Asymmetric repetition codes for cat qubits

Abstract

The invention relates to a duplication code for cat qubits, comprising a number d of data cat qubits (4) and at least (d-1) auxiliary cat qubits (6), the number d being greater than or equal to 3, each auxiliary cat qubit (6) being connected to two data cat qubits (4) via two respective CNOT gates (8), no data cat quantum bit (4) is connected to more than two auxiliary cat quantum bits (6), and the two-photon dissipation rate of the auxiliary cat quantum bits is strictly greater than the two-photon dissipation factor of the data cat quantum bit. The repetition code is achieved by performing an error correction loop on each auxiliary cat qubit (6), said error correction loop comprising at least the following steps: a) preparing an auxiliary cat qubit (6) in such a way that it is in a state "+ gt; or "-gt;" ; b) activating one of two CNOT gates (8) connected to the auxiliary cat qubit (6); c) activating another CNOT gate (8) connected to the auxiliary cat qubit (6); and d) measuring the parity of the number of photons of the auxiliary cat qubits (6).

Description

[0001] The invention relates to the fields of quantum computing, bosonic quantum error correction codes and repetition codes.

[0002] Building large quantum computers is challenging because noise generated by uncontrolled interactions between quantum computer components and their environment can destroy the delicate quantum properties that lead to quantum speedup. In fact, all algorithms that have theoretically demonstrated quantum speedup require some level of decoherence protection.

[0003] The theory of fault-tolerant quantum computing solves this problem. Quantum error-correcting codes (QECC or "Quantum errorcorrecting codes") are designed so that errors caused by the environment do not affect quantum information. These codes operate on the principle of "fighting entanglement with entanglement": since natural errors occurring in physical systems are usually local, the quantum information to be protected is encoded in a non-local entangled state, so errors are less likely to destroy it. The most popular QECC is the surface code.

[0004] At the heart of quantum fault tolerance theory is the threshold theorem: arbitrarily long quantum computations can be performed reliably as long as the noise affecting the computer's physical components is below a constant value called the fault tolerance error threshold.

[0005] In theory, QECC can provide very good protection against noise when operating below the error tolerance threshold, thus solving the problem of decoherence. However, their implementation in the physical world comes at the expense of huge physical resources to achieve sufficient protection levels. The trade-off between the degree of protection ensured by QECC and the additional components required to achieve that protection defines the so-called "resource overhead problem".

[0006] Realistic approaches to quantum computing must address this problem. To this end, continuous variable systems (such as the harmonic oscillator), where infinite-dimensional Hilbert space lends itself to protecting and processing quantum information, appear to be ahead of discrete variable systems (DV) with only finite-dimensional Hilbert space. There are many different continuous variable encodings, usually involving superpositions of certain states of the harmonic oscillator, such as position and motion-momentum eigenstates (GKP qubits), Fock states, or coherent states (cat qubits).

[0007] More specifically, the present invention relates to cat qubits.

[0008] It is well known that a pumped (or stabilized) cat qubit benefits from noise biases. More specifically, effective error channels (e.g., bit errors or “bit flip” errors) are exponentially suppressed with the “size” (i.e., the average number of photons) of the state of the Schrödinger cat of the cat qubit. According to current knowledge, this suppression should apply to a large class of physical noise processes that have a local impact on the phase space of the resonator. This includes, but is not limited to, photon losses, thermal excitations, photon phase shifts, and various nonlinearities caused by coupling to Josephson junctions. In the context of quantum superconducting circuits, recent experiments have allowed the observation of such exponential suppression of bit errors with the average number of photons in the cat state.

[0009] Due to this noise structure, quantum error correction becomes of similar complexity to classical error correction and can be performed using repetition codes. In fact, if one only wants to correct phase jumps, one can simply use a phase-jump error correction code. This can include, for example, repetition codes defined in a dual basis or any other conventional error correction code. The space of codes is defined as d-1 stabilizers The common feature space of . The logical operator for the repeating cat qubit is The logic states |+>L and |->L are determined by Given.

