System for performing quantum gate and performing quantum error correction code operation using same

Abstract

A quantum system for executing a quantum gate, comprising: a command circuit (8) for selectively applying radiation; n data resonators (4), where N is greater than or equal to 2, each data resonator (4) having a respective resonant frequency and coupled to the command circuit (8) to stabilize a respective data cat qubit; and an auxiliary resonator (6) having an auxiliary resonant frequency, coupled to the command circuit (8) to stabilize auxiliary cat qubits, and non-linearly coupled to the data resonator (4) through the command circuit (8). The command circuit (8) is arranged for performing a quantum gate by: a) applying radiation having the auxiliary resonant frequency while stabilizing the auxiliary qubit such that the data resonator (4) and the auxiliary resonator (6) are substantially simultaneously affected by Hamiltonian generated by the radiation having the auxiliary resonant frequency; b) turning off the radiation having the auxiliary resonant frequency after a selected duration. This principle is extended for execution of quantum error correction codes.

Description

[0001] The present invention relates to a system for executing quantum gates, and more particularly, to the use of such gates in the context of cat qubits.

[0002] To extract joint information from multiple data qubits, conventional quantum circuits execute a series of two-qubit gates (typically CNOT or CZ gates) between the auxiliary qubit and the multiple data qubits before measuring the auxiliary qubit. This operation is often referred to in the art as syndrome measurements, and these two-qubit gates need to be executed in series.

[0003] The implementation of these quantum gates is crucial for detecting errors and thus performing quantum error-correcting codes (or "QECC"), which is currently considered the only way to build reliable and usable quantum chips. The basic idea behind QECC is to use multiple (at least two) physical qubits to encode a logical qubit and to establish an error detection scheme to verify that the information of the multiple physical qubits does not change over time. If an error is detected, error correction is performed, either by changing the qubit state itself or by post-processing the results of quantum algorithms involving those qubits.

[0004] The characteristics used to evaluate the quality of a quantum gate are its execution time (i.e., the time it takes for the gate to run) and its associated error probability. For a cat qubit, this error probability depends on the ratio κ1 / κ2, where κ1 is the single-photon loss rate (error rate) of the qubit used to execute the gate, and κ2 is the two-photon loss rate (correction rate) of the qubit used to execute the gate.

[0005] The goal is to realize quantum gates such that error correction becomes more efficient with increasing distance between the error-correcting codes used to execute the error detection scheme (which is related to the number of physical qubits used to encode the logic qubits). Currently, to achieve this effect of cat qubits in repeating codes, the ratio of κ1 / κ2 needs to be controlled at 5 × 10⁻⁶. -3 The following is a summary. However, the current cat qubit has not yet reached this ratio requirement.

[0006] This invention aims to improve this situation. To this end, the applicant proposes a quantum system for executing quantum gates, comprising: a command circuit for selectively applying radiation; N data resonators, where N is greater than or equal to 2, each data resonator having a respective resonant frequency and coupled to the command circuit to stabilize its respective data cat qubit; and an auxiliary resonator having an auxiliary resonant frequency, coupled to the command circuit to stabilize an auxiliary cat qubit, and nonlinearly coupled to the data resonators via the command circuit. The command circuit is arranged to execute quantum gates in such a way that:

[0007] a) While stabilizing the auxiliary cat qubit, radiation with the auxiliary resonant frequency is applied, such that the data resonator and the auxiliary resonator are substantially simultaneously affected by the Hamiltonian generated by the radiation with the auxiliary resonant frequency.

[0008] b) After a selected duration, the radiation having the auxiliary resonant frequency is turned off.

[0009] The advantage of this system is that it can accelerate the measurement speed of the complex in quantum error correction codes, thereby improving the performance of the error correction codes used.

[0010] In various embodiments, the system may exhibit one or more of the following features:

[0011] - The command circuit is further arranged to apply dissipative stabilization to at least one of the data resonators during the operation a), based on the time and the state of the auxiliary qubit.

[0012] - The command circuit is further arranged to: prepare the auxiliary qubit in the "|+>" or "|->" state of the X operator before operation a), and apply the measurement operation of the operator X on the auxiliary resonator after the radiation having the auxiliary resonant frequency is turned off.

[0013] - The command circuit is arranged such that in operation a), N radiations having the auxiliary resonant frequency are applied, such that the data resonator is substantially simultaneously affected by the corresponding Hamiltonian generated by one of the N radiations, wherein an even number of the N radiations are selected to have opposite amplitudes.

[0014] The present invention also relates to a quantum system for executing quantum error-correcting codes, comprising a command circuit for selectively applying radiation, J data resonators (where J is greater than or equal to two, each data resonator having its own resonant frequency and coupled to the command circuit to stabilize its own data cat qubit), and J-1 auxiliary resonators, each auxiliary resonator having an auxiliary resonant frequency, coupled to the command circuit to stabilize the auxiliary cat qubit. Each of the J-1 auxiliary resonators is nonlinearly coupled to two corresponding data resonators of the J data resonators, and each of the J data resonators is connected to at most two auxiliary resonators of the J-1 auxiliary resonators, and the command circuit is arranged to perform J-1 quantum operations in such a way that:

[0015] 1) For each of the J-1 auxiliary resonators, prepare the auxiliary cat qubit in each of the J-1 auxiliary resonators in the "+" or "-" state of the X operator;

[0016] 2) For each of the J-1 auxiliary resonators, while stabilizing the auxiliary cat qubit, radiation having the auxiliary resonant frequency of each auxiliary resonator is applied to the command circuit, such that the data resonator connected to each auxiliary resonator and each auxiliary resonator are substantially simultaneously affected by the Hamiltonian generated by the radiation having the auxiliary resonant frequency of each auxiliary resonator.

[0017] 3) Shut down the radiation at the auxiliary resonant frequency of each of the auxiliary resonators, and

[0018] 4) Measure the operator X on each of the auxiliary resonators.

[0019] In various embodiments, the system may exhibit one or more of the following features:

[0020] - The system is configured to perform the first half of J-1 quantum operations simultaneously on each data qubit on a single auxiliary qubit nonlinearly coupled thereto, and then perform the second half of J-1 quantum operations simultaneously on the remaining auxiliary qubits.

[0021] -The system is configured to perform approximately J-1 quantum operations simultaneously; and

[0022] The command circuit is arranged to periodically repeat J-1 quantum operations.

[0023] The present invention also relates to a method for implementing a quantum gate between N data resonators (where N is greater than or equal to 2) and an auxiliary resonator, each data resonator having its own resonant frequency and carrying its own data cat qubit, the auxiliary resonator having an auxiliary resonant frequency and carrying an auxiliary cat qubit, the auxiliary resonator being nonlinearly coupled to the data resonators, the method comprising the following steps:

[0024] a) While stabilizing the auxiliary cat qubit, radiation with the auxiliary resonant frequency is applied, such that the data resonator and the auxiliary resonator are substantially simultaneously affected by the Hamiltonian generated by the radiation with the auxiliary resonant frequency.

