since the condition 8g 2 |α| < κ b is not met for |α| 2 ≳ 1 [39].Interestingly, despite the observed limitation on T X in the non adiabatic regime, it is still possible to improve gate speed and fidelity by going to large values of g 2 |α|.
It is possible to infer the Z gate fidelity from the measured evolution of W (β) during the drive.Owing to the noise bias of the cat code, the Z gate is the least faithful when applied to Clifford states | ± X⟩ and | ± Y ⟩.In contrast, we measure that | ± Z⟩ 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(−πκ z /Ω z )/2 (see Sec. H 4). Using Gaussian pulses for the memory drive leads to Z gate fidelities F = 95±2% in 26 ns for |α| 2 = 9.3 and for a drive amplitude εZ /2π = 1.625 MHz.We further improve the gate fidelity by using square pulses.Indeed, for a fixed average drive amplitude εZ , the square pulse has the smallest maximum amplitude, hence induces the least nonadiabatic errors.We then reach F = 96.5 ± 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 κ 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.We emphasize that repetition codes are more lenient with errors than surface codes [48], which is another benefit of using strongly noise-biased qubits.Moreover, the associated photon exchange rate g 2 allows us to perform Z gates with up to 96.5 % fidelity in 28 ns.We achieved a notable bit flip time T X of up to 0.3 s.Several mechanisms have been identified as possible limitations and mitigation strategies have been proposed for future realizations.The figure-of-merit κ 2 /κ 1 = 1.5 × 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, for instance using a notch filter at the memory frequency and improving the fabrication process, could dramatically decrease the phase flip rate and bolster the scaling of error correction.Besides the present design parameters are quite conservative for this first demonstration.The zero point fluctuations of the modes in the junctions (φ zpf,m = 0.0305 and φ zpf,b = 0.0648) can safely be increased, which would improve g 2 .Additionally, in the future, the large coupling rates of the autoparametric approach could be leveraged under the frequency matching condition ω b = 4ω m .This could then stabilize cat qubits that are composed of superpositions of the four coherent states |α⟩, |iα⟩, | − α⟩, and | − iα⟩, whose autonomous stabilization is still missing despite the strong interest of this approach for quantum error correction [1].

ACKNOWLEDGMENTS
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 Gerhard Kirchmair, 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 [49][50][51].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 ℏg 2 m2 ĉ † + h.c..When driving the pump mode c at 2ω m with a large enough power such that the number of photons in the pump mode exceeds the threshold n thr = κ 2 m /(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 [52][53][54].These two phases correspond to two coherent states |α⟩ and | − α⟩ 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 2ω 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 resonator frequency 2ω m .The high Q limit of mode a lowers the parametric oscillation threshold to n thr ≪ 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 |α⟩ and | − α⟩.
This realization is at the origin of the autoparametric circuit design.We started from a superconducting circuit widely used for degenerate parametric amplifica-tion: a resonator comprising a DC flux biased SQUID that is flux pumped at twice its frequency and acts as our memory mode [50].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 φ 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 ϕ 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 φ 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 the memory, hence the buffer is responsible for meeting the frequency matching condition 2ω m = ω 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.

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 due to the flux bias.Next, we will identify the eigen- modes of the linear part of the Hamiltonian.Finally, we will calculate the nonlinearities and discuss the impact of junction asymmetry.

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 = β J E J with β J < 1.Following the procedure detailed in [55], we compute the equilibrium phase drop across each junction φ1 and φ2 and φW .Note that we decompose each phase drop φ x as a sum of a constant part φx and a dynamical part φx with zero mean.Current conservation inside the loop imposes that φ1 = − φ2 so that we denote φJ = φ1 the main junction phase drop.Current conservation further dictates that φJ = arcsin(β J sin φW ). (B1) The phase drop around the loop φ = φW + φ1 − φ2 then reads φ = φW + 2 arcsin(β J sin φW ).(B2) Besides the superconducting loop is threaded by the external flux ϕ ext = φ ext φ 0 , which leads to the constraint Finding the configuration of phase drops at equilibrium thus consists in first determining φW (φ ext ) by solving the equation φ ext = φ( φW ) of which a graphical representation is shown in Fig. 6a.Then, one gets φJ from Eq. (B1) and the effective inductive energies of each junction ĒJ = E J cos( φJ ), ĒW = E W cos( φW ).The effective inductances follow as LJ = φ 2 0 / ĒJ and LW = φ 2 0 / ĒW .In case where β J > 1/2, the function φW (φ ext ) is multivalued [55] as shown in Fig. 6b.In order to avoid hysteresis effects or instabilities, we chose β J < 1/2 which is equivalent to E W < E J /2.We can check in Fig. 6a that φ( φW ) is indeed monotonous in that case.
For β J = 0.46 as in our device, at ϕ QEC = 0.311ϕ 0 that is slightly offset from the sweet spot (0.40ϕ 0 ), we find FIG.6. Flux sweet spots of the mixing element.(a) Sum φ of the phase differences across the three junctions as a function of the phase difference φW across the weak junction, calculated using Eq.(B2) for βJ = EW /EJ = 0.46 as in the experiment (blue), βJ < 1/ √ 2 (orange) and βJ > 1/ √ 2 (green).A flux bias imposes φext = φ so that these curves show which values of φW are possible.(b) Phase differences φJ (solid-dotted) and φW (dashed-dotted) as a function of φext.Same color code for the values of βJ 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 φW = π/2.In (a) it crosses φ at φ We focus on the case where φ (sweet) ext ∈ [0, π] because of the 2π periodicity of the solutions, and by symmetry around φ ext = 0. Additionally, we exclude the trivial solution φ ext = 0 because we need φJ ̸ = 0 at the sweet spot to maintain 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 ω m originates from ĒJ (φ ext ) = E J cos( φJ ).The flux sweet spot is therefore a local extremum of φJ (φ ext ).From Eq. (B1), φ (sweet) ext is therefore close to φW = π/2 since sin φ W is maximal there.Finally, using the condition that φ ext < π imposes an upper bound on β J .Indeed, for φW = π/2, Eq. (B2) comes down to π/2+2 arcsin β J < π and thus , a sweet spot will thus be visible at φ (sweet) ext = π/2 + 2 arcsin β J (blue and orange closed circles in Fig. 6b).Close to the sweet spot, at first order, φJ is constant and φW = π/2 + δφ ext where is the flux offset from the sweet spot.At second order, we then have Note that the single minimum condition β J < 1/2 that is verified in the experiment is well in this limit.The regime 1/2 < β J < 1/ √ 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 β J > 1/ √ 2, there exists a sweet spot when but it is not the lowest energy phase configuration (green open circle in Fig. 6b).

