Asymmetric repitition code for a cat qubit
Patent Information
- Application Number
- EP2023713416
- Authority / Receiving Office
- EP · EP
- Patent Type
- Applications
- Current Assignee / Owner
- Priority Date
- 2022-03-16
- Filing Date
- 2023-03-15
- Publication Date
- 2025-01-22
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Abstract
Description
[0001]
[0002] Title: Asymmetric repetition code for cat qubit
[0003] The invention relates to the field of quantum computing, bosonic quantum error correcting codes, and repetition codes.
[0004] Building a large-scale quantum computer is challenging because noise induced by uncontrolled interactions of quantum computer components with their environment destroys the fragile quantum characteristics responsible for quantum acceleration. Indeed, all algorithms with theoretically proven quantum acceleration require some level of protection against decoherence.
[0005] The theory of fault-tolerant quantum computing addresses this issue. Quantum error-correcting codes (QECCs) are designed so that environmentally induced errors do not affect quantum information. These codes operate on the principle of "fighting entanglement with entanglement": since natural errors in physical systems are generally local, the quantum information to be protected is encoded in non-local entangled states, making it unlikely that errors could corrupt it. The most popular QECC is the surface code.
[0006] The core of quantum fault tolerance theory is the threshold theorem: arbitrarily long quantum computations can be performed reliably provided that the noise affecting the physical components of the computer is below a constant value called the fault-tolerant error threshold.
[0007] In theory, CCEQs provide, when operating below the fault-tolerant error threshold, arbitrarily good protection against noise, thus solving the decoherence problem. However, their implementation in the physical world comes at the cost of enormous physical resources to achieve a sufficient level of protection. The trade-off between the degree of protection provided by a CCEQ and the increase in components required for its implementation defines what is known as the "resource overhead problem."
[0008] Realistic approaches to quantum computing must address this problem. In this regard, continuous variable systems (such as a harmonic oscillator), in which an infinite-dimensional Hilbert space is readily available to protect and process quantum information, seem to have a head start over discrete variable (DV) systems that only have a finite-dimensional Hilbert space. There are many different continuous variable encodings, usually involving the superposition of certain specific states of a harmonic oscillator, such as position and momentum eigenstates (GKP qubits), Fock states, or coherent states (cat qubits).
[0009] The invention relates more particularly to cat qubits.
[0010] Pumped (or stabilized) cat qubits are known to benefit from a noise bias. Specifically, an effective error channel (e.g., bit errors or bit flips) is suppressed exponentially with the "size"—i.e., the average photon number—of the Schrodinger cat states of cat qubits. According to current knowledge, this suppression should apply to a large class of physical noise processes having a local effect on the phase space of a harmonic oscillator. This includes, but is not limited to, photon loss, thermal excitations, photon dephasing, and various nonlinearities induced by coupling to a Josephson junction. Recent experiments in quantum superconducting circuits have observed this exponential suppression of bit errors with the average photon number in cat states.
[0011] Due to this noise structure, quantum error correction becomes similar in complexity to classical error correction and can be achieved using a repetition code. Indeed, if one wishes to correct only the phase jump, it is sufficient to use the phase jump error correcting code. This can be, for example, a repetition code defined in the dual basis, or any other classical error correcting code. The code space is defined as the common eigenspace +1 of the d-1 stabilizers s, = Xj ® x j+1 , I [[I,N - 1]].. The logical operators for the repeating cat qubit are x L = x 1: z L = jZj, Y L = ix L z L . The logical states |+)L and |— )L are given by |±) t ■■= |±)® N .
[0012] A cat qubit repetition code is constructed using d cat qubits, called data cat qubits, in which the logical information is encoded. The repetition code is implemented, i.e., errors are detected, by repeatedly measuring the repetition code's stabilizers. This is achieved using additional d-1 cat qubits, called ancillary cat qubits. The quantum circuit of the repetition code requires preparation in the |+) state - for a cat qubit, this is the Schrodinger cat state -, measurement of the Pauli X operator - for a cat qubit, this is the parity of the number of photons -, and CNOT gates between the ancillary cat qubits and the data cat qubits.