[0025] b) After a selected duration, the radiation having the auxiliary resonant frequency is turned off.

[0026] In various embodiments, the method may exhibit one or more of the following features:

[0027] - Operation a) further includes applying dissipative stabilization to at least one of the data resonators according to the time and the state of the auxiliary qubits;

[0028] The method further includes, before operation a), preparing the auxiliary qubit in a "+" or "-" state of the X operator; and further includes, c) after operation b), applying a measurement operation of the operator X to the auxiliary resonator; and

[0029] Operation a) includes applying N radiations having the auxiliary resonant frequency such that the data resonator is substantially simultaneously affected by a corresponding Hamiltonian generated by one of the N radiations, wherein an even number of the N radiations are selected to have opposite amplitudes.

[0030] The present invention also relates to a method for performing quantum error-correcting codes between J data resonators and J-1 auxiliary resonators, wherein J is greater than or equal to 2, each data resonator has its own resonant frequency and carries its own data cat qubit, each auxiliary resonator has an auxiliary resonant frequency and carries an auxiliary cat qubit, each of the J-1 auxiliary resonators is nonlinearly coupled to two corresponding data resonators of the J data resonators, and each of the J data resonators is connected to at most two auxiliary resonators of the J-1 auxiliary resonators, the method comprising performing J-1 quantum operations in the following manner:

[0031] 1) For each of the J-1 auxiliary resonators, prepare the auxiliary cat qubit in each of the J-1 auxiliary resonators to be in the "+" or "-" state of the X operator.

[0032] 2) For each of the J-1 auxiliary resonators, while stabilizing the auxiliary qubit, radiation having the auxiliary resonant frequency of each auxiliary resonator is applied, such that the data resonator connected to each auxiliary resonator and each auxiliary resonator are substantially simultaneously affected by the Hamiltonian generated by the radiation having the auxiliary resonant frequency of each auxiliary resonator.

[0033] 3) Shut down the radiation having the auxiliary resonant frequency of each of the auxiliary resonators; and

[0034] 4) Measure the operator X on each of the auxiliary resonators.

[0035] In various embodiments, the method may exhibit one or more of the following features:

[0036] - For each data qubit on a single auxiliary qubit nonlinearly coupled to it, the first half of J-1 quantum operations are performed simultaneously, and the second half of J-1 quantum operations are performed simultaneously on the subsequent remaining auxiliary qubits; and

[0037] -J-1 quantum operations are performed substantially simultaneously. Other features and advantages of the invention will become apparent from the following description of the accompanying drawings, which illustrate exemplary embodiments of the invention, wherein:

[0038] - Figure 1 A general schematic diagram of the conventional CNOT gate between the data cat qubit and the auxiliary cat qubit is shown;

[0039] - Figure 2 Explanation is shown Figure 1 A schematic diagram illustrating how quantum gates function;

[0040] - Figure 3 A schematic diagram illustrating the compensation required when interpreting the CNOT gate is shown;

[0041] - Figure 4 A general schematic diagram of the CXX gate between the two data cat qubits and the auxiliary cat qubit according to the present invention is shown;

[0042] - Figure 5 Explanation is shown Figure 4 A schematic diagram illustrating how quantum gates function;

[0043] - Figure 6 The demonstration shows Figure 4 Charts showing the performance of quantum gates;

[0044] - Figure 7 It shows Figure 4 A schematic diagram of the compensation plan for the CXX gate;

[0045] - Figure 8 It shows the use of Figure 4 A general schematic diagram of a quantum measurement system with a quantum gate;

[0046] - Figure 9 Explanation is shown Figure 8 A schematic diagram illustrating how quantum gates in measurement work. (Execution;)

[0047] - Figure 10 It shows the use of Figure 8 A general schematic diagram of a first embodiment of the quantum error correction code (QECC) for a quantum measurement system;

[0048] - Figure 11 Explanation is shown Figure 10 The diagram illustrates how QECC is executed.

[0049] - Figure 12 The demonstration shows Figure 10 The chart showing QECC performance is shown;

[0050] - Figure 13 It shows the use of Figure 8 A general schematic diagram of the second embodiment of the quantum measurement system shown;

[0051] - Figure 14 A general schematic diagram of the CX^N gates between the N data gate qubits and the auxiliary gate qubits according to the present invention is shown;

[0052] - Figure 15 Explanation is shown Figure 15 A schematic diagram illustrating how quantum gates function;

[0053] - Figure 16 It shows Figure 15 The diagram shows the compensation scheme for the CX^N gate;

[0054] - Figure 17 A general schematic diagram of a surface code execution scheme for implementing CX^N quantum gates and / or MX^N measurement operations according to the present invention is shown;

[0055] - Figure 18 and Figure 19 A general schematic diagram is shown of the execution scheme of the surface code for implementing CZ^N quantum gate and / or MZ^N measurement operations according to the present invention;

[0056] - Figure 20 A general schematic diagram is shown illustrating the execution scheme of XZZX code implementing CX^N quantum gates and / or MX^N measurement operations according to the present invention; and

[0057] - Figure 21 A general schematic diagram is shown of the execution scheme of XZZX code for implementing CZ^N quantum gate and / or MZ^N measurement operations according to the present invention.

[0058] The accompanying drawings and the following description primarily contain positive and well-defined features. Therefore, they not only aid in understanding the invention but can also be used to assist in its definition when necessary.

[0059] This invention relates to the realization of high-performance quantum gates in the context of cat qubits.

[0060] It is known that stable cat qubits benefit from noise bias. More precisely, effective error channels (e.g., bit errors or "bit flips") are suppressed exponentially with the "size" (i.e., the average number of photons) of the Schrödinger cat state of the cat qubit.

[0061] Based on current knowledge, this suppression should be applicable to a large class of physical noise processes that have a local effect on the phase space of the harmonic oscillator. This includes, but is not limited to, photon loss, thermal excitation, photon dephasing, and various nonlinearities caused by coupling to the Josephson junction.

[0062] Recent experiments in the context of quantum superconducting circuits have shown that bit-flipping errors are suppressed exponentially with the average number of photons in the cat state.

[0063] Because of this noisy structure, it can be assumed that using a single repeating code is sufficient to correct the remaining error channels. In fact, if only phase jumps need to be corrected, a phase jump error-correcting code can be used. For example, this could be a repeating code defined in the dual basis, or any other classical error-correcting code.

[0064] A repeating code for cat qubits consists of d cat qubits (called data cat qubits) that encode logical information. This is achieved by repeatedly measuring quantum operators that reveal whether errors have occurred on the data cat qubits. This is accomplished using d-1 additional cat qubits (called auxiliary cat qubits). The quantum circuitry for the repeating code requires preparing the auxiliary qubits in a "+" state (which is the Schrödinger cat state for cat-state qubits), applying two CNOT gates between the auxiliary and data cat-state qubits, and measuring the Pauli X operator of the auxiliary qubits (which is the photon number-parity measurement for cat-state qubits).

