A system for executing quantum gates and the operation of quantum error correction codes using these quantum gates.

Abstract

A system for executing a quantum gate comprises a command circuit (8) for selectively applying radiation, two or more N data resonators (4), and an ancilla resonator (6) having an ancilla resonant frequency. Each data resonator (4) has its own resonant frequency and is connected to the command circuit (8) to stabilize its respective data cat qubit. The ancilla resonator (6) is connected to the command circuit (8) to stabilize an ancilla cat qubit and is nonlinearly connected to the data resonators (4) via the command circuit (8). The command circuit (8) is configured to execute a quantum gate by a) applying radiation having an ancilla resonant frequency while stabilizing the ancilla cat qubit so that the data resonators (4) and ancilla resonators (6) follow a Hamiltonian resulting from the radiation having an ancilla resonant frequency substantially simultaneously, and b) stopping the radiation having an ancilla resonant frequency after a predetermined period. This principle can be extended to execute quantum error correction codes.

Description

[Technical Field]

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

[0002] To extract coupling information from multiple data qubits, conventional quantum circuits sequentially execute two-qubit gates (typically CNOT or CZ gates) between an ancilla qubit and multiple data qubits, and then measure the ancilla qubit. This operation, commonly called a syndrome measurement in the art, requires these two-qubit gates to be executed in series.

[0003] The realization of these quantum gates is crucial for error detection and execution of quantum error correction codes (QECCs). QECCs are currently considered the only way to build reliable and practical quantum chips. The general idea of ​​QECCs is to encode logical qubits using multiple (at least two) physical qubits 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 state of the qubits themselves or by post-processing the results of a quantum algorithm that includes these qubits.

[0004] The properties used to evaluate the quality of a quantum gate are its execution time (the time the gate operates) and the associated error probability. In the case of 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. [Overview of the project]

[0005] The goal is to realize a quantum gate where error correction becomes more effective as the distance of the error correction code (which is related to the number of physical qubits used to encode the logical qubit) used to perform the error detection scheme increases. Currently, to achieve this effect with cat qubits in iterative codes, the ratio κ1 / κ2 must be 5 × 10⁻¹⁰. -3 It needs to be less than [a certain value]. This kind of ratio has not been achieved with current cat qubits.

[0006] The present invention aims to improve this situation. To this end, the applicant proposes a quantum system for performing quantum gates. The quantum system comprises a command circuit for selectively applying radiation, N data resonators, 2 or more of which are data resonators, and an ancilla resonator having an ancilla resonant frequency. Each data resonator has its own resonant frequency and is connected to the command circuit to stabilize its respective data cat qubit. The ancilla resonator is connected to the command circuit to stabilize the ancilla cat qubit and is nonlinearly connected to the data resonators via the command circuit. The command circuit applies radiation having an ancilla resonant frequency while stabilizing the ancilla cat qubit so that the data resonators and ancilla resonators substantially simultaneously follow the Hamiltonian resulting from the radiation having an ancilla resonant frequency. b) After a predetermined period, cease radiation having an ancilla resonant frequency. It is configured to execute quantum gates.

[0007] This system has the advantage of being able to speed up syndrome measurement in QECC, thereby improving the performance of the error correction codes used.

[0008] In various embodiments, this system may present one or more of the following features: The command circuit is further configured to apply dissipative stabilization to at least one data resonator in operation a) depending on time and the state of the Ancilan cat qubit. · The instruction circuit is further configured to prepare the ancilla cat qubit in the state |+> or state |-> of the X operator before operation a), and after stopping the radiation having the ancilla resonance frequency, apply a measurement operation of the operator X to the ancilla resonator. · The instruction circuit is configured to apply N radiations having the ancilla resonance frequency in operation a), and the data resonators substantially follow each Hamiltonian resulting from one of the N radiations simultaneously, and among the N radiations, the even-numbered radiations are selected to have opposite amplitudes.