[0010] The repetition code of cat qubits is constructed using d cat qubits (called data cat qubits) in which logical information is encoded. The repetition code is realized by repeatedly measuring the stabilizer of the repetition code, that is, detecting errors. This is done using d-1 additional cat qubits (called auxiliary cat qubits). The quantum circuit of the repetition code requires the preparation of the state |+> (Schrödinger cat state) of the cat qubit, the measurement of the Pauli operator X (parity of the number of photons) of the cat qubit, and a CNOT gate between the auxiliary cat qubit and the data cat qubit.

[0011] The difficulty in implementing such a repetitive code lies in how to successfully make the code run below the error tolerance threshold, which means that the fidelity of the quantum operations in the circuit must be high enough. More specifically, when the repetitive code runs above the threshold, that is, when the fidelity of the physical operations that make up the repetitive code is not high enough, the lifetime of the logical information decreases as the number of qubits d of the physical data qubits increases: the new errors introduced by the increased quantum system cannot be compensated by the error correction strategy. However, when the code runs below the error correction threshold, that is, when the fidelity of the physical operations is high enough, the lifetime of the logical information increases exponentially with the number of physical data qubits d (consistent with the distance d of the repetitive code).

[0012] Cat qubits are realized in the laboratory either by dissipative stabilization (via artificial two-photon dissipation at a rate of κ2) or by Hamiltonian confinement, using either a Kerr-type Hamiltonian with amplitude K or a two-photon exchange Hamiltonian (or TPE, for “two photons exchange”) with amplitude g2.

[0013] Several papers have suggested using cat qubits in QECC, either with codes that are fully specialized for phase errors (or “phase flips”) or with codes that are more tolerant to phase errors than bit errors (such as bias-noise tailored codes of the rectangular surface or XZZX surface type), including “Repeating cat qubits for fault-tolerant quantum computing” by Jeremie Guillaud and Mazyar Mirrahimi, 2 December 2019, Phys. Rev., X 9, 041053, “Bias-preserving gates with stable cat qubits” by Shruti et al., 21 August 2020, SCIENCE ADVANCES, Vol. 6, No. 34, Jeremie Guillaud and Mazyar Mirrahimi, 13 April 2021, Phys. Rev., A 103, 042413, Christopher Chamberland et al., 23 February 2022, PRX Quantum 3, “Building a Fault-Tolerant Quantum Computer Using Tandem Cat Codes,” 010329, or Andrew S. Darmawan et al., “Practical Quantum Error Correction Using XZZX Codes and Kerr-Cat Qubits,” published on PRX Quantum2 on September 16, 2021, 030345.

[0014] In a cat-qubit-based architecture, the quality of the hardware is measured by the ratio between two time scales: time 1 / κ2 and 1 / κ1, where κ2 is the two-photon dissipation rate of the stable qubit and κ1 is the single-photon loss rate. There are also other sources of error, such as rate phase shifts κ Φ 、 Thermal excitation th , self- and cross-Kerr interactions, or other undesirable couplings with other quantum systems present in the vicinity of the memory, etc. Nevertheless, single-photon losses are the dominant error mechanism, and for clarity only this physical error mechanism is considered below. In the general case, everything that follows is valid, replacing the κ1 / κ2 ratio by the sum of the rates of the error mechanisms divided by κ2.

[0015] In the prior art (published work), experimentally obtained κ1 / κ2 ratios range from 1 to 10 -2However, in order for quantum error correction to work, theoretical predictions estimate that it is necessary to design this ratio to be lower than κ1 / κ2<5x10 -3 For large-scale quantum computing, it is necessary to control this ratio within [10 -5 ; 10 -4 ] within the range of ].

[0016] Therefore, there is currently no solution that can achieve cat-qubit repetition codes that are compatible with the error-tolerance threshold.