[0065] The challenge in implementing this repeating code lies in running it on hardware with a fault tolerance threshold. This means that the fidelity of quantum operations in the circuit must be extremely high for the code to have a positive impact.

[0066] More precisely, when the repetition code operates above a threshold, that is, when the fidelity of the physical operations constituting the repetition code is insufficient, the lifetime of the logical information will shorten as the number of physical data qubits d increases: new errors introduced by adding quantum systems cannot be compensated for by error correction strategies.

[0067] On the other hand, when the repetition code operates below the error correction threshold, that is, when the fidelity of the physical operation is high enough, the lifetime of the logical information increases exponentially with the number of data qubits d (also known as the distance d of the repetition code).

[0068] Several articles have proposed using cat qubits in QECC, either with code entirely dedicated to phase errors (“phase flips”) or with code that is far more tolerant of phase errors than to bit errors (e.g., bias noise-customized code of rectangular surface code type or XZZX surface code type). These solutions are described in the following literature: for example, “Repeating cat-state qubits for fault-tolerant quantum computing” by Jérémie Guillaud and Mazyar Mirrahimi, published in Physical Review X, Vol. 9, No. 041053, December 12, 2019; “Bias-hold gates with stable cat qubits” by Shruti et al., published in Science Advances, Vol. 6, No. 34, August 21, 2020; “Error rate and resource overhead of repeating cat qubits” by Jérémie Guillaud and Mazyar Mirrahimi, published in Physical Review A, Vol. 103, No. 042413, April 13, 2021; and Christopher Chamberland… "Building Fault-Tolerant Quantum Computers Using Cascaded Cat Codes" by Chamberland et al., published in PRX Quantum, Volume 3, No. 010329, February 23, 2022; and "Practical Quantum Error Correction Using XZZX Codes and Kerr-Cat Qubits" by Andrew S. Darmawan et al., published in PRX Quantum, Volume 2, No. 030345, September 16, 2021.

[0069] In all known implementations of repeating codes, the synthetic measurement used in QECC (the result of the QECC quantum operator measurement) requires two distinct time steps: the first time step executes the first CNOT gate between the first qubit and the auxiliary qubit, and the second time step executes the second CNOT gate between the second qubit and the auxiliary qubit. When measuring the auxiliary qubit in QECC, two additional time steps are required: before the CNOT gate step, to prepare the auxiliary qubit to be in a selected state; and after the CNOT gate step, to measure the state of the auxiliary qubit.

[0070] This approach has a long history. Its principle is based on the fact that multi-qubit quantum gates can be implemented using only two qubit gates. In many architectures, implementing an N-qubit gate becomes increasingly difficult as N increases, thus a preference is placed on synthesizing N-qubit gates using only two qubit gates. For repeating codes, implementing a cascade of two CNOT gates is superior to implementing a CXX gate. Incidentally, this method is not limited to CXX gates and can be extended to any multi-qubit entangled gate with more than two qubits. In other words, when two or more quantum entities interact, a common approach to performing these multi-entity interactions is to perform multiple two-entity interactions.

[0071] The field of quantum computing is relatively young, especially in the area of ​​cat qubits. From multiple perspectives, it exhibits characteristics typical of a research-intensive field. Therefore, the preferred approach is to adopt improvements that may initially seem incremental but actually require extensive physical verification and industrialization before implementation. In other words, any technology considered state-of-the-art typically remains unchanged until significant obstacles are encountered. This means that known, feasible solutions are difficult to replace.

[0072] Recently, significant progress has been made in cat qubits. In implementing quantum error-correcting codes (QECC) for cat qubits, the applicant used conventional quantum gates, but found that they caused some problems, which will be explained below.

[0073] A cat qubit refers to any implementation of a cat qubit, especially a two-photon dissipative Schrödinger cat qubit. Alternatively, other cat qubits can be used. In some embodiments, the data cat qubit and the auxiliary cat qubit can be implemented using different types of cat qubits.

[0074] Such cat qubits can be stabilized through the following exemplary scheme:

[0075] a) Dissipative stability, its jump operator Where κ2 is the two-photon dissipation rate, a is the photon annihilation operator, and α is the complex number defining the cat qubit. This hopping operator can be used to couple the buffered mode b to the cat qubit and design the Hamiltonian. To achieve this, where b is the photon annihilation operator of mode b, and the pump frequency is 2ω. a -ω b The driving frequency in buffer mode is ω bIt is important to note that dissipative stabilization of two coherent states requires designing a nonlinear transition between the two photons of the first mode a (also known as the cat qubit mode) carrying the stable quantum manifold and one photon of the second mode b (called the buffer mode), and vice versa. This stabilization scheme can suppress bit flipping exponentially with the number of photons in the two coherent states.

[0076] b) Kerhammill Where K is the complex amplitude of the Kerhammillon, α is the photon annihilation operator, and α is the complex number that defines the cat qubit.

[0077] c) Detuned Kerheimian Where K is the complex amplitude of the Ker Hamiltonian, a is the photon annihilation operator, α is the complex number defining the cat qubit, and Δ is the detuning factor.

[0078] d) Two-photon exchange (TPE) Hamiltonian Where g is the complex amplitude of the TPE Hamiltonian, α is the g-photon annihilation operator, α is the complex number defining the cat qubit, and σ is the complex number defining the cat qubit. + It is the order-up operator for a two-level buffer system.

[0079] e) Dissipative compression stabilization scheme with jump operator Where κ² is the two-photon dissipation rate, a is the photon annihilation operator, α is the complex number defining the cat qubit, and ξ = re iθ It is a complex compression parameter. This can stabilize "compressed cat qubits," also known as shift-compressed vacuum states. The hopping operator can stabilize the buffered mode b (with a decay rate of κ) by passing the buffer mode b. b Coupled to cat qubits, and designed coupling Hamiltonians To achieve this, where κ b >>g.

[0080] The above schemes a) to e) can be adjusted to be conditional on the auxiliary quantum bit state.

[0081] The theoretical implementation of establishing a CNOT gate between two stable cat qubits in modes a1 and a2 depends on the following three elements:

[0082] 1) The dissipative stability performance of the control qubit a1, its jump operator Where a1 is the photon annihilation operator of mode a1, and α is the complex number that defines the cat qubit, or the performance of one of the above stable schemes;

[0083] 2) Add a "feedforward" Hamiltonian to the circuit, also known as a "CNOT" Hamiltonian or a "longitudinal" Hamiltonian, with the following formula: Where gCX α is the complex amplitude of the Hamiltonian, a1 is the photon annihilation operator of mode a1, a2 is the photon annihilation operator of mode a2, and α is the complex number that defines the cat qubit.