[0013] The challenge in implementing this repetition code is to successfully operate the code below the fault-tolerant error threshold, which means that the fidelity of the quantum operations in this circuit must be sufficiently high. More precisely, when the repetition code is operated above the threshold, that is, when the fidelity of the physical operations composing the repetition code is insufficient, the lifetime of the logical information decreases as the number of physical data qubits d increases: the new errors introduced by the addition of quantum systems are not compensated by the error correction strategy.On the other hand, when the code is operated below the error correction threshold, that is to say when the fidelity of the physical operations is sufficient, the lifetime of the logical information increases exponentially with the number of physical data qubits d (which coincides with the distance d of the repetition code).
[0014] Cat qubits are implemented in the laboratory either by dissipative stabilization (via artificial two-photon dissipation at rate K2) or by Hamiltonian confinement, using a Kerr-type Hamiltonian of amplitude K or a two-photon exchange (or TPE) Hamiltonian of amplitude g2.
[0015] Several papers propose using cat qubits in a CCEQ, either in a code entirely dedicated to phase-flip, or in a code that tolerates phase errors much better than bit errors (such as a biased noise tailored code, such as a rectangular surface code or XZZX surface code), among which "Repetition Cat Qubits for Fault-Tolerant Quantum Computation" by Jérémie Guillaud and Mazyar Mirrahimi, Phys. Rev. X 9, 041053, 12 / 12 / / 2019, "Bias-preserving gates with stabilized cat qubits" de Shruti et al., SCIENCE ADVANCES, Vol 6, Issue 34, 21 / 08 / 2020, "Error rates and resource overheads of repetition cat qubits" de Jérémie Guillaud and Mazyar Mirrahimi, Phys. Rev. A 103, 042413, 04 / 13 / 2021, "Building a Fault-Tolerant Quantum Computer Using Concatenated Cat Codes" by Christopher Chamberland et al.; PRX Quantum 3, 010329, 02 / 23 / 2022, or “Practical Quantum Error Correction with the X / ZX Code and Kerr-Cat Qubits” by Andrew S. Darmawan et al., PRX Quantum 2, 030345, 09 / 16 / 2021.
[0016] In cat qubit-based architectures, hardware quality is measured by the ratio of two timescales: time 1 / K2, where K2 is the two-photon dissipation rate that stabilizes the qubit; and 1 / Kq, where K is the one-photon loss rate. There are other sources of errors such as phase shift at rate K^, thermal excitations n thself-Kerr and cross-Kerr interactions, or other unwanted couplings with other quantum systems present in the vicinity of the memory, etc. However, one-photon loss is the dominant error mechanism and for the sake of clarity, we only consider this physical error mechanism in what follows. In the general case, all that follows is valid by replacing the ratio K ]K by the sum of the error mechanism rates divided by K .
[0017] In the prior art (published works), the experimentally obtained K-K ratio varies from 1 to 10' 2 (smaller being better). However, for quantum error correction to work, theoretical predictions suggest that it is necessary to design this ratio to be less than K ]K < 5 x 10 -3. For large-scale quantum computing, it will be necessary to bring this ratio into a range of the order of [10 -5 ; 10' 4 ].
[0018] Currently, there is therefore no solution that allows the implementation of a cat qubit repetition code that is compatible with the fault tolerance error threshold.
[0019] The invention improves the situation. To this end, it proposes a cat qubit repetition code characterized in that it comprises a number d of data cat qubits and at least (d-1) ancillary cat qubits, the number d being greater than or equal to 3, each ancillary cat qubit being connected to two data cat qubits by two respective CNOT gates in such a way that no data cat qubit is connected to more than two ancillary cat qubits, the two-photon dissipation rate of the ancillary cat qubits being strictly greater than the two-photon dissipation rate of the data cat qubits.This repetition code is implemented by performing for each ancillary cat qubit error correction cycles comprising at least the following operations: a) preparation of the ancillary cat qubit in an eigenstate of the operator X: "l+>" or "l->", b) activation of one of the two CNOT gates connected to this ancillary cat qubit, c) activation of the other CNOT gate connected to this ancillary cat qubit, and d) measurement of the parity of the number of photons of this ancillary cat qubit.