Eigenmodes
Now that the equilibrium phase differences and the effective inductive energies ĒJ and ĒW 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 Hence, incorporating the constraints of Eq. (B3), we find 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 where φ m and φ 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 with E L,m = 2 ĒJ and E L,b = 2 ĒJ + 4 ĒW .The mode frequencies and zero-point fluctuations of the buffer and memory mode are given by so that, in second quantization, the linear part of the Hamiltonian is with annihilation operators defined by their relation to the phase differences The frequencies of both the memory and buffer modes depend on ϕ ext through the dependency of φJ and φW .
In the actual circuit, stub resonators are connected to the ring of junctions so that the mode frequencies and zero point fluctuations are modified.ϕ ext is then chosen so that the condition ω b = 2ω m is matched, defining the value of ϕ QEC as shown in Fig. 1.Numerically, fitting the frequency dispersion versus flux with the full model gives us φ zpf,m = 0.0305 and φ zpf,b = 0.0648 at ϕ QEC .

Nonlinearities
Around the DC solution of the system, the full potential ) can be used to compute the nonlinearities.Using the former change of variable we get The potential energy is represented in Fig. 7 for the circuit parameters E W /h ≈ 115 GHz and E J /h ≈ 250 GHz for several values of the external flux bias ϕ ext .As expected from the change of variable, the global minimum is located at (φ m , φ b ) = (0, 0) for any flux bias φ ext .Other minima exist owing to the periodicity of the Josephson potential but there is a potential barrier of 2E J ≈ k B × 22 K to overcome in order to transit from one solution to the other as can be seen in Fig. 7.By expanding the sin and cos functions up to fourth order in the phases, we obtain the parameters of the Hamiltonian.b.Fourth order terms Expanding even further the Hamiltonian to fourth 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: Note that it is the equivalent Josephson energies ( ĒJ , ĒW ) that appears and not the bare ones (E J , E W ). The corresponding Hamiltonian reads The effect of these spurious terms is taken into account in the simulations used in Figs. 3 and 4 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 FIG. 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 (-500 µm and 500 µm), 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 (-175 µm and 425 µm) leading to the highest memory coupling quality factor, the current field is a mix of slot-line and co-planar geometries.
the input to disable the coupling to the memory mode.Memory and buffer modes only couple to propagating modes having the same symmetries.
A first input line design that can benefit from these symmetries is a slotline made of a gap, separating two ground planes [56].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 [57].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 antisymmetric 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 slotline 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).The memory coupling rate to the transmission line is given by the sum of the coupling rates to the coplanar 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 µm and a negative one l − = −175 µm) for which the total coupling rate to the slot-line and coplanar 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 chose.The bottom stub of the memory mode in Fig. 8 is terminated to the ground by three elements in parallel: two identical geometrical inductances L (horizontal lines) and a Josephson junction of Josephson energy E J in series with a capacitor C t (large pad in pink).The important parts of this circuit are schematized in Fig. 11.Focusing on the sole memory mode for the autoparametric circuit, one models it by an inductor L m and a capacitor C m in series.With the last scheme of Fig. 11, one sees how the small inductance L/2 inductively couples the memory and transmon modes.A simple way to compute this coupling term consists in writing Kirchoff's law We see that the inductive energy of the small inductance L/2 reads Expanding this square term leads to a renormalization of the frequencies of the memory mode and of the transmon mode and the cross-product gives the coupling term we wanted to compute: ) where θt = φt /φ 0 and g mt is given by LE J 2ℏLmφ0 φ ZPF,m up to a factor of order 1.The coupling rate between the memory and transmon modes, g mt , was measured from the system's low-energy spectrum to be 2π × 225 MHz.This value agrees well with the coupling rate predicted by electromagnetic simulations.

Appendix D: Cabling
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 ω IF,q /2π = 100 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 ω q , ω r , and ω m,tomo /ω m,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 Buffer Memory FIG.
12. 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 signal driving the buffer mode at ω b is transmitted through the alternative port of the device.It's combined with a DC signal directly generated by one DAC of the OPX using a 3 GHz frequency diplexer, facilitating a swift transition from ϕ tomo to ϕ 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) [58].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. 12).
with κ 2 = 4g 2 2 /κ b .Our task is to measure the Wigner function W of the encoded state.The conventional technique for measuring W (β) [36][37][38] begins by displacing the memory by D (−β) before applying an unconditional π/2 pulse on the transmon qubit.The system then remains idle for a time period of π/χ ≈ 2.8µs, during which the qubit gains information about the parity of the number of photons in the memory.Subsequently, a second π/2 pulse is applied to map the memory parity into the states |g⟩ or |e⟩ of the qubit.
However, this method would not be effective at ϕ QEC due to the large 2-photon dissipation.This would impede proper displacements D (−β).Moreover, this dissipation broadens the memory energy levels by κ 2 ≫ χ, effectively neutralizing the dispersive coupling between the memory and transmon qubit.
To overcome these challenges, the Wigner tomography is performed at ϕ tomo where ω b,tomo ̸ = 2ω m,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 ϕ QEC and ϕ tomo , we employ a fast flux line that sets the desired magnetic flux in approximately 20 ns.
While the memory dynamics at ϕ tomo are primarily dominated by the self-Kerr rate χ m,m /2π ≈ 220kHz, which only marginally impacts the system during the 20 ns it takes to switch the flux, it is crucial to keep the drive ϵ d (α) on before shifting from ϕ QEC to ϕ tomo .The memory dynamics at ϕ QEC are indeed dominated by the 2-photon dissipation with a rate κ 2 , which significantly impacts the system in 20 ns.To prevent state distortion prior to the Wigner tomography, the drive ϵ d (α) is thus maintained during the flux change.This drive at ω b,QEC does not affect the memory at ϕ tomo , where the frequency matching condition is no longer satisfied.

Phase correction of the stabilized cat
Owing to the change in memory frequency when the flux is switched between ϕ QEC and ϕ 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 ω LO,m = 2π × 3.988481 GHz and another one at twice this frequency ω LO,b = 2ω LO,m (see Fig. 13).They are generated using 2 channels of an Anapico APUASYN20 so that their phases are locked.
where t is the time spent at ϕ QEC .This can be seen as cat states whose direction in phase-space changes over time (Fig. 14a).
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. 14b).This is of particular interest for the measurement of T X where we need to measure the evolution of W (±α), 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 D(β) applied on the memory during Wigner tomography are performed by applying a drive at frequency ω m, tomo .We calibrate how the displacement amplitudes β depend on the voltage amplitude V d at the level of the DAC by 3 methods (Fig. 15).We then verify how good is the match between the proportionality factor µ = dβ/dV d they provide.