[0084] 3) The dissipative stability performance of the target qubit a2 depends on the state of the control qubit a1 and its jump operator. Where κ2 is the two-photon dissipation rate; a1 is the photon annihilation operator of mode a1, a2 is the photon annihilation operator of mode a2, and α is a complex number defining the cat qubit, or the performance of one of the above stable schemes a) to e), the form of which is suitable for conditions with the auxiliary qubit state.

[0085] Theoretically, the optimal implementation would use all three elements mentioned above. However, the applicant's experiments show that the first two elements strike a good balance between ease of implementation and quality of results.

[0086] The application of element 3 replaces the conventional stabilization performed on the target qubit to stabilize the cat state. If element 3 is absent, the target qubit will not undergo any cat state stabilization when the gate element is applied. The same principle applies to all embodiments described below: the static stabilization function of the target cat qubit is disabled during gate or measurement operations, and this function can be replaced by element 3 during gate operations.

[0087] In fact, although theoretically CNOT gates can be implemented using only elements 1 and 3 without the feedforward Hamiltonian, in practice, the fidelity of CNOT gates would be very poor without this element, so implementing element 2 is crucial.

[0088] Figure 1 A general schematic diagram of a conventional CNOT gate is shown. In this diagram, both the data cat qubit 4 and the auxiliary cat qubit 6 are controlled by the command circuit 8 and connected together through the CNOT gate 10.

[0089] Generally, cat qubits are resonant modes that remain stable within a specific circuit that receives specific radiation controlled by a command circuit. Therefore, a cat qubit can typically be designated as a resonator with a specific resonant frequency (i.e., the frequency of the cat qubit), which is controlled by the command circuit. Each cat qubit can be controlled by a specific command circuit, or a single circuit can be arranged to control all cat qubits in a given circuit. In the example described herein, a single command circuit 8 controls all data cat qubits 4 and auxiliary cat qubits 6. Furthermore, when referring to the resonator in which the cat qubits are encoded, the cat qubit can be referred to as a "mode".

[0090] Figure 2A block diagram of the operation of the CNOT gate (also known as the CX gate) is shown. In the first operation 200, the feedforward Hamiltonian (element 2) is enabled, and the stability of the data cat qubit 4 is disabled. Since the auxiliary cat qubit 6 requires stability, element 1 is always applied. Alternatively, operation 200 may include enabling the time-dependent dissipation on both the feedforward Hamiltonian (element 2) and the data cat qubit (element 3), or only activating the time-dependent dissipation on the data cat qubit (element 3). After a selected duration of operation 200 (i.e., the gate duration, discussed below), the feedforward Hamiltonian (or a variant of operation 200 thereof) is disabled in operation 210, and the stability of the data cat qubit 4 is restored.

[0091] The stability of the auxiliary cat qubit and the data cat qubit can be achieved using one of the five stabilization schemes mentioned above.

[0092] The applicant’s earlier work showed that the method used to design feedforward Hamiltonians also introduced a very strong deterministic spurious effect. Therefore, it is necessary to eliminate this deterministic spurious effect in order to achieve arbitrary results.

[0093] More precisely, when developing terms for feedforward Hamiltonians, the term... The detuning of the data cat's qubits does not require physical implementation; rather, it can be achieved by redefining the α value of the data cat's qubits in software. Meanwhile, the project... It is a linear shift of the auxiliary cat qubit, which is a conventional implementation method.

[0094] The most difficult part of entanglement to achieve in interactions is Typically, modes a1 and a2 are adjacent on the chip and both participate in nonlinear circuit elements (usually an asymmetric threaded superconducting quantum interference device, or "ATS," as described in the 2020 paper "Exponential suppression of bit flips in oscillator-coded qubits" by Lescanne R. et al. in *Nature Physics*), allowing this term to be pumped ∈ p (t)=∈ p cos(ω1t) provides direct drive. However, applying a pump at this frequency produces a linear drive on the auxiliary cat qubit a1, which requires precise compensation.

[0095] Figure 3This illustrates the effect on the Wigner function of the auxiliary cat qubit. In the stable state without a feedforward Hamiltonian, the Wigner distribution of the auxiliary cat qubit is shown on the eastward axis. When the feedforward Hamiltonian is applied, it causes a displacement as indicated by the solid arrows. This displacement, combined with the stability of the auxiliary cat qubit, leads to decoherence and / or leakage in the cat qubit code space. Therefore, a compensation (as indicated by the dashed arrows) needs to be designed. For example, this can be achieved by enabling a resonant drive on the auxiliary cat qubit, precisely compensating for the pumping effect of the entangled portion of the feedforward Hamiltonian.

[0096] While such compensation is feasible, it becomes a burden when executing multiple CNOT gates because it requires designing multiple CNOT gates almost simultaneously without creating antagonistic effects.

[0097] In trying to find a solution for correctly implementing the feedforward Hamiltonian, the applicant discovered that the "N-1 interaction between two quantum entities" paradigm could be improved, contrary to everyone's expectations, and that the CNOT gate could be directly implemented without the aid of successive CNOT steps.

[0098] More specifically, the applicant has discovered a method in quantum error-correcting codes to shorten the error correction cycle when the gates are used to implement quantum operator measurements, thereby improving the performance (threshold), as described below. As a further advantage, the applicant has found that this can be done simultaneously with solving the feedforward Hamiltonian compensation problem.

[0099] Figure 4 A general schematic diagram of a quantum system for performing the CNOTNOT quantum gate 2 according to the present invention is depicted. In the following text, the terms "CNOTNOT gate," "CNOTNOT quantum gate," "CXX gate," or "CXX quantum gate" are used interchangeably. The same principle applies to the terms "MNOTNOT operation," "MNOTNOT quantum operation," "MXX operation," or "MXX quantum operation."

[0100] In the example described here, the quantum system 2 used to execute the CXX gate connects two data cat qubits 4 and one auxiliary cat qubit 6. In the context of this invention, "data cat qubit" and "target qubit" are interchangeable, referring to the physical qubit that needs to read, control, or correct data. Similarly, "control cat qubit" and "auxiliary cat qubit" are interchangeable, referring to the physical qubit used to read the photon parity of the data cat qubit to achieve the aforementioned purpose. In other words, a so-called data cat qubit refers to a physical qubit containing the quantum information that QECC attempts to protect. A so-called auxiliary cat qubit refers to the complement of the data cat qubit in QECC, i.e., the physical qubit used to detect errors in the data cat qubit.