[0020] This device is particularly advantageous because it allows the implementation of a cat qubit repetition code compatible with the fault tolerance error threshold.
[0021] According to various embodiments, the invention may have one or more of the following characteristics:
[0022] - the two-photon dissipation rate of the ancillary cat qubits is chosen to be substantially equal to a multiple of the two-photon dissipation rate of the data qubits, which multiple is chosen from the group comprising 2, 5, 20, 30 and 50, - operations a) to d) are repeated for each ancillary cat qubit a number of times substantially equal to the ratio of the two-photon dissipation rate of the data cat qubits to the two-photon loss rate of the ancillary cat qubits, then followed by a refresh operation whose duration is of the order of the inverse of the two-photon dissipation rate of the data cat qubits,
[0023] - error correction cycles are implemented substantially simultaneously for all ancillary cat qubits,
[0024] - for each ancillary cat qubit, operations a) to d) are performed sequentially,
[0025] - data cat qubits are stabilized in a 3D cavity mode using an ATS circuit,
[0026] - ancillary cat qubits are resonant cat qubits, and
[0027] - Ancillary cat qubits are cat qubits stabilized in a mode of a 2D resonator using an ATS circuit.
[0028] The invention also relates to a surface code for cat qubit characterized in that it comprises a number d of data cat qubits and at least (d-1) ancillary cat qubits, the number d being greater than or equal to 3, each ancillary cat qubit being connected to four data cat qubits by four respective CNOT gates in such a way that no data cat qubit is connected to more than four ancillary cat qubits, the two-photon dissipation rate of the ancillary cat qubits being strictly greater than the two-photon dissipation rate of the data cat qubits, said surface code being implemented by performing for each ancillary cat qubit cycles comprising at least the following operations: a) preparing the ancillary cat qubit in state X "l+>" or "l->", b) sequential activation of the CNOT gates connected to this ancillary cat qubit, and c) measuring the parity of the number of photons from this ancillary cat qubit.
[0029] Other characteristics and advantages of the invention will appear better on reading the following description, taken from examples given for illustrative and non-limiting purposes, taken from the drawings in which: - figure 1 represents a block diagram of a repetition code according to the invention,
[0030] - Figure 2 represents an error probability diagram for a CNOT gate on the one hand in an ancillary cat qubit and on the other hand in a data cat qubit of the repetition code of Figure 1,
[0031] - figure 3 represents an implementation diagram of the repetition code of figure 1 according to a first embodiment,
[0032] - figure 4 represents an implementation diagram of the repetition code of figure 1 according to a second embodiment,
[0033] - Figure 5 shows deformation curves of the data cat qubits in the implementation of Figure 4,
[0034] - Figure 6 represents a top view of another real embodiment implementing the repetition code of Figure 1,
[0035] - Figure 7 represents a side view of the repetition code of Figure 5,
[0036] - Figure 8 represents a top view of another real embodiment implementing the repetition code of Figure 1,
[0037] - Figure 9 represents a side view of the repetition code of Figure 5,
[0038] - Figure 10 represents a phase error probability for a state-of-the-art symmetric repetition code,
[0039] - Figure 11 represents a phase error probability for an asymmetric repetition code according to the invention with the correction cycle of Figure 4, and
[0040] - figure 12 represents a block diagram of a surface code according to the invention.
[0041] The drawings and the description below contain, for the most part, elements of a certain character. They may therefore not only serve to better understand the present invention, but also contribute to its definition, if necessary.
[0042] Figure 1 represents a block diagram of a repetition code 2 according to the invention. In the example described here, the repetition code 2 is of dimension 3, that is to say that it comprises three data cat qubits 4 and two ancillary cat qubits 6.
[0043] By cat qubit is meant any implementation of a cat qubit, and in particular a two-photon dissipative Schrodinger cat qubit. Alternatively, other cat qubits may be used. In some embodiments, the data cat qubits 4 and the ancillary cat qubits 6 may be implemented with distinct types of cat qubits.