[0017] The present invention improves this situation. To this end, the present invention provides a repetition code for cat qubits, characterized in that it includes a number of data cat qubits d and at least (d-1) auxiliary cat qubits, the number d is greater than or equal to 3, each auxiliary cat qubit is connected to two data cat qubits through two corresponding CNOT gates, so that no data cat qubit is connected to more than two auxiliary cat qubits, and the two-photon dissipation rate of the auxiliary cat qubit is strictly greater than the two-photon dissipation rate of the data cat qubit. The repetition code is implemented by performing an error correction cycle on each auxiliary cat qubit, the cycle comprising at least the following operations:

[0018] a) prepare the auxiliary cat qubit to be in a state suitable for operator X: “|+>” or “|->”,

[0019] b) activating one of the two CNOT gates connected to this auxiliary cat qubit,

[0020] c) activating another CNOT gate connected to this auxiliary cat qubit, and

[0021] d) Measure the parity of the photon number of the auxiliary cat quantum bit.

[0022] The device is particularly advantageous because it allows the realization of cat-qubit repetition codes that are compatible with fault-tolerant error thresholds.

[0023] According to various embodiments, the present invention may have one or more of the following features:

[0024] - the two-photon dissipation rate of the auxiliary cat qubit is selected to be substantially equal to a multiple of the two-photon dissipation rate of the data qubit, the multiple being selected from the group consisting of 2, 5, 20, 30, and 50,

[0025] - Repeat operations a) to d) for each auxiliary cat qubit, the number of repetitions being roughly equal to the ratio of the two-photon dissipation rate of the data cat qubit to the two-photon loss rate of the auxiliary cat qubit, and then perform a refresh operation, the duration of which is within the range of the inverse of the two-photon dissipation rate of the data cat qubit,

[0026] -apply error correction cycles to all auxiliary cat qubits roughly simultaneously,

[0027] - For each auxiliary cat qubit, perform operations a) to d) in sequence,

[0028] -DataCat qubits are stabilized in 3D cavity mode using ATS circuits,

[0029] - the auxiliary cat qubit is a resonant cat qubit, and

[0030] -The auxiliary cat qubit is a cat qubit stabilized in a 2D resonator mode using an ATS circuit.

[0031] The present invention also relates to a surface code for cat qubits, characterized in that it comprises a number d of data cat qubits and at least (d-1) auxiliary cat qubits, the number d being greater than or equal to 3,

[0032] Each auxiliary cat qubit is connected to four data cat qubits through four corresponding CNOT gates, so that no data cat qubit is connected to more than four auxiliary cat qubits,

[0033] The two-photon dissipation rate of the auxiliary cat qubit is strictly greater than the two-photon dissipation rate of the data cat qubit.

[0034] The surface code is implemented by executing a loop including at least the following operations on each auxiliary cat qubit:

[0035] a) Prepare the auxiliary cat qubit to be in state X: “|+>” or “|->”,

[0036] b) sequentially activate the CNOT gates connected to this auxiliary cat qubit, and

[0037] c) Measure the parity of the number of photons of the auxiliary cat qubit.

[0038] Other characteristics and advantages of the invention will better emerge on reading the following description of examples given as non-limiting illustrations with reference to the accompanying drawings, in which:

[0039] - Figure 1 shows a block diagram of a repetition code according to the present invention;

[0040] - Figure 2 Shows Figure 1 Error rate graphs of CNOT gates in (on the one hand) the auxiliary cat qubit and (on the other hand) the data cat qubit of the repetition code;

[0041] - Figure 3 The first embodiment of the present invention is shown Figure 1 Implementation diagram of repetition code;

[0042] - Figure 4 The second embodiment of the present invention is shown Figure 1 Implementation diagram of repetition code;

[0043] - Figure 5 Shown in Figure 4 Distortion curves of datacat qubits in the context of implementation methods;

[0044] - Figure 6 Shows the implementation Figure 1 A top view of another practical embodiment of a repeating code;

[0045] - Figure 7 Shows Figure 5 A side view of a repeating code;

[0046] - Figure 8 Shows the implementation Figure 1 A top view of another practical embodiment of a repeating code;

[0047] - Fig. 9 Shows Figure 5 A side view of a repeating code;

[0048] - Fig.10 shows the phase error rate of the symmetric repetition code of the prior art;

[0049] - Fig.11 It shows that the Figure 4 Phase error rate of an asymmetric repetitive code with a correction period in ; and

[0050] - Fig.12 A block diagram of a surface code according to the present invention is shown.

[0051] The drawings and descriptions below contain elements of a specific nature in nature, and therefore, they are not only used to better understand the invention, but also to help define the invention where appropriate.