[0009] The present invention also relates to a quantum system for performing a quantum error-correcting code. The quantum system includes an instruction circuit for selectively applying radiation, J data resonators where J is two or more, and J - 1 ancilla resonators. Each data resonator has its respective resonance frequency and is connected to the instruction circuit to stabilize its respective data cat qubit. Each ancilla resonator has an ancilla resonance frequency and is connected to the instruction circuit to stabilize the ancilla cat qubit. Each of the J - 1 ancilla resonators is non-linearly connected to two of the respective J data resonators, and each data resonator of the J data resonators is connected to two auxiliary resonators of at most J - 1 ancilla resonators. The instruction circuit (8) 1) For each ancilla resonator of the J - 1 ancilla resonators, prepare the ancilla cat qubit in each ancilla resonator of the J - 1 ancilla resonators in the state + or state - of the X operator; 2) For each ancilla resonator of the J - 1 ancilla resonators, while stabilizing the ancilla cat qubit, apply radiation having the ancilla resonance frequency of each ancilla resonator to the instruction circuit (8), and cause the data resonators (4) and ancilla resonators (6) connected to each ancilla resonator to substantially follow the Hamiltonian resulting from the radiation having the ancilla resonance frequency of each ancilla resonator simultaneously; 3) Stop the radiation having the ancilla resonance frequency of each ancilla resonator; 4) Apply a measurement for the operator X to each ancilla resonator. is configured to perform J - 1 quantum operations.

[0010] In various embodiments, this system can present one or more of the following features. · This system is configured to simultaneously perform the first half of J - 1 quantum operations, where the quantum operations are performed on a single ancilla cat qubit non - linearly coupled to each data cat qubit, and then the second half of the J - 1 quantum operations are simultaneously performed on the remaining ancilla cat qubits. · This system is configured to perform J - 1 quantum operations substantially simultaneously. · The instruction circuit (8) is configured to periodically perform J - 1 quantum operations.

[0011] The present invention also relates to a method for performing a quantum gate between N data resonators, where N is 2 or more, and an ancilla resonator. Each data resonator has its own resonance frequency and holds its own data cat qubit. The ancilla resonator has an ancilla resonance frequency, holds an ancilla cat qubit, and is non - linearly connected to the data resonators. The method includes the following operations. a) While stabilizing the ancilla cat qubit, applying radiation having the ancilla resonance frequency such that the data resonators and the ancilla resonator substantially simultaneously follow the Hamiltonian resulting from the radiation having the ancilla resonance frequency. b) After a predetermined period, stopping the radiation having the ancilla resonance frequency.

[0012] In various embodiments, this method can present one or more of the following features. · Operation a) includes applying dissipative stabilization to at least one data resonator according to time and the state of the ancilla cat qubit. The method further includes, before operation a), preparing an ancilla cat qubit to the state + or state - of operator X, and further including, after stopping the emission having an ancilla resonant frequency, applying a measurement operation of operator X to the ancilla resonator. Operation a) involves applying N radiations having an ancilla resonant frequencies, where the data resonators substantially simultaneously follow their respective Hamiltonians resulting from one of the N radiations, and among the N radiations, even-numbered radiations are selected to have opposite amplitudes. The present invention also relates to a method for performing quantum-corrected error coding between J data resonators, which are 2 or more, and J-1 ancilla resonators. Each data resonator has its own resonant frequency and holds its own data cat qubit. Each ancilla resonator has an ancilla resonant frequency and holds an ancilla cat qubit. Each of the J-1 ancilla resonators is nonlinearly connected to two data resonators of each of the J data resonators, and each data resonator of the J data resonators is connected to two auxiliary resonators of up to J-1 ancilla resonators. The present method is 1) For each of the J-1 ansila resonators, prepare the ansila cat qubit in each of the J-1 ansila resonators to state + or state - of the X operator, 2) For each of the J-1 ancilla resonators, while stabilizing the ancilla cat qubit, radiation having the ancilla resonant frequency of each ancilla resonator is applied to the command circuit so that the data resonators and ancilla resonators connected to each ancilla resonator follow substantially simultaneously the Hamiltonian resulting from the radiation having the ancilla resonant frequency of each ancilla resonator. 3) Stop the radiation having the ancilla resonant frequency of each ancilla resonator, 4) Applying measurements to the operator X of each ancillary resonator, This involves performing J-1 quantum operations.

[0013] In various embodiments, this method may present one or more of the following features: The first half of the J-1 quantum operations is performed simultaneously for each data cat qubit on a non-linearly coupled single ancillane cat qubit, and then the second half of the J-1 quantum operations is performed simultaneously on the remaining ancillane cat qubits. • J-1 quantum operations are performed virtually simultaneously. [Brief explanation of the drawing]

[0014] Other features and advantages of the present invention will become apparent from the following description of the drawings illustrating embodiments of the present invention.