Ramsey interferometry
The first calibration method relies on a Ramsey sequence [59].Starting from the memory in its vacuum state, a drive of amplitude V d is applied to displace the memory to a coherent state |β⟩.Accounting for the residual thermal occupation of the memory mode, the mean number of photons is n = β 2 + n th .The dispersively coupled qubit is then prepared in an equal superposition of ground and excited state by applying an unconditional π/2 pulse.After a varying time t, the superposition accumulates a phase χ m † mt that depends on the memory photon number m † m.A second unconditional ±π/2 pulse is then applied on the qubit, which is then measured to give 2 average signals S ± .The difference between these 2 signals then evolves as [59] S + − S − = cos (n sin (χt)) e n(cos(χt)−1)−t/T2 . (F1) From this measurement (Fig. 15a), we can extract the cross-Kerr coupling rate χ/2π = 170kHz between mem-ory and transmon qubit.We also obtain the thermal population n th = 0.011±0.002and µ = 31.33±0.85V −1 .

Thermal state tomography
Another calibration method is to perform a Wigner tomography of the memory thermal state, using the independently measured average occupation n th = 0.01.The density matrix can be written as a Boltzmann distribution ρth = n n n th (1+n th ) n+1 |n⟩ ⟨n| and the Wigner function associated where Therefore, we obtain e −2|β| 2 /(1+2n th ) .
(F3) Using a conversion factor µ = 31.31± 0.14 V −1 rescales the displacement amplitudes from voltages V d into complex amplitudes β for the measured Wigner function in (Fig. 15b), in such a way that the standard deviation σ, of this Gaussian distribution is σ = √ 1 + 2n th /2, with n th = 0.01.

Measurement of cat states fringes
Our last method to calibrate the conversion factor is based on the Wigner tomography of a cat state [60].The particular features of the cat Wigner function allow to directly estimate µ, assuming the distortion due to memory self-Kerr or thermal population is negligible.The Wigner function of an even cat state of size α, |C + α ⟩, can be written as + 2 cos (4Im (α * β)) e −2|β| 2 .

(F4)
Introducing δV α and δV I the drive voltages corresponding to respectively a displacement of 2α (distance between the two Gaussian distributions in Fig. 15c) and the periodicity of the fringes π/2α (seen in Fig. 15d), this yields The values of δV α and δV I are measured via cuts of the Wigner function in the direction and orthogonal to the cat state (Fig. 15c,d).The conversion factor obtained via this method is µ = 31.41±0.04V −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 [60].

Calibration
To conclude this section, the three methods are compatible with µ = 31.4± 0.1 V −1 .We use this value of µ = 31.4V −1 for the whole article.
Appendix G: Measurement of κ2 and κ1

Determination of κ2 using engineered relaxation of cat qubits
In order to measure the rate κ 2 , we first prepare |C + α ⟩ or |C − α ⟩ by driving the buffer with a drive ϵ d (α) at ϕ QEC .Turning off ϵ d (α) while remaining at ϕ QEC then ensures the memory loses pairs of photons to the environment at the rate κ 2 .|C ± α ⟩ then converges to a state in the manifold {|0⟩ , |1⟩} with the same parity as the initial state.An example of such an evolution starting from |C + α ⟩, α = 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 κ 2 from these dynamics, the initial density matrix describing the memory is approximated by α 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.However, the obtained description of the initial density matrix is 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 (G1), the evolution of the memory state is then simulated using the Hamiltonian and loss operators The single photon loss rate κ 1 /2π ∼ 14 kHz is extracted from the decay of the single photon state |1⟩ → |0⟩ (see section G 2).Using the memory self-Kerr rate of 2π × 220 kHz measured at ϕ tomo , we use its predicted flux dependence in Eq. (B11) to estimate that the self-Kerr rate at ϕ QEC is χ m,m /2π ∼ 206 kHz.Minimizing the difference between measured and simulated Wigner functions at all times t, then allows us to fit the value of κ 2 that best reproduces the memory dynamics.The uncertainty shown in Fig. 2a is then calculated using the result of the minimization method ∆κ 2 ≤ √ tol × 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 not being verified for α ≳ 1, we observe a deviation between the 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. 16).In practice, we use the same model as in Sec.H 2 apart from the detunings ∆ m = 1 MHz and ∆ b = 0. However this bipartite evolution does not provide an effective value of κ 2 acting on the memory mode, hence the benefit to stick with the simpler model.
We have tried other methods to estimate κ 2 , in particular by extracting m † m from the measured Wigner function and compare it with the theoretical expression given in [61].However, reconstructing 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) [62] in order to circumvent this issue.The measurement of κ 1 is done by observing the decay from Fock state |1⟩ to the vacuum.If we prepare the state |C − α ⟩ in the memory, and let it evolve under the action of two-photon dissipation at a rate κ 2 ≫ κ 1 (Fig. 17a), the parity of the memory is preserved so that the memory state ends up in the subspace generated by {|0⟩ , |1⟩} with the same parity as |C − α ⟩: that is the Fock state |1⟩.All that is necessary to measure κ 1 is then to monitor the memory parity πW (0)/2 as it evolves towards 1, here corresponding to the vacuum.
In order to prepare |C − α ⟩, we prepare |C + α ⟩ 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 |1⟩ to |0⟩ (Fig. 17b).
The evolution of W (0) in (Fig. 17b) is fitted by an exponential relaxation at a rate κ 1 /2π = 14 kHz.Repeating this measurement over the course of months, we found that it is not stable.The rate κ 1 /2π typically varies by ±2 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. 18a).This is performed by driving the memory with a drive at frequency ω m , effectively implementing a displacement whose Hamiltonian reads Ĥz /ℏ = −iϵ Z e iθz m + h.c..
Here, θ z and ϵ Z are the drive phase and amplitude.Applying this drive while simultaneously driving the buffer at ϕ QEC (Fig. 18b) implements a quantum Zeno dynamics of the cavity which is restricted to states in the cat qubit subspace.The effect of Ĥz on the Wigner function of the memory then simply consists in shifting the phase of the interference fringes as seen in Fig. 18c, effectively inducing the desired dynamics on the logical qubit.

Calibration of the drive phase
The first calibration needed for this scheme is to set the value of θ z to π.This maximizes the gate speed for a given drive amplitude ϵ 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 Ω z .Note that measuring W (0) alone would leave the sign of Ω z undetermined, which is why we measure a vertical cut of the Wigner function (Fig. 19a).
Doing this measurement for different value of θ z , (Fig. 19b) shows an evolution Ω z (θ z ) ∝ cos (θ z ).This is expected as only the vertical component of the drive Re ϵ z e iθz effectively displaces the fringes of the cat Wigner function (Fig. 4c).The horizontal component Im ϵ z e iθz is disabled by the 2 photon dissipation and does not affect the system, which can be seen as a cancellation of Ω z for θ z = ±π/2.