[0101] The theoretical implementation of the CXX gate between the three stable cat qubits in modes a1, a2, and a3 depends on the following three elements:

[0102] 1) To demonstrate the dissipative stability of the control qubit a1, its jump operator is: Where a1 is the photon annihilation operator of mode a1, and α is the complex number that defines the auxiliary cat qubit;

[0103] 2) Add a "feedforward" Hamiltonian to the circuit, also known as a "CXX" Hamiltonian, whose formula is: Where T is the CXX gate duration, a1 is the photon annihilation operator for the auxiliary cat qubit 6, a2 and a3 are the photon annihilation operators for the data cat qubit, and α is the complex number defining the cat qubit. For simplicity, the photon populations of the auxiliary cat qubit and the data cat qubit are equal, but they can be different: α1 2 Corresponding to a1, α2 2 Corresponding to a2, a3 2 Corresponding to a3. The feedforward Hamiltonian can be expressed in terms of frequency. The pump is used to achieve this, and it acts as a pull force applied to data cat qubit 4. During the application of the feedforward Hamiltonian, the stability of data cat qubit 4 is turned off.

[0104] 3) One or more dissipative stabilizers can be used as drag to one or two data cat qubits 4, depending on the state of the auxiliary cat qubit a1, whose jump operator is: and jump operator and This can be achieved by providing corresponding buffer modes b2 and b3 for Datacat qubit 4, and by engineering the Hamiltonian. And at a frequency of and Apply a pump at a frequency of This is achieved by applying a driver at the specified location. For jump operators... The same effect can also be achieved by replacing a2 and b2 in the above equation with a3 and b2.

[0105] In theory, the optimal implementation uses all three elements. However, the applicant's experiments show that the first two elements offer a good trade-off between ease of implementation and quality of results. Similar to the case of the CNOT gate described above, stabilization schemes 1) and 3) can be implemented together with any of the five stabilization schemes mentioned above.

[0106] As shown in the figure, two data cat qubits 4 are connected to the auxiliary cat qubit 6 via a CXX gate 11. However, unlike existing techniques that use two consecutive CNOT gates, there is only a single CXX gate 11 here, and as shown in the figure, both data cat qubits 4 are connected to the CXX gate 11 simultaneously, so the CXX gate is executed synchronously on both data cat qubits 4.

[0107] Figure 5 Showing Figure 4 A block diagram of the CXX gate operation. In the first operation 500, the feedforward Hamiltonian (element 2) on both the data cat qubit 4 and the auxiliary cat qubit 6 is enabled. Since the auxiliary cat qubit needs to be stable, element 1 is always applied, while the stability of the data cat qubit 4 is disabled. Alternatively, operation 500 may include enabling the feedforward Hamiltonian (element 2) and time-dependent dissipation on the data capture qubit. as well as (Element 3), or simply activate the time-dependent dissipation on the data capture qubit. and (Element 3). After a selected duration of operation 500 (i.e., the gate duration, which will be discussed below), the feedforward Hamiltonian (or a variant of operation 500) is turned off in operation 510, and the stability of data qubit 4 is restored.

[0108] To demonstrate that the CXX strategy delivers superior overall performance, the analytical error of the CXX gate will be calculated and compared with the performance of two consecutive CX gates. Figure 6 The numerical simulation shown is used to verify the analytical calculation results.

[0109] In the following demonstration, the CXX gate is again considered to be implemented between three cat qubits, with modes a1, a2, and a3. Qubits a2 and a3 are the target qubit 4, and a1 is the control qubit. The CXX gate between the three cat qubits can be implemented using the master equation. The implementation is achieved, where a_1, a2, and a3 are the annihilation operators for the control qubit and the two target qubits, respectively. and

[0110] In the cat qubit architecture, the quality of the hardware is measured by a ratio of two time scales: time 1 / κ2, where κ2 is the two-photon dissipation rate of the stable qubit; and time 1 / κ1, where κ1 is the single-photon loss rate. Other error sources exist, such as the rate of κ... Ф Phase shifts, thermal excitations, self-Kerr and cross-Kerr interactions, or other poor couplings with other quantum systems near the memory, etc.

[0111] However, single-photon loss is the primary error mechanism, and for clarity, only this physical error mechanism will be considered below. In general, all of the following holds true by replacing the ratio κ1 / κ2 with the sum of the error mechanism rates divided by κ2.

[0112] To analyze the master equation, a hybrid basis is used, where the control qubit is described by a shifted Fock basis, and the two target qubits are described by a standard Fock basis. The Hamiltonian can be represented as follows:

[0113]

[0114] In the rotating coordinate system and the fully displaced Focke's equations, the master equation is given by the following:

[0115]

[0116] Among them, the jump operator on the target qubit is composed of and Provided.

[0117] By considering only the first excited state of each of the three qubits, neglecting internal coupling, and ignoring the last term of the two target jump operators, the master equation becomes:

[0118]

[0119] By adiabatic elimination of the excited state, the following master equation is obtained:

[0120]

[0121] By integrating from 0 to time T and neglecting higher-order terms, the equation is obtained as follows:

[0122]

[0123] By returning to the initial frame and removing the single perfect CXX, the faulty channel of CXX can be obtained:

[0124]

[0125] By performing integration and focusing on the diagonal terms, the Z-error rate of CXX is obtained from... (where the second item corresponds to a non-adiabatic error) and Provided.

[0126] Figure 6 Showing the use Figure 4 The results of simulating quantum circuits using quantum gates were compared with the non-adiabatic errors identified above. The short dashed lines represent values. The reciprocal of the value κ2 (at different α) 2 The curve showing the change of value (under α). For each α value... 2 The simulated error levels for specific values ​​of κ2 (circles, triangles, and squares) were plotted. The plot shows that the theoretically determined non-adiabatic errors are expected to be met in practical implementations, and the error budget calculations below demonstrate these advantages. In this simulation, κ1 was set to 0 because its effects are well understood and do not require checking.

[0127] To specifically evaluate the gain brought by using the CXX gate according to the present invention, we will compare the error budgets of two cascaded CNOT gates. The total time for implementing CXX or two cascaded CNOT gates is fixed at T. In the following, The average number of photons in the cat state.

[0128] *Incorrect budget for cascading CNOT gates

[0129] The error rate of the auxiliary qubit is

[0130] The error rate of two data qubits is

[0131] The correlation error rate between the data qubit and the auxiliary qubit is

[0132] Therefore, the total error rate is

[0133] *CXX door's erroneous budget

[0134] The error rate of the auxiliary qubit is

[0135] The error rate of two data qubits is

[0136] The correlation error rate between the data and the auxiliary qubits is

[0137] Therefore, the total error rate is

[0138] Therefore, for any time T, the error budget of CXX is less than the error budget of two cascaded CNOT gates.

[0139] This means that the CXX gate according to the present invention can achieve two advantages: either faster operation while maintaining the same error rate as two CNOT gates, or higher fidelity while maintaining the same time consumption as two CNOT gates.