[0015] [Figure 1] A schematic diagram of a conventional CNOT gate between a data cat qubit and an ancilla cat qubit is shown. [Figure 2] Figure 1 shows a diagram illustrating how the quantum gate is executed. [Figure 3] This diagram illustrates the need for compensation when a CNOT gate is executed. [Figure 4] A schematic diagram of a CXX gate between two data cat qubits and an ancilla cat qubit according to the present invention is shown. [Figure 5] Figure 4 shows a diagram illustrating how the quantum gate is executed. [Figure 6] Figure 4 shows a graph illustrating the performance of the quantum gate. [Figure 7] Figure 4 shows a diagram of the compensation scheme for the CXX gate. [Figure 8] Figure 4 shows a schematic diagram of a measurement quantum system using quantum gates. [Figure 9] Figure 8 shows a diagram illustrating how the measurement quantum gate is executed. [Figure 10] Figure 8 shows a schematic diagram of the first embodiment of a quantum error correction code (QECC) using a measurement quantum system. [Figure 11] Figure 10 shows a diagram illustrating how QECC is executed. [Figure 12] Figure 10 shows a graph illustrating the performance of QECC. [Figure 13] Figure 8 shows a schematic diagram of a second embodiment of QECC using the measurement quantum system. [Figure 14] A schematic diagram of the CX^N gate between N data cat qubits and an ancilla cat qubit according to the present invention is shown. [Figure 15] Figure 15 shows a diagram illustrating how the quantum gate is executed. [Figure 16] Figure 15 shows a diagram of the compensation scheme for the CX^N gate. [Figure 17] A schematic diagram of the execution scheme of a surface code that implements the CX^N quantum gate and / or MX^N measurement operation according to the present invention is shown. [Figure 18] A schematic diagram of the execution scheme of a surface code that implements the CZ^N quantum gate and / or MZ^N measurement operation according to the present invention is shown. [Figure 19] A schematic diagram of the execution scheme of a surface code that implements the CZ^N quantum gate and / or MZ^N measurement operation according to the present invention is shown. [Figure 20] A schematic diagram of the execution scheme of the XZZX code that implements the CX^N quantum gate and / or MX^N measurement operation according to the present invention is shown. [Figure 21] A schematic diagram of the execution scheme of the XZZX code that implements the CZ^N quantum gate and / or MZ^N measurement operation according to the present invention is shown. [Modes for carrying out the invention]

[0016] The drawings and the following description consist mostly of positive and clearly defined features. As a result, they are not only useful for understanding the invention, but can also be used to contribute to its definition when necessary.

[0017] This invention relates to the realization of a high-performance quantum gate in the context of cat qubits.

[0018] Stabilized cat qubits are known to benefit from noise bias. More precisely, the effective error channel (e.g., bit errors or "bit inversions") is exponentially suppressed in proportion to the "size" of the Schrödinger cat state of the cat qubit, i.e., the average number of photons.

[0019] Current understanding suggests that this suppression effect should be applied to a wide range of physical noise processes that have local effects on the phase space of a harmonic oscillator. This includes, but is not limited to, photon loss, thermal excitation, photon phase dephase, and various nonlinearities induced by coupling with Josephson junctions.

[0020] Recent experiments in quantum superconducting circuits have observed this exponential suppression of bit flip errors associated with the average number of photons in the cat state.

[0021] Due to this noise structure, it is considered sufficient to use a single repeating code to correct the remaining error channels. In fact, if only phase jumps need to be corrected, it is sufficient to use only a phase jump error correction code. This may be, for example, a repeating code defined in the dual basis, or other classical error correction codes.

[0022] A cat qubit iterative code is constructed using d cat qubits (called data cat qubits) that encode logical information. This is achieved by repeatedly measuring a quantum operator that determines whether an error has occurred in the data cat qubit. This is done using d-1 additional cat qubits (called ancilla cat qubits or ancilari cat qubits). The quantum circuit of the iterative code requires preparing the ancilla qubits in a "+" state (for cat qubits, this is the Schrödinger cat state), placing two CNOT gates between the ancilla cat qubits and the data cat qubits, and measuring the Pauli X operator of the ancilla qubits (for cat qubits, this is the photon number parity).

[0023] The challenge in implementing this iterative code is executing it on hardware that operates below the error tolerance threshold. In other words, for the code to have a positive effect, the fidelity of the quantum operations in this circuit must be extremely high.