Comparison of κZ with the theoretical model
The decay rate κ Z of the oscillations around Z (shown in Fig. 4) is obtained by simulating the master equation where is the Lindblad superoperator.The last four 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 where ∆ b = ω b − ω d is the detuning of the drive with respect to the buffer frequency, and Here, the Gaussian drive envelope reads where the time window of the pulse is set to w = T /6, with T being the total time of the pulse, and εZ corresponding to the average drive amplitude.The case of a square pulse is easily extended by choosing ϵ Z (t) = εZ over the same time window.As for the experiment, we fit the decaying oscillations of the photon number parity to extract the rotation frequencies Ω Z and decay rate κ Z corresponding to each drive amplitude ϵ Z .
We numerically observe that the value of κ Z is mostly sensitive to the loss rate κ 1 , the ratio 4g /2π ≈ 6.5 MHz.Interestingly, the decay rate κ Z strongly depends on g 2 and we use it to extract this parameter experimentally.For perfect frequency matching 2ω m = ω b , the best fit to the simulation is obtained for g 2 /2π = 6MHz (solid line in Fig. 20).
To illustrate the sensitivity of the simulations to the value of g 2 , we also compute κ Z (α) for g 2 /2π = 5.5MHz and g 2 /2π = 6.5MHz.The clear deviations in Fig. 20a show that under the assumption that 2ω m = ω b , g 2 can be determined with a much better precision than 2π × 0.5 MHz from the measured decay rates κ Z .
The uncertainty on g 2 is actually dominated by the values it can take over the range of conceivable detunings ∆ m .In order to get a higher bound on this uncertainty, we choose three values for the resonance condition: (2ω m −ω b )/2π = −5, 0, 5 MHz, and search for the rate g 2 that best reproduces the experiment.The fitted g 2 rates are between 2π × 6.0 MHz, and 2π × 6.5 MHz (Fig. 20b).We therefore claim that g 2 /2π = 6 ± 0.5MHz.
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 /2π = 6 MHz which best fits the gate rates and decay rates for the detuning (2ω m − ω b )/2π = 3.5 MHz.

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 [63], meaning they do not convert phaseflip errors into bit-flip errors.
In order to verify the bias-preserving nature of the Z gate, we measure the dependence of T X when continuously driving the memory with a varying drive amplitude ϵ Z .Similarly to the measurement presented in Fig. 3, the flux is first set to ϕ tomo and the memory displaced by D (α) in order to prepare the desired state |α⟩.The flux is then changed to ϕ QEC and 2 drives are sent to the buffer and memory modes, with amplitudes of ϵ d (α) and ϵ 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 (±α) is then finally measured for various waiting times, and their difference W (α) − W (−α) ∝ e −t/TX is fitted to extract T X (Fig. 21a).The measured dependence of T X on εZ is shown in Fig. 21b for different amplitudes α.Despite a rather large measurement uncertainty, no visible decrease of T X can be observed for |ε Z /2π| < 6 (MHz), after which the displacement becomes strong enough to overcome the stabilization provided by the two-photon dissipation.This induces increased number of bit-flip errors, leading to a decrease of T X .
The Z rotation presented in Fig. 4a is measured with εZ /2π = 1.625It can thus be assumed the drive did not induce additional bit-flip errors.Comparing the measured T X ∼ 10 ms for α 2 = 9.3 with the Z (π) 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 96.5% thus primarily originates from phase-flip errors.

Fidelity of the Z gate
To evaluate the fidelity of the Z gate, we use the evolution of W (β) shown in Fig. 4c and directly estimate the evolution of the logical ⟨σ x,L ⟩, ⟨σ y,L ⟩ and ⟨σ z,L ⟩ from the Wigner functions.These operators are defined in the cat encoding as . The measured evolution of the three logical Bloch vector coordinates is shown in Fig. 22a.Interestingly, the evolution of ⟨σ z,L ⟩ (t) during the gate shows no visible evolution (Fig. 22b), which is expected from its bias preserving property Fig. 21b.Using the formalism of the standard process matrix χ [64], the impact of the Z gate on an initial density matrix ρ is modeled as where the fixed set of operators is chosen as Êm m ∈ {1, σx,L , σy,L , σz,L }.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. (H2) and only consider the simpler error model that provides a lower bound on the fidelity by neglecting the χ z,1 and χ 1,z terms The parameter χ z,z = (1−ε) then corresponds to the gate fidelity owing to the definition F = Tr (χ χ opt ).Indeed, the matrix χ opt is the χ matrix describing an ideal Z gate, with all its terms null except for χ z,z = 1.Using the general form of a density matrix describing a qubit from its Bloch vector (x y z) the density matrix after application of the gate reads The parameters ε can then be estimated with the evolution of ⟨σ x,L ⟩ (t) and ⟨σ y,L ⟩ (t).We first check that both the evolution of ⟨σ x,L ⟩ = W (0)π/2 and ⟨σ y,L ⟩ matches the fit of W (0) used in Fig. 4b and Fig. 4c, up to a scaling factor and a dephasing.Finally, we compute F = 1/2 + e −πκz/Ωz /2 = 95 ± 2% for the 26 ns long Gaussian pulse with |α| 2 = 9.3 and for a drive amplitude εZ /2π = 1.625 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 ± 2% in 28 ns.Note that the infidelity in the preparation of |C + α ⟩ is due to a preparation time of 500 ns starting from |0⟩, similar to the phase-flip time of the cat qubit 1/Γ Z ≈ 500 ns for |α| 2 = 9.3.This time should be optimized in future measurements.Figure 3 shows the measured T x and Γ Z for various cat sizes α.An example of a phase-flip rate and bit-flip time measurement is shown in (Fig. 24) for α ≈ 2.6.
The phase-flip rate corresponds to the loss of coherence of the cat qubit, through which any superposition of |±α⟩ decays to a mixture of these 2 coherent states.In order to probe it, a cat state |C + α ⟩ is first prepared in the memory by applying a drive ϵ d (α) at ϕ QEC , starting from an empty cavity.The cat qubit decoherence towards (|α⟩ ⟨α| + |−α⟩ ⟨−α|) /2 is then monitored by simply measuring W (0). Fitting this evolution with an exponential decay at a rate Γ Z gives the value of the phase-flip rate (Fig. 23a).
The bit-flip time T X characterizes the typical time it takes for the populations in |α⟩ and |−α⟩ to equilibrate.We measure this value for various amplitudes α by first displacing the memory at ϕ tomo to prepare |α⟩, before applying a drive ϵ d (α) at ϕ QE .The state |α⟩ is then protected by the Zeno dynamics.We monitor the values of W (α) and W (−α) over time, and fit W (α) − W (−α) with an exponential decay at a rate 1/T X which gives the value of the bit-flip time (Fig. 23b).Solid line: fit with an exponential decay of the measured data, the bit-flip time is deduced from the fit of W (α) − W (−α). Insert: associated pulse sequence.