[0140] Another advantage of implementing CXX in a single step instead of using two consecutive CNOT gates is that the aforementioned compensation problem can be solved by selecting a specific phase arrangement of the feedforward Hamiltonian. More precisely, any form of The Hamiltonian can produce the expected effect. The applicant found that for H CX The specific case of the "-" sign in the last term of the formula is determined by the corresponding term of the feedforward Hamiltonian. and Unexpected displacements (such as) Figure 3 (As shown) they exactly complement each other, such as Figure 7 As shown, no further engineering design compensation is needed. In practice, due to experimental limitations, very small compensation may still be required, but since most of the compensation has been offset, engineering design is much easier.

[0141] In alternative embodiments, compensation can be engineered using conventional methods.

[0142] Figure 8 Demonstrated the use of Figure 4 A general diagram for measuring XX syndrome using CXX quantum gates. Figure 8 The system and Figure 4 The system is very similar to that in the previous one, except that it includes a control qubit preparation process 12 and a control qubit measurement process 14.

[0143] By providing these elements, the measurement of auxiliary qubit 6 can measure the joint photon number parity between data qubits 4 (which is a stabilizer for the repeating code on the two-component cat qubit). It is worth noting that the measurement operation of this invention is the same as that of the CXX gate, but is not limited to "XX" measurements. In other words, another gate using the same structure (i.e., the synchronous CNOT gate) but with different markings according to a specific convention is still within the scope of this invention. Therefore, the wording of CXX is intended to be easier to understand in light of existing standards, but the real goal of this invention is to implement a gate (which can flip the state of the control bit based on whether the joint photon number parity of the target qubit is odd). The same principle applies to MXX for any measurement operation that relies on the CXX gate or its equivalent.

[0144] Figure 9 The execution was displayed Figure 8 A flowchart of the measurement operation. (and) Figure 5 Similarly, the difference lies in that operation 900, which prepares the auxiliary cat qubit 6 to be in a "+" state, precedes the feedforward Hamiltonian operation 910, followed by operation 920, which measures the auxiliary cat qubit 6. In operation 910, if the embodiment uses conditional dissipative stabilization... and Then conditional dissipative stability and It will be activated simultaneously. In operation 900, you can also prepare to control qubit 6 to put it in the "|->" state.

[0145] therefore, Figure 9 As shown, the XX syndrome measurement operation according to the present invention is completed within three time steps, while the conventional XX syndrome measurement requires four time steps due to continuous CNOT operations. The advantage of reducing one time step has been demonstrated above.

[0146] Figure 10 Demonstrated the use of Figure 8 The overall timeline of the QECC measurement operation is shown. For simplicity, only the auxiliary cat qubit 6, its preparation process 12, and its measurement process 14 are shown. The command circuit 8 and the data cat qubit 4 are deliberately omitted; the data cat qubit is represented by the MXX operation circuitry.

[0147] When two operations 11 are vertically aligned, it indicates that these operations are performed simultaneously. When they are vertically offset, it indicates that they are offset in time, and one operation is executed before the other. As mentioned above, performing the MXX operation is achieved by turning on the corresponding feedforward Hamiltonian while turning off the conventional stability (and possibly the corresponding conditional dissipation stability) on the datacat qubit 4.

[0148] According to the present invention, the QECC is configured such that each data cat qubit 4 is connected to two different auxiliary cat qubits 6 through its own MXX operation, except for the data cat qubits 4 located at both ends of the QECC. Figure 10 The distance between the codes shown is 3. In another embodiment, the data cat qubits 4 can be arranged in a circle such that all data cat qubits 4 are connected to two different auxiliary cat qubits 6.

[0149] Although Figure 10 The example shown is a QECC with a distance of 3, but this QECC can have a larger distance (e.g., 5 or more). In this case, the MXX operation is performed synchronously in the alternative. This means that in the first time step, each data cat bit performs an MXX operation, but only establishes a connection with one of its non-linearly coupled auxiliary cat bits. After the first time step is completed, the system performs the MXX operation on the remaining auxiliary cat bits.

[0150] from Figure 10 Initially, a QECC with a distance of 5 is used, which means that the next row of auxiliary cat qubits will be aligned with the first row of auxiliary cat qubits, and the last row of auxiliary cat qubits will be aligned with the second row of auxiliary cat qubits.

[0151] Figure 11 Showing Figure 10 Operational block diagram of QECC with a mid-range distance J. In operation 1100, the MXX operation is performed for each auxiliary cat qubit with an odd number of rows (2i+1). Then, in operation 1110, the MXX operation is performed for each auxiliary cat qubit with an even number of rows 2i.

[0152] Figure 12 Two charts are shown, comparing the performance of traditional QECC (left) and QECC according to the present invention (right). In the QECC design evaluation, the error tolerance threshold can be viewed as the east-west axis; the error rate improves with increasing code distance.

[0153] exist Figure 12 In this case, an 8-photon cat qubit was used, and it was assumed that all operations took the same amount of time, 1 / κ2. The logical Z error rate was plotted as κ1 / κ2 as a function of different code distances. The left side uses two CNOT gates to detect the Z error rate, and the right side uses CXX gates. The threshold was improved by approximately 25%, from 0.289 to 0.366, which is a significant improvement.

[0154] Figure 13 Showing Figure 10A less preferred alternative to the QECC code in this embodiment. In this example, all CXX operations 11 are synchronous. While this may seem better at first glance, it is actually... Figure 10 Compared to the implementation shown, the proposed solution may have potential flaws when implemented in the real world.

[0155] First, we will demonstrate this by distinguishing between two implementation schemes. In the first implementation scheme, the CXX gate is implemented using elements 1 and 2. In the second implementation scheme, the CXX gate is also implemented using element 3.

[0156] In the first implementation scheme, Figure 13 The sequence shown is technically feasible. However, the fact that the data cat qubit acts as the target qubit for the two auxiliary qubits presents some problems. This is because the first implementation requires turning on the drivers to obtain two different frequencies (the frequencies of the two different auxiliary cat qubits) of feedforward Hamiltonians. If multiple drivers of different frequencies are turned on simultaneously, there is a risk of crosstalk, frequency congestion, and each feedforward Hamiltonian needs to be driven with a low amplitude to prevent excessive energy being supplied to the circuit, which could lead to increased circuit temperature.

[0157] The applicant has determined that, at this stage, it is best to use offset feedforward Hamiltonians, each with full power. Therefore, Figure 10 The embodiments described are preferred.

[0158] In the second implementation scheme (adding element 3), using the current implementation scheme of CXX described above, it is impossible to execute two CXX gates simultaneously on the same data qubit. To achieve this, a more complex dissipation mechanism is required, and the applicant is currently unaware of how to design such a mechanism. Therefore, Figure 10 The embodiments described above are again preferred.