[0024] More precisely, if the iterative code operates beyond a threshold, i.e., if the fidelity of the physical operations constituting the iterative code is insufficient, the lifetime of the logical information decreases as the number of physical data qubits d increases. New errors introduced by the addition of quantum systems are not compensated for by error correction policies.

[0025] On the other hand, if the iterative code operates below the error correction threshold, i.e., if the fidelity of the physical operation is sufficient, the lifetime of the logical information increases exponentially in proportion to the number of data qubits d (also called the distance d of the iterative code).

[0026] Several papers have proposed using cat qubits in QECCs, either in codes that are completely phase-inverted ("phase-reversal" codes) or codes that are far more resistant to phase errors than bit errors (e.g., biased noise-specialized codes of the rectangular surface code type or XZZX surface code type). These solutions are, for example, "Repetition Cat Qubits for Fault-Tolerant Quantum Computation", Jeremie Guillaud, Mazyar Mirrahimi, Phys. Rev. 21 / 08 / 2020, "Error rates and resource overheads of repetition cat qubits" Jeremie Guillaud, Mazyar Mirrahimi, Phys. Rev. A 103, 042413,13 / 04 / 2021, "Building a Fault-Tolerant Quantum Computer Using Concatenated Cat Codes", Christopher Chamberland et al; PRX Quantum 3, 010329, This is described on 23 / 02 / 2022, or in "Practical Quantum Error Correction with the XZZX Code and Kerr-Cat Qubits", Andrew S. Darmawan et al, PRX Quantum 2, 030345, 09 / 16 / 2021.

[0027] In all known iterative code implementations, the syndrome measurement used in QECC (the result of the quantum operator measurement of QECC) requires two distinct time steps: a first time step in which a first CNOT gate is executed between the first qubit and the ancilla qubit, followed by a second time step in which a second CNOT gate is executed between the second qubit and the ancilla qubit. When performing the ancilla qubit measurement for QECC, there is an additional time step before the CNOT gate step to prepare the ancilla qubit to a selected state, and another time step after the CNOT gate step to measure the state of the ancilla qubit.

[0028] This scheme was developed quite some time ago. Its principle relies on the fact that multi-qubit quantum gates can be realized using only two-qubit gates. In many architectures, realizing N-qubit gates becomes more difficult as N increases, so it is preferable to synthesize N-qubit gates using only two-qubit gates. For iterative codes, it is preferable to realize a cascade of two CNOT gates rather than implementing a CXX gate. This technique is not limited to CXX gates but generalizes to any multi-qubit entanglement gate between more than two qubits. In other words, if two or more quantum entities are set up to interact, the standard way to perform these multi-entity interactions is to perform multiple two-entity interactions.

[0029] The field of quantum computing is extremely young, especially in the realm of cat qubits. In many ways, it's still a research area. As a result, the mainstream of progress appears to be making incremental changes, but in reality, their verification and practical application require significant research in physics. In other words, what is considered cutting-edge technology today will not fundamentally change until a major obstacle is discovered. This means that known, effective solutions are not easily replaced.

[0030] Significant progress has recently been made in cat qubits. In the process of working on the implementation of QECC for cat qubits, the applicant used conventional quantum gates, but found that these caused the problems described below.

[0031] Here, "cat qubit" refers to any implementation of a cat qubit, particularly a two-photon dissipative Schrödinger cat qubit. Alternatively, other cat qubits may be used. In some embodiments, data cat qubits and ancilla cat qubits may be implemented with different types of cat qubits.

[0032] Such a cat qubit can be stabilized by the following exemplary scheme. a) Jump operator Dissipative stabilization in TIFF2026510324000002.tif6150. Here, κ2 is the two-photon dissipation rate, a is the photon annihilation operator, and α is a complex number defining the cat qubit. This jump operator couples buffer mode b to the cat qubit, and the Hamiltonian H∝a²b † This can be achieved by designing the following. Here, b is the photon annihilation operator for mode b, with a frequency of 2ω a -ω b The pump and frequency ω b This involves driving to the buffer mode. For the record, the dissipative stabilization of the two coherent states requires performing a nonlinear transformation between the two photons of the first mode a (also known as the cat qubit mode) that holds the stabilized quantum manifold and the one photon of the second mode b, known as the buffer mode. Such a stabilization scheme allows for exponential suppression of bit flips depending on the number of photons in the two coherent states mentioned above.