[0044] By data cat qubit, we mean that this physical qubit contains the quantum information that the repetition code 2 seeks to protect. By ancillary cat qubit, we mean the complement of the data cat qubit in the repetition code, i.e., this physical qubit is used to detect phase errors of the data cat qubit.
[0045] The repetition code 2 has a classic cat qubit repetition code architecture: a number d greater than or equal to 3 of data cat qubits 4 and a number d-1 of ancillary cat qubits 6. In some embodiments, it is also possible to have d ancillary cat qubits 6 for d data cat qubits 4. Each ancillary cat qubit 6 is connected by a CNOT gate 8 to two data cat qubits 4. The connections between the data cat qubits 4 and the ancillary cat qubits 6 are such that a data cat qubit 4 is connected to at most two ancillary cat qubits 6. Each ancillary cat qubit 6 is also connected to a photon number parity operator measurement device 10, which is used to detect phase errors after the error repetition code cycle has been implemented.
[0046] As explained in the introduction, the implementation of the repetition code is done as follows:
[0047] - the preparation in the state |+) or |— ), which are the eigenstates of the Pauli operator X, for each ancillary cat qubit 6,
[0048] - each CNOT gate 8 is applied, the CNOT gates 8 associated with a same ancillary cat qubit 6 being applied sequentially, i.e. one after the other without overlap, so that the cat qubit 6 receives the state of the data cat qubit 4 to which it is connected by a respective CNOT gate 8, and
[0049] - the parity measurement of the number of photons is performed for each ancillary cat qubit. In all known systems, the same cat qubits are used for both the data cat qubits 4 and the ancillary cat qubits 6. Efforts are focused on improving the ratio K ]K of the cat qubits. This is nevertheless complicated, and, currently, all efforts to improve K2 result in a proportional increase in K . More precisely, there are several technologies for stabilizing cat qubits. Generally speaking, technologies that allow for a very low Kq rate (such as 3D cavities) are technologies for which it is difficult to couple strongly to the cat qubit, and therefore to obtain high K2 values.Conversely, some technologies (such as 2D resonators, or resonant cat circuits) allow much higher K2 values to be obtained because it is easier to couple strongly to the cat, but these circuits are much more difficult to isolate from the environment and therefore have a higher one-photon loss rate.
[0050] The Applicant has discovered that it could be interesting to use different cat qubits for the data cat qubits 4 and the ancillary cat qubits 6. Indeed, the roles of the data cat qubits 4 and the ancillary cat qubits 6 are very different in the repetition code 2. As will be explained in principle and then in detail with Figure 2, there is an advantage in using, for the same fixed value of K1 / K2, a higher value of K2 for the ancillary cat qubit 6 compared to K2 for the data cat qubit 4.
[0051] The interest of such a choice can be explained as follows. The typical time T of an error correction cycle as described with reference to Figure 1 depends mainly on the time scale set by the two-photon dissipation rate of the ancillary cat qubits 6. Thus, T = / ancillary with G in the range [0,1; 10]. However, the ' K 2 Typical error rate on data cat 4 qubits during this cycle is K ^ onnees .
[0052] Assume that the ratio of one-photon loss to two-photon dissipation is the same for both the ancillary cat qubits 6 and the data cat qubits 4. Then, we have ô = K ancillary K ancillary > K data ^data ma ^ s exis an asymmetry between the ancillary cat qubits 6 and the data cat qubits 44 = =
[0053] ..ancillary / ..data
[0054] K 1 / f <l
[0055] However, it is the total probability of phase shift error on the qubits in data chain 4 that matters most during a code cycle, and this probability is given by
[0056] Figure 2 represents the consequences of this asymmetry.
[0057] As can be seen in this figure, one can numerically evaluate the errors introduced into the system (ancillary cat qubits and data cat qubit) by performing a CNOT gate between two cats. In the presence of a single-photon loss at a fixed relative rate 5 = 10 -3 for different asymmetries A (along the X axis) and with two different numbers of photons a 2 = 4, then 7. Of course, other values of ô and a could be considered. For example, the range of interest for a 2 is of the order of 2 to 20-30 photons maximum for a large-scale quantum computer. The CNOT gate time is fixed
[0058] 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 some bit error component on at least one or both of the cats: these errors are exponentially suppressed with a 2 ; as can be seen on the green curve on a logarithmic scale.