[0052] Figure 1 A block diagram of a repetition code 2 according to the invention is shown. In the example described herein, the repetition code 2 has 3 dimensions, ie the repetition code comprises three data cat qubits 4 and two auxiliary cat qubits 6.

[0053] A cat qubit is understood to be any implementation of a cat qubit, in particular a two-photon dissipative Schrödinger cat qubit. Alternatively, other cat qubits may be retained. In some embodiments, the data cat qubit 4 and the auxiliary cat qubit 6 may be composed of different types of cat qubits.

[0054] The data cat qubit should be understood as the physical qubit containing the quantum information that the repetition code 2 attempts to protect. The auxiliary cat qubit should be understood as the supplement of the data cat qubit in the repetition code, that is, the physical qubit is used to detect the phase error of the data cat qubit.

[0055] The repetition code 2 has a conventional repetition code architecture for cat qubits: a number d of data cat qubits 4 greater than or equal to 3 and a number d-1 of auxiliary cat qubits 6. In some embodiments, for d data cat qubits 4, there may also be d auxiliary cat qubits 6. Each auxiliary cat qubit 6 is connected to two data cat qubits 4 via a CNOT gate 8. The connection between the data cat qubit 4 and the auxiliary cat qubit 6 is such that one data cat qubit 4 is connected to a maximum of two auxiliary cat qubits 6. Each auxiliary cat qubit 6 is also connected to a device 10 for measuring a parity check operator for the number of photons, which is used to detect phase errors after implementing a cycle of the error repetition code.

[0056] As mentioned in the introduction, the repetition code is implemented as follows:

[0057] - for each auxiliary cat qubit 6, prepare it to be in a state |+> or |-> suitable for Pauli operator X,

[0058] - applying each CNOT gate 8, the CNOT gates 8 associated with the same auxiliary cat qubit 6 are applied sequentially, i.e. one after the other without overlapping, so that the cat qubit 6 receives the state of the data cat qubit 4 to which it is connected via the corresponding CNOT gate 8, and

[0059] -Measure the photon number parity for each auxiliary cat qubit.

[0060] In all known systems, the same cat qubit is used for the data cat qubit 4 as well as for the auxiliary cat qubit 6. Efforts have focused on improving the ratio κ1 / k2 of the cat qubits. However, this is complicated and, currently, all efforts to improve κ2 result in an increase in the ratio of κ1. More specifically, there are several techniques that allow stabilizing the cat qubit. In general, technologies that allow to have very low ratios k1 (such as 3D cavities) are technologies that are difficult to couple strongly to the cat qubits, thus obtaining high values ​​of κ2. Conversely, some technologies (such as 2D resonators or resonant cat-type circuits) allow to obtain higher values ​​of κ2 because it is easier to couple strongly to the cat, but these circuits are more difficult to isolate from the environment and therefore have higher single-photon loss rates κ1.

[0061] The applicant has found that it may be interesting to use different cat qubits for the data cat qubit 4 and the auxiliary cat qubit 6. In fact, in the repetition code 2, the data cat qubit 4 and the auxiliary cat qubit 6 play very different roles. As will be explained in principle and then in Figure 2 As explained in detail in , for the same fixed value of κ1 / κ2, there is an advantage in using a higher value of κ2 for auxiliary cat qubit 6 compared to κ2 for data cat qubit 4.

[0062] The significance of this choice can be explained as follows. Figure 1 The typical time T of the described error correction cycle depends mainly on the time scale set by the two-photon dissipation rate of the auxiliary cat qubit 6. Therefore, where C is in the range [0.1; 10]. However, the typical error rate on the datacat qubit 4 in this cycle is

[0063] We assume that the ratio between single-photon loss and two-photon dissipation is the same for auxiliary cat qubit 6 and data cat qubit 4. Then, we have But there is an asymmetry between auxiliary cat qubit 6 and data cat qubit 4

[0064] However, what matters most in the code cycle is the total probability of phase shift error on the datacat qubit 4, which is given by Given.

[0065] Figure 2 shows the results of this asymmetry.