Dependence of TX and ΓZ on photon number for various buffer drive frequencies
The cat stabilization works by driving the buffer on resonance at ϕ QEC .What happens if we drive it off resonant?
Figure 24a shows the measured W (0) as a function of flux and drive frequency ω d after 5µs of stabilization.The red regions where W (0) ≈ 2/π correspond to a memory unaffected by the drive so that it is in the vacuum state.In contrast, a white region where W (0) ≪ 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 an autoparametric version of that between ω b (ϕ ext ) and 2ω m (ϕ ext ).
The dependence of T x and Γ Z on mean photon number α 2 is measured for 4 buffer drive frequencies at ϕ QEC (dots in Fig. 24a).A similar behavior for T x α 2 can be seen for the 4 different drive frequencies, with an initial exponential increase before reaching a maximum for some optimal cat size (Fig. 24b).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 X .
Note that the 1D-cuts of the Wigner functions W (β) from which are extracted the bit-flip times (not shown here), exhibit a broadening of the Gaussian distribution around ±α with increasing α 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 Γ Z α 2 is measured for the same four drive frequencies (Fig. 24c).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 Γ Z = 2 |α| 2 /T 1,eff , with T 1,eff the effec-tive memory lifetime.This effective lifetime matches the memory lifetime measured by looking at the decay from |1⟩ to |0⟩ at the optimal drive frequency (see section G 2).The effective memory lifetime drastically deteriorates as the drive frequency detuning increases.The memory pure dephasing rate κ m φ , used for numerical simulations of the bit-flip time evolution, is measured with a Ramsey-like experiment.A state close to (|0⟩ + |1⟩) /2 is prepared in the memory by first displacing the memory to the coherent state |α⟩, with α = 2.1, and letting it decay under the loss operator L2 = √ κ 2 m2 .By Zeno effect, the 2-photon loss constraints the memory dynamics to the {|0⟩ , |1⟩} 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 ∆ m between the memory drive frequency and ω m,QEC , in the frame of the drive frequency.The memory state evolves under the Hamiltonian and loss operator Note that the dephasing operator on the memory when Zeno blockade is disabled reads κ m φ m † m.The correspondence is thus κ m φ = 2Γ φ .For readout, the obtained state is then mapped into the {|α⟩ , |−α⟩} manifold by driving the buffer mode with a drive ϵ d (α), mapping the eigenvectors of (|1⟩ ⟨0| + |0⟩ ⟨1|) to the 2 coherent states.The Wigner function W (±α) (Fig. 25) is then measured, and the data fitted to extract the detuning ∆ m = 3 MHz, and dephasing rate of the memory κ m φ /2π ≈ 0.16MHz.

Population of the higher excited states of the transmon
The transmon used for the Wigner tomography, dispersively coupled to the memory mode, has been shown to be one of the main factors limiting the bit-flip time T X at large photon numbers.In this experiment, the 2 photon dissipation rate κ 2 is much greater than the dispersive shift κ 2 ≫ χ.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].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 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 |0⟩ , |1⟩ , |2⟩ , |3⟩, and |4⟩ from the others, allowing to measure the transmon population for each amplitude α of the cat qubit (Fig. 26b).
As can be seen in Fig. 26c, the transmon populations in states of higher energy than |3⟩ get excited for |α| 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 |3⟩ to a resonance between the memory and the higher levels of the transmon.The negative anharmonicity ω 12 −ω 01 = −2π ×181 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 below that of the transmon.This phenomenon, which also happens with a simple Duffing oscillator, was recently investigated theoretically in Ref. [65] and experimentally in Ref. [66].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 [65,67], 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. 26c.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 X .

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 neglect the Purcell filter of the readout resonator.The Hamiltonian of the system we consider reads where nt and θt are the transmon charge and phase operators, n g is the offset charge, ĉ is the annihilation operator of the readout resonator, ω c the frequency of the readout resonator, g mt /2π and g ct /2π 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 = 169.4MHz and Josephson energy E J /h = 22.85 GHz, as well as the values of the coupling rates g ct /2π = 67 MHz and g mt /2π = 225 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 memory and transmon, we included the readout resonator to correctly fit the spectrum of the system.
By diagonalizing the Hamiltonian in Eq. (I2), 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. 27).In the inset, one recognizes the dispersive shift χ/2π = 0.170 MHz when the transmon is in its first excited state |1⟩, 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 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 change slightly.Indeed, using a semiclassical analysis (Eq.(S22) of [23]), in the limit where that is the minimal condition for a 2D-manifold to be stabilized.The impact on the bit-flip time of the detuning ∆ m induced by the transmon higher excitation is illustrated in Fig. 28, 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 ∆ m /2π = 2.75 MHz.
The detuning associated with the first and second excited states (below 20 photons) is less than 1 MHz, mak- ing the condition |∆ m | < 3.6|α 0 | 2 MHz largely satisfied.This is numerically verified in Fig. 28.In this case, the bit flip rate becomes T −1 The small detunings associated with the first and second excited states make the weighted contribution p i T −1 X (∆ i ), i = 1, 2 negligible compared to p 0 T −1 X (∆ 0 ).However, detunings as large as 30 MHz become difficult for the dissipation scheme to compensate.For |∆ m | = 30 MHz, the minimal condition to generate a cat state in the cavity is |α 0 | 2 > 8.5 photons.Moreover, even if the memory becomes populated, this large detuning comes with a large displacement on the buffer mode, β ∼ ∆ 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. 28, where the bit-flip time corresponding to a detuning of 9 MHz does not improve over the bare memory lifetime until |α 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 X from the measured values of the state population: it is given by the inverse of the rate at which this layer of states is populated γ hyb .Let us call p hyb the population of the hybridized states, represented as a black dotted line in Fig. 26, and p 1 the population of the first excited state.Note that the state |2⟩ merges with the rising plateau of states around |α| 2 = 20 photons.The rate at which the hybridized layer gets populated thus reads γ hyb = γ hyb→1 p hyb .Using the measured γ −1 1→0 = 18 µs and assuming γ hyb→1 ≈ γ 2→1 = 2γ 1→0 , we obtain the red dotted line in Fig. 3b.