[0059] What is interesting is that the behavior of the three remaining quantum errors which are pure phase changes - which are not exponentially suppressed: the phase changes on the Z qubit a of the ancillary cat qubit (the topmost curve of the graph, almost flat), those on the Z qubit d of the data cat qubit (the lowest curve in the graph, almost flat), and the correlated phase changes on the two Z a Z d(the curve which presents the greatest variation between the other two curves except for a 2 = 4 where it passes under the curve of the qubit Z d ). Figure 2 is plotted on a logarithmic scale. A linear scale representation would show that the errors on the data cat qubit decrease linearly with increasing asymmetry A.
[0060] As explained above, the phase errors on the data cat qubits (the two overlapping orange curves) decrease linearly when the asymmetry A is decreased. On the other hand, the pure phase shift on the control qubit increases a little, but this only leads to measurement errors that are of little harm to error correction.
[0061] Therefore, the Applicant proposes a first embodiment of a repetition code 2 according to Figure 1, with an asymmetry such that the ratio Z1 = K ancillary ^data j ans une p|a g e . 50] with all possible values in this range, including 2, 5, 20, 30, and 50.
[0062] This repetition code 2 implements the correction cycle shown in Figure 3, i.e. the repetition of the following operations: a) preparation of the ancillary cat qubit 6 in state X "l+>" or "l->", b) activation of one of the two CNOT gates 8 connected to this ancillary cat qubit 6, c) activation of the other CNOT gate 8 connected to this ancillary cat qubit 6, and d) measurement of the parity of the number of photons of this ancillary cat qubit 6.
[0063] In Figure 3, three successive correction cycles are shown, with time flowing from left to right. Thus, the leftmost elements represent operations that occur first, with two vertically aligned elements indicating simultaneous or near-simultaneous operations. Operations related to the data cat qubits 4 are referred to as "data cat qubit", and operations related to the ancillary cat qubits 6 are referred to as "ancilla cat qubit".
[0064] The Applicant has discovered that this embodiment is very advantageous and allows to improve the error tolerance threshold by more than 2 times. It has also discovered that this implementation can cause a slight distortion ("leakage" in English) of the data cat qubits, because the 4-asymmetry of the two-photon dissipation rates leads to gates that are too fast compared to the time scale of the data cat qubits, given by K ^onnees _
[0065] Indeed, the gate time is fixed around T CNOT When
[0066] A is increased, so the gate is fast compared to the typical I / K2 timescale onnees This leads to a potentially large amount of data cat qubit warping. The fact that data cat qubits are warped - a significant portion of the probability density of data cat qubits is no longer in the cat qubit subspace - could lead to two undesirable effects: the introduction of bit errors that are not exponentially suppressed; and time correlations in measurement errors resulting in higher logic error rates.
[0067] The Applicant has therefore implemented a second embodiment of the repetition code 2, the correction cycle of which is shown in Figure 4. In this figure, K ancillaryest not ^ K has et K data^ no ^é et i e tem p S I e reconvergence mentioned below is referenced "refreshing time".
[0068] As can be seen in this figure, this cycle is quite similar to that of Figure 3, but it has a refresh time between A classic correction cycles.
[0069] The idea of this second embodiment is to insert "waiting times", or "refresh / reconvergence times", after performing a certain number of error correction cycles (one error correction cycle of the repetition code being composed of the preparation of the ancillary cat qubits, the application of the two CNOT gates and the measurement of the ancillary cat qubits).
[0070] During these refresh times, the two-photon pumping of the data cat qubits is turned on for a time long enough for the state of these qubits to return to the cat qubit subspace, i.e., for a time long enough for the deformation of the data cat qubits induced by the previous measurement cycles to be removed.