[0159] Although Figures 4 to 6 The previous embodiment used only two data cat qubits, but the applicant discovered that CXX can be transformed into CX^N by using N data cat qubits (each qubit connected to the same auxiliary cat qubit). Like CXX, CX^N can be implemented in a single time step. Figure 14 A general diagram of the CX^N gate is shown. Data cat qubits 4(1) to 4(N) are connected to auxiliary cat qubit 6 via CX^N gate 16.

[0160] Figure 15 Showing Figure 14The operational block diagram of the CX^N gate. The first operation 1500 includes enabling N feedforward Hamiltonians (element 2) on each data cat qubit 4 (1) to 4 (N) and auxiliary cat qubit 6. Since the auxiliary cat qubit needs to be stable, element 1 is always applied. Alternatively, operation 1500 may include enabling the feedforward Hamiltonians (element 2) and time-dependent dissipation on the data cat qubits. and (Element 3), or simply activate the time-dependent dissipation on the data cat qubits. and (Element 3). After the duration selected in operation 1500 (i.e., the gate duration, which will be discussed below), the feedforward Hamiltonian (or a variant of operation 1500) is turned off in operation 1510.

[0161] You can select the pump phase to enable N feedforward Hamiltonians so that no compensation is required. Figure 16 One possible compensation scheme is shown, in which N feedforward Hamiltonians are selected for pumping, such that the driver is precisely compensated on the auxiliary cat qubit, as indicated by the arrows in the figure. With all feedforward Hamiltonians having the same amplitude, the reference... Figure 7 This description boils down to the fact that the number of feedforward Hamiltonians in the "+" state is equal to the number of feedforward Hamiltonians in the "-" state. As a possible example, The alternating signs ensure compensation. Even if the amplitudes are not strictly equal, the phase compensation scheme produces excellent results in first-order analysis. When N is odd, one solution is to make the even number of feedforward Hamiltonians cancel each other out and design a single (or multiple) conventional compensations.

[0162] CZ^N gates can also be implemented within a single time step without cascading N CZ gates. It's worth noting that the implementation of CZ gates is quite simple: the dissipation of the data qubits and auxiliary qubits remains constant, and the gates are connected via a frequency of ω. a -ω b The pump starts the so-called beam splitter Hamiltonian. Unlike CNOT gates, this pump does not produce deterministic spurious effects on the control qubits, and therefore requires no compensation.

[0163] While the above primarily focuses on the case of repeating codes, the CX^N gates and MX^N operations of this invention can be generalized to all codes implementing X or Z stabilizer measurements. The following section will demonstrate how to apply them to surface codes and XZZX codes.

[0164] The surface code includes measurements of the weighted-4X stabilizer and the weighted-4Z stabilizer performed sequentially. In existing implementations, both the X-stabilizer and Z-stabilizer measurements are performed in parallel using four gates, with a dissipation time of four time steps.

[0165] Figure 17 Demonstrated the use of CX 4 (and CX at the boundary) 2 The CNOT gate enables X-stabilizer measurements. If the CNOT gate implementation uses only the feedforward Hamiltonian and the dissipation on the auxiliary cat qubit (i.e., elements 1 and 2), X-stabilizer measurements can be achieved in a single time step. If the CNOT gate is implemented using the time-dependent dissipation on the data cat qubit (element 3), then two time steps are required.

[0166] In this diagram, the data cat qubits are represented by medium-sized circles and connected to the auxiliary cat qubits via CNOT symbols (circles with crosses indicate the data cat qubits play a target role), which are represented by small circles (large circles surrounding the small circles indicate the auxiliary cat qubits play a control role). This indicates that half of the auxiliary cat qubits are inactive because they are used for the operation of the Z stabilizer.

[0167] Figure 18 This demonstrates the first successful Z-stabilizer measurement using CXX (and CX at the boundary) gates. If the CNOT gate implementation uses only the feedforward Hamiltonian and dissipation on the auxiliary cat qubit (i.e., elements 1 and 2), the Z-stabilizer measurement can be achieved in a single time step. If the CNOT gate is implemented using time-dependent dissipation on the data cat qubit (element 3), two time steps are required.

[0168] Figure 19 Demonstrated the use of CZ 4 (And CZZ at the boundary) gates provide an alternative for Z-stabilizer measurements. Regardless of which of these three "elements" is used to perform this gate, it can be implemented within a single time step. In the figure, the CZ operation is represented by black dots.

[0169] Therefore, using conditionally stable datacat qubits (elements 1, 3, and optional element 2), the X stabilizer can be measured over two time steps, and the Z stabilizer can be measured over a single time step using CZ. 4 The gate is used for measurement. This requires a total of three time steps. Using unstable datacat qubits (elements 1 and 2 only), both the X-stabilizer and the Z-stabilizer can be measured in a single time step (for the latter, CZ is used). 4 (Or CXX gate). This requires a total of two time steps.

[0170] CX^N and CZ^N gates can also be used for XZZX code. Here, each auxiliary qubit measures the XZZX stabilizer. In existing implementations, this requires four time steps to complete.

[0171] Figure 20 and Figure 21 Two sequences that achieve the XZZX measurement are shown. Figure 20 The first sequence shown can be implemented using conditionally stable datacat qubits (elements 1, 3, and optional element 2), which requires splitting the CXX operation into two time steps. If unstable datacat qubits (elements 1 and 2 only) are used, the CXX operation can be performed in a single time step. Figure 21 The second sequence shown uses the CZZ operation, which can be executed within a single time step regardless of which of the three elements is used. This allows for a maximum of three time steps using stable datacat qubits (elements 1, 3, and optional element 2). Using unstable datacat qubits (elements 1 and 2 only), both CXX and CZZ can be executed within a single time step, resulting in a total of two time steps.

[0172] Similar to CXX gates, MXX operations, and QECC, this invention allows for saving at least one time step.

[0173] The above only considers the case of a CXX gate between two cat qubits. However, the applicant has found that, in principle, the present invention is sufficient to function as long as the target qubit is a cat qubit, and the control qubit can typically be a conventional qubit, or any two-level system with two quantum states |0> and |1>. In some embodiments, the conventional control qubit can be a transmission qubit or a flux qubit.

[0174] In this scheme, the CXX gate is implemented using Hamiltonians. This is implemented where a2 and a3 are annihilation operators for the two target cat qubits. Since the control qubit is a conventional qubit, no stabilization is performed on it. In some implementations, the two target cat qubits are stabilized via conditional dissipation, which depends on the state of the auxiliary qubit, determined by a hopping operator. and To describe.

[0175] Alternatively, any of the above stabilization schemes a) to e) can be used, the form of which is suitable for conditions based on the auxiliary qubit state.

[0176] In this scheme, no compensation is required for the control qubits. However, it is still worthwhile to study the implementation of this multi-qubit quantum gate instead of a series of two-qubit gates because it can achieve a shorter gating time.

[0177] The same directness and generality apply to the case of CX^N quantum gates between conventional control qubits and N target cat qubits.