Main limitation of the bit-flip time
Before studying other possible limiting factors on the bit-flip time, let us first briefly review the experimental data that cannot be explained solely by the heated transmon.As a reminder, the transmon higher excited state population may have been a limitation for bit-flip times above the red dots in Fig. 3d.However, measurements taken at various drive detunings reveal earlier saturation of the bit-flip time, almost two orders of magnitude lower than 0.3 s (see Fig. 24b), which cannot be explained by the presence of a transmon.
In this section, we show that the self-Kerr effect of the memory steers the system away from resonance as the photon number increases, resulting in a smaller bit-flip time, which corroborates the dependence of the bit-flip time on the drive detuning.
Furthermore, we discuss possible mechanisms that could limit the bit-flip time even when the drive is on resonance (∆ m = 0).

a. Impact of memory self-Kerr on bit-flip time TX
The protection against bit-flip is diminished away from resonance.The memory self-Kerr results in an effective detuning on the memory as the photon number increases.Each cat size corresponds to an optimal drive detuning for which the resulting effective detuning on the memory cancels.This is illustrated in Fig. 29a, which shows the bit-flip times extracted from the simulation of Eq. (H1), at the values of the drive detunings ∆ b corresponding to the experimental values of Fig. 24b and beyond.We observe a qualitatively similar behavior between simulations and measurement of the bit-flip times, in particular for the curves corresponding to larger detunings.
Note that the self-Kerr rate of the buffer mode (not measured) is omitted here, as otherwise we observe that the required Hilbert space dimension for accurate enough simulations becomes prohibitively large for even moderate |α|.
An effective detuning results in a displacement of the buffer mode, and computing this displacement can be used to estimate the effective detuning.In the interaction picture, the master equation on the memory mode Eq. (H1) reads Assuming a steady-state solution of the form ρ = |α⟩ ⟨α| ⊗ |λ⟩ ⟨λ|, with α ̸ = 0, and taking the trace of the above equation, we obtain where θ m = arg(α).
In the limit κ The displacement amplitude on the buffer depends on the effective memory detuning, which in turn depends linearly on the cat photon number due to the self-Kerr effect.In Fig. 29b are shown the amplitudes of the buffer mode corresponding to the curves of Fig. 29a.For a given cat size, the drive detuning giving the optimal bit-flip time corresponds approximately to the smallest buffer amplitude and therefore to the smallest effective detuning.
b. Impact of memory dephasing on bit-flip time TX We expect pure dephasing on the memory (T m φ = 1/κ m φ = 1 µs, see Fig. 25) to impact the bit-flip time T X .In the adiabatic regime where 8|α|g 2 ≪ κ b , the bit-flip time scales as T m φ |α| −2 exp 2|α| 2 [1].However, in our case, the adiabaticity criteria are not met for |α| ≳ 1.As illustrated in Fig. 30, the memory dephasing noise limits the scaling of T X with the photon number but does not lead to a saturation (dashed lines).
Moreover, if one assumes the dephasing rates of the memory and buffer modes are limited by flux noise, we can estimate the buffer dephasing rate as Using the measured value κ m φ /2π = 0.16 MHz, we estimate κ b φ /2π = 9.6 MHz.When taking this large dephasing rate into account, the bit-flip time is predicted to saturate even without the transmon (solid lines in Fig. 30).During the stabilization, the driven buffer ideally stays in the vacuum state.However, the coupling to the environment may lead to the thermalization of the buffer and memory bipartite system, which induces bit-flip errors.Indeed, • If the buffer state becomes thermal, buffer photons are converted to memory photons by the twophoton exchange term, thus creating a heating term of the form m †2 on the memory, thus affecting the bit-flip time T X [27].
• The cross-Kerr effect (χ m,b ) between memory and buffer modes leads to an effective dephasing rate of the memory.For instance, we estimate that with 0.2 thermal photons in the buffer, the memory dephasing rate increases by 0.5 MHz [68].
We envision a few origins for the thermalization.
• When the buffer is displaced to a finite amplitude λ = ⟨ b⟩ ss ̸ = 0 due to drive detuning (see Fig. 29b), dephasing noise on the buffer can be upconverted to thermal photons via the drive.Indeed, dephasing noise can be seen as small random rotations in phase space resulting in a small diffusion of a coherent state along the circle of radius |⟨ b⟩ ss |.This diffusion competes with single photon loss of the buffer, resulting in an effective temperature of where κ b φ is the dephasing rate of the buffer.We find n b th = 0.24|λ| 2 .In Fig. 29c, we show the marginals of the Wigner functions of the steady state in the memory as a function of |α| 2 .The memory heating can be seen as a broadening of the Gaussian distribution.
• The large expected self-Kerr rate of the buffer χ bb /2π ≈ 10 MHz results in a small squeezing on the buffer when it is displaced.The single photon loss channel on a squeezed buffer yields an additional effective thermal occupation given by n b th = sinh 2 (r), where r is the effective squeezing parameter.
a coupler [17,18] since no exponential scaling could be demonstrated owing to parasitic Hamiltonian terms.
The saturation due to the transmon higher excited states that we observe in T X could be canceled by using the technique introduced in Ref. [39] on an ATS where the readout transmon is removed.Finally, the main remaining caveat seems to be the buffer thermalization that limits the bit-flip scaling time in the autoparametric cat.We note that detuning the buffer drive by the right amount for a given cat qubit size should cancel out this effect (see Fig. 29).

Appendix K: CNOT gate
In order to correct the remaining phase flip errors of the autoparametric cat, one can use a chain of coupled autoparametric cats in order to perform phase flip error correction using a 1D repetition code.Such error correction code needs a two-qubit gate between cat qubits.Here we propose to use a CNOT gate and explain how it would operate.
A CNOT gate can be performed between two autoparametric cats (named control and target) as long as they are coupled through a four-wave mixing element (Josephson junction, SQUID, ATS, quarton [71],. . ., see Sec.K 1).A scheme for a repetition code coupling autoparametric cats with 4-wave mixers is shown in Fig. 31.where α c and α t are the control and coherent state amplitudes of the target cat qubit.Thus the CNOT gate can be viewed as a π rotation of the target coherent state conditioned on the control state.However, an interesting feature of bosonic code, key to implement bias preserving gates, is the ability to redefine the logical basis so that the computational basis before and after a gate can differ [26,72] where âc and ât are annihilation operators of the target and control memory and ϕ c = arg(α c ).

CNOT Hamiltonian engineering
The Hamiltonian we propose for the CNOT gate is generated using a four-wave mixing interaction coming from a non-linear coupler.One can write this interaction as Ĥ =ℏg 4 (ξ(t) + ϕ c (â † c + âc ) + ϕ t (â † t + ât )) 4 , (K4) where g 4 is the four-wave mixing interaction strength, ϕ c (respectively ϕ t ) the phase zero point fluctuations of the control (respectively target) memory across the nonlinear coupler, âc (respectively ât ) the annihilation operator of the control (respectively target) memory and ξ(t) the amplitude of an RF drive on the non-linear coupler.Using a drive amplitude ξ(t) = ξ 0 cos(ω c t) at the frequency of the control memory ω c and in the frame rotating at the control memory frequency ω c , the Hamiltonian (K4 ) under rotating wave approximation leads to the interaction which can be written as: ĤCNOT = ℏg CNOT (e i arg(ξ0) â † c + e −i arg(ξ0) âc )â † t ât , (K6) with g CNOT = 12g 4 ϕ c ϕ 2 t |ξ 0 |.To get the Hamiltonian of Eq. (K3), one has to tune the phase of ξ 0 such that arg(ξ 0 ) = arg(α c ).We note that the larger two-photon coupling rate g 2 of the autoparametric cat would permit to increase the drive amplitude ξ 0 without affecting the stabilization of the control cat qubit.Ultimately, we expect larger gate fidelities and lower gate times owing to the autoparametric cats.