[0066] As shown in the figure, the error introduced into the system (auxiliary cat qubit and data cat qubit) can be numerically evaluated by performing a CNOT gate between the two cats. The error with two different photon numbers α for different asymmetries Δ (according to the X-axis) is presented. 2 =4, and α 2 =7, with a fixed relative speed of δ=10 -3 There is a single photon loss case. Of course, other values ​​of δ and α can be considered. For example, for a large quantum computer, α 2 The range of interest is in the range of 2 to 20-30 photons at most. The CNOT gate time is set to

[0067] The resulting errors can be divided into two categories. There are 12 quantum errors (IX, IY, XI, XX, XY, XZ, YI, YX, YY, YZ, ZX, ZY) that contain a given bit error component on at least one or both cats: these errors increase exponentially α 2 inhibition; as can be verified on the logarithmic scale green curve.

[0068] Significantly, the remaining three quantum errors behave as pure phase changes - not exponentially suppressed: qubit Z of the auxiliary cat qubit a The phase change (the top curve in the figure, which is almost flat) of the DataCat qubit Z d The phase change (the bottom curve in the figure, which is almost flat), and both are related to Z a Z d Phase change (this curve has the largest change with the other two curves, except α 2 = 4, the curve extends to the quantum bit Z d below the curve). Figure 2 A linear scale shows that the error on the datacat qubit decreases linearly as the asymmetry Δ increases.

[0069] As mentioned above, when the asymmetry Δ decreases, the phase error on the data cat qubit (the two orange curves in superposition) decreases linearly. On the other hand, the pure phase shift on the control qubit increases slightly, but this only leads to measurement errors that hardly harm the error correction.

[0070] Therefore, the applicant proposed Figure 1 The first embodiment of the repetition code 2 has an asymmetry such that the ratio In the range [1; 50] and all possible values ​​in that range, including 2, 5, 20, 30, and 50.

[0071] This repeat code 2 implements Figure 3 The calibration cycle shown in , is to repeat the following operations:

[0072] a) Prepare the auxiliary cat qubit 6 to be in state X “|+>” or “|->”,

[0073] b) activating one of the two CNOT gates 8 connected to the auxiliary cat qubit 6,

[0074] c) activating another CNOT gate 8 connected to the auxiliary cat qubit 6, and

[0075] d) Measure the parity of the number of photons of the auxiliary cat quantum bit 6.

[0076] Figure 3Three consecutive correction cycles are shown in Figure 1, with time passing from left to right. Therefore, the leftmost element represents the operation that occurred first, and the two vertically aligned elements represent operations that occurred simultaneously or nearly simultaneously. The operation associated with data cat qubit 4 is called the "data cat qubit", while the operation associated with auxiliary cat qubit 6 is called the "auxiliary cat qubit".

[0077] The applicant has found this embodiment to be very advantageous and allows to increase the fault tolerance threshold by more than a factor of two.

[0078] It is also found that this implementation may lead to slight distortion ("leakage" in the data cat qubit) because the asymmetry of the two-photon dissipation rate Δ causes the gate to be too fast compared to the time scale of the data cat qubit, due to Given.

[0079] In fact, the gate time is set at As Δ increases, the gate thus changes with the typical time scale This can lead to large distortions in the datacat qubits. The fact that the datacat qubits are distorted (a large fraction of the probability density of the datacat qubits no longer lies in the subspace of the cat qubits) can lead to two undesirable effects: the introduction of bit errors that are not suppressed exponentially; and higher logic error rates due to temporal correlations in the measurement errors.

[0080] Therefore, the applicant implemented a second embodiment of the repetitive code 2, whose correction period is as follows: Figure 4 In this figure, Expressed as Expressed as The reconvergence mentioned below is called "refresh time".

[0081] As shown in the figure, this cycle is Figure 3 The cycle is very similar, but it has a refresh time between Δ regular correction cycles.

[0082] The idea behind this second embodiment is to insert a "wait time" or "refresh / reconvergence time" after performing a given number of error correction cycles (an error correction cycle of a repetitive code includes the preparation of the auxiliary cat qubit, the application of two CNOT gates and the measurement of the auxiliary cat qubit).