[0178] As previously described in the two-target-cat-qubit scheme, the CX^N gate between a regular qubit and N cat qubits can be applied to quantum error-correcting codes.

Claims

1. A quantum system for performing a quantum gate, comprising: Command circuit (8) for selective application of radiation; N data resonators (4), with N greater than or equal to 2, each having a respective resonance frequency and coupled to said command circuit (8) for stabilizing a respective data cat quantum bit; and an auxiliary resonator (6) having an auxiliary resonance frequency, coupled to said command circuit (8) for stabilizing an auxiliary cat quantum bit, and nonlinearly coupled to said data resonators (4) through said command circuit (8), said command circuit (8) being arranged to perform a quantum gate by: a) applying radiation having said auxiliary resonance frequency while stabilizing said auxiliary cat quantum bit, so that said data resonators (4) and said auxiliary resonator (6) are simultaneously affected by the Hamiltonians generated by said radiation having said auxiliary resonance frequency; b) switching off said radiation having said auxiliary resonance frequency after a selected duration.

2. The quantum system of claim 1, wherein, said command circuit (8) being further arranged to apply dissipative stabilization to at least one of said data resonators (4) during said operation a) as a function of time and of the state of said auxiliary quantum bit.

3. Quantum system according to one of the preceding claims, wherein said command circuit (8) being further arranged to cause said auxiliary quantum bit to be prepared in the "|+>" or "|->" state of the X operator before operation a) and to apply a measurement operation of the operator X to said auxiliary resonator (6) after switching off said radiation having said auxiliary resonance frequency.

4. Quantum system according to one of the preceding claims, wherein, said command circuit (8) being arranged in operation a) to apply N radiations having said auxiliary resonance frequency, so that said data resonators (4) are simultaneously affected by the respective Hamiltonians generated by one of said N radiations, and an even number of said N radiations is selected to have opposite amplitudes.

5. A quantum system for executing a quantum error-correcting code, comprising: Command circuit (8) for selective application of radiation; J data resonators (4), with J greater than or equal to 2, each having a respective resonance frequency and coupled to said command circuit (8) for stabilizing a respective data cat quantum bit; and J-1 auxiliary resonators (6), each having an auxiliary resonance frequency, coupled to said command circuit (8) for stabilizing an auxiliary cat quantum bit, each of said J-1 auxiliary resonators being nonlinearly coupled to two respective data resonators of said J data resonators, and each of said J data resonators being connected at most to two of said J-1 auxiliary resonators, said command circuit (8) being arranged to perform J-1 quantum operations by: 1) causing the auxiliary cat quantum bit in each of said J-1 auxiliary resonators to be prepared in the "+" or "-" state of the X operator for each of said J-1 auxiliary resonators; 2) for each of said J-1 ancilla resonators, while stabilizing said ancilla cat qubit, applying said radiation having the ancilla resonant frequency of said each ancilla resonator to said command circuit (8) so that the data resonator (4) connected to said each ancilla resonator and said each ancilla resonator (6) are simultaneously affected by the Hamiltonian generated by said radiation having the ancilla resonant frequency of said each ancilla resonator; 3) switching off said radiation having the ancilla resonant frequency of said each ancilla resonator; and 4) performing a measurement on the operator X on said each ancilla resonator.

6. The quantum system of claim 5, wherein, The system is arranged to perform the first half of said J-1 quantum operations simultaneously for each data qubit on a single ancilla qubit to which it is nonlinearly coupled, and subsequently to perform the second half of said J-1 quantum operations simultaneously on the remaining ancilla qubits.

7. The quantum system of claim 5, wherein, The system is arranged to perform said J-1 quantum operations substantially simultaneously.

8. The quantum system of one of claims 5 to 7, wherein, The command circuit (8) is arranged to repeat said J-1 quantum operations periodically.

9. A method of performing a quantum gate between N data resonators (4) and an ancilla resonator (6), where N is greater than or equal to 2, each data resonator (4) having a respective resonant frequency and carrying a respective data cat qubit, and said ancilla resonator having an ancilla resonant frequency and carrying an ancilla cat qubit, said ancilla resonator (6) being nonlinearly coupled to said data resonators (4), said method comprising the operations of: a) while stabilizing said ancilla cat qubit, applying radiation having said ancilla resonant frequency so that said data resonators (4) and said ancilla resonator (6) are simultaneously affected by the Hamiltonian generated by said radiation having said ancilla resonant frequency; b) switching off said radiation having said ancilla resonant frequency after a selected duration.

10. The method of claim 9, wherein, Operation a) further comprises applying dissipative stabilization to at least one of said data resonators (4) as a function of time and state of said ancilla qubit.

11. The method of claim 9 or 10, further comprising, prior to operation a), preparing said ancilla qubit in an X operator "+" or "-" state; and, after operation b), further comprising c) applying a measurement operation on the operator X to said ancilla resonator (6).

12. The method according to one of claims 9 to 11, wherein Operation a) comprises applying N radiations having said ancilla resonant frequency so that said data resonators (4) are simultaneously affected by the respective Hamiltonians generated by one of said N radiations, wherein an even number of said N radiations are selected to have opposite amplitudes.

13. A method of performing a quantum error correction code between J data resonators (4) and J-1 ancilla resonators (6), wherein J is greater than or equal to 2, each data resonator (4) has a respective resonant frequency and carries a respective data cat quantum bit, each ancilla resonator has an ancilla resonant frequency and carries an ancilla cat quantum bit, each of the J-1 ancilla resonators is non-linearly coupled with two respective data resonators of the J data resonators, and each data resonator of the J data resonators is connected at most with two ancilla resonators of the J-1 ancilla resonators, the method comprising performing J-1 quantum operations by: 1) for each ancilla resonator of the J-1 ancilla resonators, preparing an ancilla cat quantum bit in the "+” or "-” state of the X operator in the each said ancilla resonator of the J-1 ancilla resonators; 2) for each ancilla resonator of the J-1 ancilla resonators, while stabilizing the ancilla quantum bit, applying a radiation having the ancilla resonant frequency of the each ancilla resonator (6) such that the data resonators (4) connected to the each ancilla resonator (6) and the each ancilla resonator (6) are simultaneously affected by a Hamiltonian resulting from the radiation having the ancilla resonant frequency of the each ancilla resonator (6); 3) turning off the radiation having the ancilla resonant frequency of the each ancilla resonator (6); and 4) performing a measurement on the operator X on the each ancilla resonator (6). The first half of the J-1 quantum operations is performed simultaneously for each data quantum bit on a single ancilla quantum bit with which it is non-linearly coupled, and wherein the second half of the J-1 quantum operations is subsequently performed simultaneously on the remaining ancilla quantum bits.

14. The method of claim 13, wherein, The J-1 quantum operations are performed substantially simultaneously.

15. The method of claim 13, wherein, The J-1 quantum operations are performed substantially simultaneously.

Images