CNOT sequence
The CNOT gate described by Eqs.(K1) and (K2) can be performed with the following sequence 1.Turn off the stabilization of the target cat.One can do it by turning off the buffer drive and moving away from the ϕ QEC flux point.
3. Turn back on the stabilization of the target with a buffer drive phase shifted by π such that the stabilized sub-space is {|iα t ⟩ , |−iα t ⟩}.The stabilization has to be kept during a time larger than 1/|α t | 2 κ 2,t , where κ 2,t is the target two-photon dissipation rate, such that the target memory state is projected onto the cat qubit manifold {|iα t ⟩ , |−iα t ⟩}.

FIG. 2 .
FIG. 2. (a) Dots: Measured two-photon relaxation rate κ2 as a function of flux bias close to ϕQEC.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 |α| = 2.5 and at the flux indicated by the star in (a).Bottom: results of the simulation using κ2/2π = 2.16 MHz.The buffer mode is adiabatically eliminated in these simulations to provide an effective value of κ2 (see Sec. G).

FIG. 3 .
FIG. 3. (a) Pulse sequence of the phase flip rate measurement.(b) Dots: measured phase flip rate ΓZ (linear scale) of the cat code as a function of photon number |α| 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|α| 2 κ1.(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 |α| 2 .All values are obtained by fitting the difference W (α) − W (−α) to an exponential decay in time.Error bars represent statistical uncertainties (see Sec.I 1).Solid black line: expected bit-flip time with κ m φ /2π = 0.16 MHz under the adiabatic elimination of the buffer: e 2|α| 2 /(|α| 2 κ m φ ).Dashed blue line: simulated bit-flip time with the same κ m φ , assuming a detuning 2ωm − ω b = 2π × 3.5 MHz.Red dots: bound below which TX is not limited by excitation of higher states of the transmon (Fig. 26).

FIG. 4 .
FIG. 4. (a) Pulse sequence used to perform a Z rotation in the cat qubit {|α⟩, | − α⟩} basis.(b) Dots: Measured oscillations of W (0) as a function of time t using the pulse sequence of Fig.3awhere the cat code is stabilized with a photon number |α| 2 = 9.3.An additional displacement drive at ωm starts 240 ns after the buffer drive is turned on.Here, its amplitude ϵZ (t) is Gaussian shaped with a mean amplitude εZ /2π = 1.625 MHz and its phase is chosen to displace in the direction indicated by the arrow in c).Line: fit to oscillations at a frequency ΩZ /2π = 19.8MHz, which are decaying at a rate κZ /2π = 0.62 MHz.(c) Measured Wigner functions W (β) after a Z rotation of angle θ = 2π, 3π/2, and π from top to bottom.(d) Dots: Inferred rotation frequency ΩZ as a function of cat code amplitude α, and for various mean drive amplitudes εZ /2π = 0.32, 0.625, 0.965, 1.295, 1.625, 1.955, 2.285, and 2.66 MHz from bright to dark orange.Lines: expected rotation frequency ΩZ = 4Re(εZ α) around Z. (e) Dots: Inferred decay rate κZ as a function of |α| for the same drive amplitudes.Lines: simulated decay rate with g2/2π = 6 MHz as a fit parameter and the same detuning as in Fig.3d.

FIG. 5 .
FIG. 5. Simplified circuit diagram of the device.Only the modes of the ring of junctions are considered for simplicity.For each junction φx = φx + φx, respectively the oscillating part and the equilibrium part.
colored lines).In (b) it crosses φW where φJ has a sweet spot at the same values of φ (sweet) ext (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 φext (vertical green line) another configuration is favored (green closed circle) which is not a flux sweet spot.φJ = 0.42 and φW = 1.11 leading to ĒJ /h = 228 GHz and ĒW /h = 51 GHz.At ϕ tomo = 0.168ϕ 0 , we find φJ = 0.25 and φW = 0.56 leading to ĒJ /h = 242 GHz and ĒW /h = 97 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 φ 0 φ

FIG. 7 .
FIG. 7. Potential energy U (φm, φ b ) /EW for EW /h = 115 GHz and EJ /h = 250 GHz at six different flux biases.For each flux bias, an offset Umin(φext) is subtracted to Eq. (B10) in order to set the potential global minimum to 0 and better highlight the height of the potential barrier.
FIG.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.Inset: larger view of the coupler and its connections to the stubs.

FIG. 10 .
FIG.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 µm and -175 µm.

2 .
FIG.11.Equilavent electronic schemes for the inductive coupling between the transmon mode and the memory mode.
MHz, ω IF,r /2π = 75 MHz, ω IF,m,tomo /2π = 40 MHz or ω IF,m,QEC = (ω m,tomo + ω IF,m,tomo ) − ω b,QEC /2, and ω IF,b,QEC = 2ω IF,m,QEC .ω IF,m,tomo and ω IF,m,QEC are the modulation frequencies used to respectively drive the memory at ϕ tomo or ϕ QEC , ω m,tomo and ω b,QEC are the frequencies of the memory mode at ϕ tomo and the buffer mode at ϕ QEC .The above condition on ω IF,m,QEC and ω IF,b ensures the phase stability of the encoded cat in the frame rotating at the memory frequency.

Appendix E: Wigner measurement of cat states 1 .
The use of a fast flux lineAs explained in the main text, the preparation of the cat state |C α + ⟩ ∝ |α⟩ + | − α⟩ is as simple as starting from the memory vacuum state at ϕ QEC and turning on a drive with the right amplitude |ϵ d | = α 2 g 2 at twice the memory frequency ω d = 2ω m,QEC .This drive, resonant with the buffer mode, injects photons with energy ℏω b,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

FIG. 13 .
FIG. 13.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.

FIG. 15 .
FIG. 15.(a) Ramsey interferometry.Dots: measured signal S+ − S− between 2 Ramsey-like experiments for various voltages V d = 0, 10, 20, 30, 40, mV from top to bottom.Lines: Fit of the measurements to Eq. (F1) leading to a photon number n = 0.01, 0.10, 0.35, 0.81, 1.45.The residual thermal population is thus n th = 0.01.(b) Measured Wigner function of the memory in thermal equilibrium with its environment.The conversion used between V d and |β| to plot it is made by a Gaussian fit of the measurement with the Wigner function of a thermal state with n th = 0.01 photons on average.(c) Dots: Cuts of the Wigner tomography of a stabilized cat qubit along β ∈ R after 100 µs of dephasing.Line: Theoretical prediction.(d) Same plot along β ∈ iR, 500 ns after the buffer drive is turned on.