[0071] The typical duration of this data chat qubit refresh time is of the order of / K ^ onneesThese waiting times necessarily introduce additional phase errors on the data cat qubits, since these are still subject to error mechanisms during these "long" reconvergence times. However, the Applicant has discovered that chaining "fast" error correction cycles, then refreshing the data cat qubits following the deformation induced by these fast cycles, before starting again, made it possible to obtain a code performance superior to the symmetric operating mode where the error correction cycles are slower but do not deform or only slightly deform the data cat qubits, therefore not requiring a refresh time. The number of error correction cycles that can be performed before a refresh is necessary is typically of the order of the asymmetry 21 = K ^ ncillaire / K ^ onnees _
[0072] Figure 6 shows curves representing the deformation of the cat data qubits that takes place when implementing the repetition code with the correction cycle of Figure 4. In this figure, K ncillaire is noted note On the ordinate, the "Leakage" reference designates the deformation rate, that is to say the probability density of the data cat qubits which is outside the cat subspace as a function of the cycle time. This figure varies between 0 and 1: 0 when the state of the qubit is perfectly in the subspace, and 1 when it is no longer there at all. The larger this figure is, the more the cat qubit is deformed. On the abscissa, the time, in unit K^T .
[0073] This figure therefore shows the deformation rate obtained for three different ratios 21. It appears that after each refresh period, the deformation rate returns to its floor, which shows that the correction cycle in Figure 4 is indeed effective in removing the deformation induced by the speed of the quantum gates.
[0074] The Applicant has discovered that this second embodiment is very advantageous and makes it possible to improve the error tolerance threshold by more than 4 times. Figure 10 illustrates the variation of the phase error probability as a function of the dimension d of the repetition code, with a ratio A equal to 1, as a function of the value of the ratio ô. As the ratio A is equal to 1, we are dealing with a "classic" repetition code, described as "symmetric", and this diagram constitutes the performance of the state of the art.
[0075] To give some benchmarks for comparison, for ô equal to 0.001 and d equal to 3, the probability is approximately 0.02. For ô equal to 0.003, for all values of d, the probability is between 0.02 and 0.06.
[0076] Figure 11 illustrates the variation of the phase error probability as a function of the dimension d of the repetition code, with a ratio A equal to 21, as a function of the value of the ratio ô. As the ratio A is equal to 1, the repetition code is that of the invention and is described as "asymmetric".
[0077] To take the values in relation to the state of the art noted with figure 10:
[0078] - for ô equal to 0.001 and d equal to 3, the probability is now 0.002 (a gain of ratio 10 compared to the state of the art),
[0079] - for ô equal to 0.003, the probability is 0.02 for d equal to 3 (a gain of ratio 3 compared to the state of the art), 0.005 for d equal to 5 (a gain of ratio 10 compared to the state of the art), and 0.001 for d equal to 7 (a gain of ratio 50 compared to the state of the art).
[0080] Figures 6 and 7 represent a "physical" implementation of the repetition code 2 of Figures 1 to 5.
[0081] In this first embodiment implemented in the laboratory, the repetition code 2 comprises an electromagnetic isolation device 60 which comprises a base 62 made of artificial magnetic conductor from which rise a plurality of projections forming a bed of nails 64. The electromagnetic device 60 comprises a portion which is arranged above the bed of nails to produce an electromagnetic bandgap structure. The bed of nails has a regular pattern of nails 64. The device 60 has been the subject of a patent application whose filing number is FR2111275. The electromagnetic isolation device 60 has the role of confining the electromagnetic field in the 3D cavities of the data cat qubits 4 described below in order to limit the radiation losses - which reduces the loss to a photon - of the data qubits which are enclosed therein.
[0082] The repetition code 2 thus comprises three data cat qubits 4 which are here cat qubits stabilized in the mode of a 3D cavity allowing to reach typical values ^2 — = 50kHz — 100 kHz. The typical values of K_1 reached by this
[0083] As can be seen in Figure 7, these qubits are implemented in a cavity 66 in the center of which a finger 68 is expressed in the same direction as the nails 62. The cat data qubit 4 is realized using an ATS (Asymmetrically Threaded SQUID) circuit 70.