[0083] During these refresh times, the two-photon pumping of the data cat qubits is turned on for a period of time that is long enough to return the state of these qubits to the subspace of the cat qubits, that is, long enough to suppress the distortion of the data cat qubits caused by the previous measurement cycle.

[0084] The typical duration of this refresh time for a DataCat qubit is These waiting times necessarily introduce additional phase errors on the data cat qubits, since these errors are always subject to the error mechanism during these "long" convergence times. Nevertheless, the applicants have found that linking "fast" error correction cycles and then refreshing the data cat qubits after the distortions induced by these fast cycles, and then starting over again, can achieve higher code performance than the symmetric operation mode, in which the error correction cycles are slower but do not or barely distort the data cat qubits, so no refresh time is required. The number of error correction cycles that can be performed before a refresh is needed is typically in the asymmetric range of Inside.

[0085] Figure 6 Shown in use Figure 4 The curve of the data cat qubit distortion that occurs during the correction cycle of the repetitive code. In this figure, Expressed as Expressed as On the ordinate axis, reference "leakage" represents the distortion rate, or the probability density of the data cat qubit being outside the cat's subspace as a function of the cycle time. This number varies between 0 and 1: 0 when the qubit's state is completely in the subspace, and 1 when it is not. The larger this number, the more distorted the cat qubit is. On the abscissa axis, time, in units of

[0086] Thus, the figure shows the distortion ratios obtained for three different ratios Δ. It can be seen that after each refresh cycle the distortion ratio returns to its lowest value, which indicates that Figure 4 The correction period is effective in suppressing the distortion caused by the rapidity of the quantum gates.

[0087] The applicant has found that the second embodiment is very advantageous and can increase the fault tolerance threshold by more than 4 times.

[0088] Fig.10 The variation of the phase error rate with the repetition code dimension d, where the ratio Δ is equal to 1, is shown for the ratio value δ. Since the ratio Δ is equal to 1, we are dealing with a "conventional" repetition code, identified as "symmetric", and the figure contains the performance of the prior art.

[0089] To give a few comparisons, the probability is about 0.02 for δ equal to 0.001 and d equal to 3. For δ equal to 0.003, the probability is between 0.02 and 0.06 for all values ​​of d.

[0090] Fig.11The variation of the phase error rate with the repetition code dimension d is shown, where the ratio Δ is equal to 21, and with the ratio value δ. Since the ratio Δ is equal to 1, the repetition code is a repetition code of the present invention and is deemed "asymmetric".

[0091] Consider again Fig.10 Values ​​related to the prior art reported in:

[0092] - For δ equal to 0.001 and d equal to 3, the probability is now 0.002 (a gain of 10 compared to the prior art)

[0093] - For δ equal to 0.003 and d equal to 3, the probability is 0.02 (a gain of 3 compared to the prior art), for d equal to 5, the probability is 0.005 (a gain of 10 compared to the prior art), and for d equal to 7, the probability is 0.001 (a gain of 50 compared to the prior art).

[0094] Figure 6 and Figure 7 Shows Figures 1 to 5 The "physical" implementation of the repetition code 2.

[0095] In this first embodiment implemented in the laboratory, the repetitive code 2 comprises an electromagnetic isolation device 60, which comprises a base 62 made of an artificial magnetic conductor, from which rises a plurality of protrusions forming a bed of nails 64. The electromagnetic device 60 comprises a portion mounted above the bed of nails to form a forbidden electric tape structure. The bed of nails has a regular pattern of nails 64. The device 60 is the subject of a patent application with application number FR2111275.

[0096] The electromagnetic isolation device 60 is intended to confine the electromagnetic field within the 3D cavity of the data cat qubit 4 described below to limit the radiation losses of the data qubit enclosed therein (which reduces single photon losses).

[0097] The repetitive code 2 thus comprises three data cat qubits 4, which here stabilize the cat qubits in a 3D cavity mode, thereby allowing to reach typical values The typical value of κ1 achieved by this design is κ1 -1 ≈100μs-10ms.