FIG. 16 .
FIG. 16.Evolution of the Wigner functions of the memory starting close to a cat state C + α , under the effect of twophoton dissipation at ϕQEC, 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 g2/2π = 6 MHz (Fig. 3e)

FIG. 17 .
FIG. 17.(a) Measured Wigner functions of the memory starting close to C − α 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 C − α .Note the much longer timescale for this single photon decay compared to the sub µs time needed to prepare Fock state |1⟩ in (a).Dashed line: Fit of the exponential relaxation to vacuum.

FIG. 19
FIG. 19.(a) Measured Wigner function W (β) of the memory as a function of β ∈ iR and time t.The displacement drive parameters are εZ /2π = 1.25 MHz and θz = π/2 (b) Measured oscillation frequency Ωz around the z axis of the Bloch sphere as a function of θz.

FIG. 20 .
FIG. 20. (a)Triangles: measured decay rate κZ of the oscillations around Z as a function of |α| 2 for four drive amplitudes ϵZ corresponding to distinct colors as in Fig.4d.Lines: simulated decay rates κZ using Eq.(H1) with three values of the rate g2/2π indicated as an inset.(b) Triangles: same measurement as above.Lines: simulated κZ for three values of the detuning between the buffer and the memory (2ωm − ω b )/2π = −5, 0, 5 MHz covering its the uncertainty range, and the corresponding optimal values of g2/2π = 6.5, 6.0, 6.0 MHz.

FIG. 21 .
FIG. 21.(a) Pulse sequence for the TX measurement, while continuously applying the Z gate on the memory with an average drive amplitude εZ .(b) Dots: Measured TX as a function of the average memory drive amplitude εZ for different cat qubit sizes α = 2, 2.4, and 2.7 (yellow to brown).Lines: measured TX at εZ = 0.

FIG. 22 .
FIG. 22. (a) Trajectory of the cat qubit during the Z gate estimated from the Wigner functions of Fig. 4c.(b) Dots: Mean value of σz,L as a function of the gate time.Line: Linear fit of ⟨σz,L⟩ (t).(c) Dots: Mean value of σx,L as a function of the gate time.Line: fit used in Fig. 4b to oscillations at a frequency ΩZ /2π = 19.8MHz, decaying at a rate κZ /2π = 0.62 MHz.A scaling factor of π/2 is applied to respect the relation W (0) = 2 ⟨σx,L⟩ /π.(d) Dots: Mean value of σy,L as a function of the gate time.Line: fit used in (c) dephased by π/2.
Appendix I: Complementary data and analysis about bit-flip and phase-flip rates 1.Measurement of TX, ΓZ

FIG. 23 .
FIG. 23.Determination of Tx and ΓZ for α 2 ≈ 6.5.(a) Dots: measured W (0) as a function of cat stabilization time t.Solid line: fit with an exponential decay e −Γzt .Insert: associated pulse sequence.(b) Dots: measured W (−α), W (α) and W (α) − W (−α) 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 (α) − W (−α). Insert: associated pulse sequence.

FIG. 24 .
FIG. 24.(a) Anticrossing of the autoparametric memorybuffer system.Measured W (0) after applying a buffer drive at ω d for 5µs while at ϕext.(b) Dots: measured Tx as a function of α 2 for 4 different drive frequencies.The color matches with the dots shown in (a).(c) Dots: measured ΓZ as a function of α 2 for 4 different drive frequencies.Dashed line: Linear fit of the measured data with ΓZ = 2 |α| 2 /T 1,eff .where the fit parameter T 1,eff is shown as an inset.

FIG. 26 .
FIG. 26.(a) Histogram of 10 7 measurements of the readout quadratures when the transmon and memory are in equilibrium with the cold environment.(b) Same histogram when a cat qubit space is stabilized for 100 µs with a mean photon number |α| 2 ≈ 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.dotted line: sum of the populations in states higher than |3⟩.

FIG. 27 .
FIG.27.Dots: computed memory frequency shift as a function of mean photon number n in the memory for various transmon states.The vertical lines are due to state mistrack-Inset: zoom on the small dispersive shifts for low transmon excitation.

FIG. 28 .
FIG. 28.Effect of memory detuning ∆m on the bit-flip time TX .The bit-flip time is plotted as a function of the drive amplitude for several values of ∆m.The drive amplitude is converted to the corresponding photon number at ∆m = 2.75 MHz, which approximately corresponds to the detunings set for Fig. 3b.The simulations are done under the assumption that κ m φ /2π = 0.16 MHz, κ b φ = 60 × κ m φ , χm,m/2π = 0.22 MHz, χ m,b /2π = 1.6 MHz and without any self-Kerr on the buffer mode.The crosses indicate the experimentally measured bit-flip times shown in Fig. 3d.

FIG. 29 .
FIG. 29.(a) Simulated bit-flip times for various values of the drive detuning ∆ b = ω b − ω d (color code as an inset in panel b), among which the values of Fig. 24b.The memory detuning from the buffer is 2ωm − ω b = 2π × 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 circles).(b) Corresponding amplitudes of the buffer mode in the steady state as a function of average photon number |α| 2 in the memory for the same detunings.(c) Simulated probability distributions of the quadrature defined by the direction of α in the memory phase space P(Re(β)) = |⟨Re(β)|ψ⟩| 2 , plotted for several values of |α| 2 (arbitrary units and offset for each α).The detuning is set to ∆ b /2π = −4 MHz.As the photon number increases, the distribution broadens.All the simulations are done under the assumption that κ m φ /2π = 0.16 MHz, κ b φ = 60 × κ m φ , χm,m/2π = 0.22 MHz, χ m,b /2π = 1.6 MHz and without any self-Kerr on the buffer mode.

FIG. 30 .
FIG. 30.Simulated bit-flip times as a function of the memory dephasing rate κ m φ for two values of the buffer dephasing rate, κ b φ /2π = 0 MHz (dashed lines) and κ b φ /2π = 9.6 MHz (solid lines).The simulations are done under the assumption that χm,m/2π = 0.22 MHz, χ m,b /2π = 1.6 MHz and without any self-Kerr on the buffer mode.

FIG. 31 .
FIG.31.Possible layout for a repetition code using the autoparametric cat.The coupling between two neighboring cats is provided by a four-wave mixing element (such as a Josephson junction, SQUID, ATS or Quarton) driven at the frequency of the control cat to perform a CNOT gate.

TABLE I .
State-of-the-art in key figures of the autoparametric cat, cats stabilized by an ATS, and Kerr-cats qubit.* The gate fidelity is here given for a π/2 gate instead of the π gate in the main text in order to better compare with other works.Irrelevant g 2 = E J ϵ 0 φ ZPF,b φ ZPF,m

TABLE II .
Estimated parameters of the device and the associated measurement method.