[0084] The ATS 70 circuit comprises a symmetrical SQUID (Superconducting Quantum Interference Device) which is short-circuited in its center by a large inductance, which forms two loops and which is surrounded on either side by two pads 72 made of superconducting material. The ATS 70 circuit is powered by three lines 74, two of which carry the pumping current of the data cat qubit 4 as well as the flows for the ATS 70 circuit to stabilize the qubit. The third line allows a current return for the flows. The ATS 70 circuit has been the subject of a US patent application published under the number US 2021 / 0234086.
[0085] The state of the data cat qubit 4 can be read by means of a conventional device 76. In the example described here, the device 76 is formed by a transmon comprising two pads and a Josephson junction coupled to a transmon measurement resonator coupled to a transmission line. Alternatively, the device 76 could be implemented in many other conventional ways in the state of the art. The CNOT gate 8 is implemented using a transmission line 78 which connects one of the pads 72 to the ancillary cat qubit 6. This transmission line serves as a “coupler” and makes it possible to increase the participation of the mode of the ancillary cat qubit 6 in the Josephson junctions of the ATS 70, which makes it possible to obtain a non-linear coupling between the ancillary cat qubit 6 and the data cat qubit 4. The CNOT gate 8 can be activated by adding one or more signals on the lines 74.
[0086] The ancillary cat qubit 6 is in the example described here a 2D resonant cat qubit 80 which allows to reach typical values — = 500 kHz — 10 MHz. The values typical of K achieved by this design are / 1"1 — 100 dd.s. More particularly, the 2D resonant cat qubit has two modes wa and wb such that wb = 2wa, the Schrodinger cat state being stabilized in the wa mode and the wb mode being an auxiliary mode. The ancillary cat qubit is also powered by a transmission line allowing a direct current to be applied to control the flux applied at the ancillary cat circuit, and also allowing coupling to the auxiliary mode localized on the ancillary cat circuit. The 2D resonant cat qubit has been the subject of a European patent application published under the number EP21306965.1. A device 81 makes it possible to read the state of the ancillary cat qubit 6 and implements the device 10. The device 81 can be implemented in a similar manner to the device 76.
[0087] In this figure, some of the nails 64 have been intentionally not shown below the cavities 66 to better show the lines 74. However, these are much smaller in reality than the nails 64 and are arranged between them, and the pattern of the nail bed 64 is indeed periodic, except where there are the cavities 66.
[0088] Figures 8 and 9 show an alternative implementation. For simplicity, only the differences from Figures 6 and 7 are described. In this alternative, the cat 6 qubits are still in 2D, but this time they are no longer resonant and are stabilized by an ATS 82.
[0089] In the above, the data cat qubits are always identical to each other. In order to limit the risks of crosstalk due to the phenomenon of frequency crowding, the frequencies of the modes of the data cat qubits could be distinct between spatially neighboring qubits. This crosstalk would have the consequence that the control of a qubit would influence a neighboring qubit whose resonant mode is at a close frequency. In addition, the data cat qubits could be realized other than with cat qubits stabilized in the mode of a 3D cavity. The embodiments of Figures 6 to 9 show the generality of the invention and should not be interpreted in a restricted manner.
[0090] The invention also relates to architectures where the cat qubits are used in an error-correcting code other than the repetition code. More particularly, the invention also relates to surface code architectures based on cat qubits.
[0091] Indeed, the strategy of using a repetition code to correct the phase errors of the cat qubits is relevant when the noise bias of said cat qubits is very large, i.e. when the probability of having a single bit error in the entire system (in all the cat qubits of the system, and for the entire typical operating time of the system; i.e. during the typical execution time of a quantum algorithm on this architecture) is very low compared to the probability of logical phase error.
[0092] When the noise bias is not large enough (e.g. because the size of the cat qubits, measured in number of photons, is not sufficient; or when the bit error rate is saturated and no longer increases with the size of the cats, which happens at large number of photons when the physical phenomena that cause bit errors are no longer local), it may be interesting to use a surface code instead of the repetition code, which exclusively corrects only phase errors, or another quantum error correcting code that also allows to tolerate a small number of bit errors.