[0098] like Figure 7 As shown, these qubits are implemented in a cavity 66, with a finger 68 at the center of the cavity extending in the same direction as the nail 62. The data cat qubit 4 is formed by an ATS (Asymmetric Threaded SQUID) circuit 70.

[0099] The ATS circuit 70 includes a SQUID ("superconducting quantum interference device"), the center of which is short-circuited by a large inductor, forming two loops, and is surrounded on both sides by two gaskets 72 made of superconducting material. The ATS circuit 70 is powered by three lines 74, two of which carry the pump current for the data cat qubit 4 and the current for the ATS circuit 70, allowing the qubit to be stabilized. The third line enables the current to return. The ATS circuit 70 is the subject of a U.S. patent application with publication number US2021 / 0234086.

[0100] The state of the data cat qubit 4 can be read by a conventional device 76. In the example described herein, the device 76 is formed by a transmon comprising two pads and a Josephson junction coupled to a measurement resonator of the transmon, which is coupled to a transmission line. Alternatively, the device 76 can be composed in many other conventional ways known in the art.

[0101] The CNOT gate 8 is realized by a transmission line 78 connecting one of the pads 72 to the auxiliary cat qubit 6. This transmission line acts as a "coupler" and allows increasing the participation of the modes of the auxiliary cat qubit 6 in the Josephson junction of the ATS 70, which allows obtaining a nonlinear coupling between the auxiliary cat qubit 6 and the data cat qubit 4. The CNOT gate 8 can be activated by adding one or more signals on the line 74.

[0102] In the example described herein, the auxiliary cat qubit 6 is a 2D resonant cat qubit 80, which allows to reach typical values The typical value of κ1 achieved by this design is κ1 -1 ≈1-100μs. More specifically, the 2D resonant cat qubit has two modes wa and wb, so that wb=2wa, so that Schrödinger's cat state is stable in the wa mode, while the wb mode is an auxiliary mode. The auxiliary cat qubit is also powered by a transmission line, allowing the application of direct current to control the current applied at the auxiliary chat circuit, and also allowing coupling to auxiliary modes located on the auxiliary chat circuit. The 2D resonant cat qubit is the subject of a European patent application with publication number EP21306965.1. Device 81 allows the state of the auxiliary cat qubit 6 to be read and device 10 to be realized. Device 81 can be constructed in a manner similar to device 76.

[0103] In this figure, some nails 64 are intentionally not shown below the cavities 66 in order to better illustrate the lines 74. However, these nails are actually much smaller than the nails 64 and are arranged between the nails, and the pattern of the nail 64 bed is periodic, except where there are cavities 66.

[0104] Figure 8 and Fig. 9 For simplicity, only the Figure 6 and Figure 7 In this variant, the cat qubit 6 is always 2D, but this time it is not resonant and is stabilized by ATS82.

[0105] In the foregoing, the data cat qubits are always identical to each other. To limit the risk of crosstalk due to frequency crowding phenomena, the frequencies of the data cat qubit modes can differ between spatially adjacent qubits. This crosstalk will cause the control of one qubit to affect neighboring qubits whose resonant modes are at close frequencies. Furthermore, the data cat qubits can be different from the cat qubits that are stabilized in the 3D cavity mode. Figures 6 to 9 The examples are presented to illustrate the generality of the present invention and should not be interpreted in a limiting sense.

[0106] The present invention also relates to an architecture in which cat qubits are used for error correction codes other than repetition codes. More specifically, the present invention also relates to a surface code architecture based on cat qubits.

[0107] In fact, the strategy of using repetition codes to correct phase errors of cat qubits makes sense when the noise deviations of the cat qubits are very high, i.e., when the probability of a single bit error in the entire system (among all the cat qubits in the system, and over the entire typical operating duration of the system; i.e., the typical time for executing quantum algorithms on this architecture) is very low compared to the probability of a logical phase error.

[0108] When the noise deviation is not high enough (for example, because the size of the cat qubit measured in terms of the number of photons is not high enough; or when the bit error rate saturates and no longer increases with the size of the cat, when the physical phenomena causing the bit errors are no longer local, which happens with a large number of photons), it might make sense to use a surface code instead of a repetition code, which only corrects phase errors, or to use another quantum error correction code that can also tolerate a small number of bit errors.

[0109] For these embodiments, it is also advantageous to use an auxiliary cat qubit whose two-photon dissipation rate is strictly greater than that of the data cat, for the reasons discussed above. In fact, in the case of surface codes, the stabilizer (the quantum operator to be measured on the data qubit to detect errors) is of the form and For Stabilizer S jFor the measurement, the error correction cycle performed consists of preparing the auxiliary cat, applying 4 CNOT gates, and then measuring the operator X of the auxiliary cat. As in the case of the repetition code, the typical speed at which this cycle is performed depends mainly on the stabilization rate of the auxiliary cat (two-photon dissipation). Therefore, using two different techniques for the auxiliary cat and the data cat can improve the performance of the code and thus increase the value of the error correction threshold.

[0110] Fig.12 A block diagram of such a surface code is shown.

Claims

1. A repetition code for a cat quantum bit, characterized in that The repetitive code comprises a number d of data cat quantum bits (4) and at least (d-1) auxiliary cat quantum bits (6), the number d being greater than or equal to 3, Each auxiliary cat qubit (6) is connected to two data cat qubits (4) via two corresponding CNOT gates (8), such that no data cat qubit (4) is connected to more than two auxiliary cat qubits (6), The two-photon dissipation rate of the auxiliary cat quantum bit is strictly greater than the two-photon dissipation rate of the data cat quantum bit, The repetition code is implemented by performing an error correction cycle on each auxiliary cat quantum bit (6) including at least the following operations: a) preparing the helper qubit (6) to be in a state suitable for operator X: "|+>" or "|->", b) activating one of the two CNOT gates (8) connected to the auxiliary cat qubit (6), c) activating another CNOT gate (8) connected to the auxiliary cat qubit (6), and d) Measuring the parity of the photon number of the auxiliary cat quantum bit (6).

2. The repetition code according to claim 1, wherein: The two-photon dissipation rate of the auxiliary cat qubit (6) is selected to be substantially equal to a multiple of the two-photon dissipation rate of the data qubit, the multiple being selected from the group consisting of 2, 5, 20, 30 and 50.

3. The repetition code according to claim 1 or 2, wherein: Repeat operations a) to d) for each auxiliary cat quantum bit (6), the number of repetitions being substantially equal to the ratio of the two-photon dissipation rate of the data cat quantum bit (4) to the two-photon loss rate of the auxiliary cat quantum bit (6), and then perform a refresh operation, the duration of which is within the inverse range of the two-photon dissipation rate of the data cat quantum bit (4).

4. A repetition code according to any one of the preceding claims, wherein The error correction cycle is performed on all auxiliary cat qubits (6) substantially simultaneously.

5. The repetition code according to claim 4, wherein: For each auxiliary cat qubit (6), operations a) to d) are performed sequentially.

6. A repetition code according to any one of the preceding claims, wherein The data cat qubit (4) is stabilized in a 3D cavity mode using an ATS circuit (70).

7. A repetition code according to any one of the preceding claims, wherein The auxiliary cat qubit (4) is a resonance cat qubit (80).

8. The repetition code according to any one of claims 1 to 6, wherein: The auxiliary cat qubit (4) is a cat qubit stabilized in a 2D resonator mode using an ATS circuit (82).

9. A surface code for a cat quantum bit, characterized in that The surface code comprises a number d of data cat quantum bits (4) and at least (d-1) auxiliary cat quantum bits (6), the number d being greater than or equal to 3, Each auxiliary cat qubit (6) is connected to four data cat qubits (4) through four corresponding CNOT gates (8), such that no data cat qubit (4) is connected to more than four auxiliary cat qubits (6), The two-photon dissipation rate of the auxiliary cat quantum bit is strictly greater than the two-photon dissipation rate of the data cat quantum bit, The surface code is implemented by executing a loop including at least the following operations on each auxiliary cat quantum bit (6): a) preparing the auxiliary cat qubit (6) to be in state X "|+>" or "|->", b) sequentially activating the CNOT gates (8) connected to the auxiliary cat qubit (6), and c) Measuring the parity of the number of photons of the auxiliary cat qubit (6).

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