[0093] For these implementations, the use of ancillary cat qubits whose two-photon dissipation rate is strictly larger than the two-photon dissipation rate of the data cats also presents an advantage, for the same reasons as mentioned above. Indeed, in the case of the surface code, the stabilizers (the quantum operators to be measured on the data qubits to detect errors) are of the form Sj = Xj gx j+1 gx j+2 gx i+2 and s'j = Zj gz j+1 gz j+2 gz, +3. For the measurement of the stabilizers s, , an error correction cycle comprising the preparation of the ancillary cats, the application of 4 CNOT gates, and then the measurement of the operator X of the ancillary cats is performed. As in the case of the repetition code, the typical speed at which this cycle is performed depends mainly on the stabilization rate (two-photon dissipation) of the ancillary cats. The use of two different technologies for the ancillary cats and the data cats therefore makes it possible to improve the performance of the code, and thus to increase the value of the error correction threshold.
[0094] Figure 12 shows a block diagram of such a surface code.
Claims
Demands
1. A cat qubit repetition code characterized in that it comprises a number d of data cat qubits (4) and at least (d-1) ancillary cat qubits (6), the number d being greater than or equal to 3, each ancillary cat qubit (6) being connected to two data cat qubits (4) by two respective CNOT gates (8) such that no data cat qubit (4) is connected to more than two ancillary cat qubits (6), the two-photon dissipation rate of the ancillary cat qubits being strictly greater than the two-photon dissipation rate of the data cat qubits, said repetition code being implemented by performing for each ancillary cat qubit (6) error correction cycles comprising at least the following operations: a) preparing the ancillary cat qubit (6) in an eigenstate of the operator X: "l+>" b) activation of one of the two CNOT gates (8) connected to this ancillary cat qubit (6),c) activation of the other CNOT gate (8) connected to this ancillary cat qubit (6), and d) measurement of the parity of the number of photons of this ancillary cat qubit (6).
2. Repetition code according to claim 1, wherein the two-photon dissipation rate of the ancillary cat qubits (6) is chosen to be substantially equal to a multiple of the two-photon dissipation rate of the data qubits, which multiple is chosen from the group comprising 2, 5, 20, 30 and 50.
3. A repetition code according to claim 1 or 2, wherein operations a) to d) are repeated for each ancillary cat qubit (6) a number of times substantially equal to the ratio of the two-photon dissipation rate of the data cat qubits (4) to the two-photon loss rate of the ancillary cat qubits (6), and then followed by a refresh operation whose duration is on the order of the inverse of the two-photon dissipation rate of the data cat qubits (4).
4. Repetition code according to any one of the preceding claims, wherein error correction cycles are implemented substantially simultaneously for all ancillary chat qubits (6).
5. Repetition code according to claim 4, wherein, for each ancillary cat qubit (6), operations a) to d) are performed sequentially.
6. A repeat code according to any one of the preceding claims, wherein the data chat qubits (4) are stabilized in a 3D cavity mode using an ATS circuit (70).
7. Repetition code according to any one of the preceding claims, wherein the ancillary cat qubits (4) are resonant cat qubits (80).
8. Repeat code according to any one of claims 1 to 6, wherein the ancillary cat qubits (4) are cat qubits stabilized in a mode of a 2D resonator using an ATS circuit (82).
9. A surface code for a cat qubit characterized in that it comprises a number d of data cat qubits (4) and at least (d-1) ancillary cat qubits (6), the number d being greater than or equal to 3, each ancillary cat qubit (6) being connected to four data cat qubits (4) by four respective CNOT gates (8) such that no data cat qubit (4) is connected to more than four ancillary cat qubits (6), the two-photon dissipation rate of the ancillary cat qubits being strictly greater than the two-photon dissipation rate of the data cat qubits, said surface code being implemented by performing, for each ancillary cat qubit (6), cycles comprising at least the following operations: a) preparing the ancillary cat qubit (6) in state X "l+>" or "l->", b) sequentially activating the gates CNOT (8) linked to this ancillary cat qubit (6),and c) measure of the parity of the number of photons of this ancillary cat qubit (6).