[0033] b) Kerr Hamiltonian TIFF2026510324000003.tif6150 Here, K is the complex amplitude of the Kerr Hamiltonian, a is the photon annihilation operator, and α is the complex number that defines the cat qubit.

[0034] c) Detuned Kerr Hamiltonian TIFF2026510324000004.tif6150 Here, \(K\) is the complex amplitude of the Kerr Hamiltonian, \(a\) is the photon annihilation operator, \(\alpha\) is the complex number defining the cat qubit, and \(\Delta\) is the detuning coefficient.

[0035] d) Two-photon exchange (TPE) Hamiltonian TIFF2026510324000005.tif6150 Here, \(g\) is the complex amplitude of the TPE Hamiltonian, \(g\) is the photon annihilation operator, \(\alpha\) is the complex number defining the cat qubit, and \(\sigma\) + is the raising and lowering operator of the two-level buffer system.

[0036] e) Jump operator TIFF2026510324000006.tif7150 Dissipative squeezing stabilization at. Here, \(\kappa_2\) is the two-photon dissipation rate, \(a\) is the photon annihilation operator, \(\alpha\) is the complex number defining the cat qubit, and \(\xi = re\) iθ is the complex squeezing parameter. This stabilizes the "squeezed cat qubit", also called the displaced squeezed vacuum state. This jump operator couples the buffer mode \(b\) (decay rate \(\kappa\) b ) to the cat qubit and is realized by constructing the coupling Hamiltonian \(g(L_2b\) b + h.c.) with \(\kappa\) † ≫ \(g\).

[0037] The above schemes a) - e) are applied conditional on the state of the ancilla qubit.

[0038] The theoretical realization of a CNOT gate between two stabilized cat qubits in modes \(a_1\) and \(a_2\) relies on the use of the following three elements.

[0039] 1) For the control qubit \(a_1\), the jump operator Dissipative stabilization in TIFF2026510324000007.tif6150, or execution of one of the aforementioned stabilization schemes. Here, a1 is the photon annihilation operator for mode a1, and α is a complex number defining the cat qubit.

[0040] 2) In the circuit, Add the "feedforward" Hamiltonian (also called the "CNOT" Hamiltonian or "longitudinal" Hamiltonian) represented by the formula in TIFF2026510324000008.tif6150. Here, g CX is the complex amplitude of the Hamiltonian, a1 is the photon annihilation operator for mode a1, a2 is the photon annihilation operator for mode a2, and α is the complex number that defines the cat qubit.

[0041] 3) A jump operator for target qubit a2, which depends on the state of control qubit a1. Dissipative stabilization in TIFF2026510324000009.tif11150, or execution of one of the aforementioned stabilization schemes a) to e) in a form adapted to be conditional on the state of the ancilla qubit, where κ2 is the two-photon dissipation rate, a1 is the photon annihilation operator for mode a1, a2 is the photon annihilation operator for mode a2, and α is a complex number defining the cat qubit.

[0042] The best theoretical implementation uses the three elements described above. However, the applicant's experiments have shown that the first two elements offer a good compromise between ease of implementation and quality of results.

[0043] Element 3, when applied, replaces the conventional stabilization performed on the target qubit to stabilize the cat state. If Element 3 is not present, the target qubit is not subject to cat state stabilization while an element is applied for performing a gate. This applies to all embodiments described below. During a gate operation or measurement operation, the stationary stabilization of the target cat qubit is turned off and may be replaced by Element 3 during the gate operation.

[0044] In fact, theoretically, it is possible to implement a CNOT gate using only elements 1 and 3 without using the feedforward Hamiltonian. However, in practice, implementing element 2 is extremely important because the fidelity of the CNOT gate decreases without this term.

[0045] Figure 1 shows a schematic diagram of a conventional CNOT gate. In this diagram, both the data cat qubit 4 and the ancilla cat qubit 6, controlled by the command circuit 8, are interconnected by the CNOT gate 10.

[0046] Generally speaking, a cat qubit is a stabilized resonant mode in a specific circuit that receives a specific radiation controlled by a command circuit. As a result, a cat qubit is generally designated as a resonator with a specific resonant frequency (the frequency of the cat qubit) controlled by a command circuit. Each cat qubit can be controlled by a specific command circuit, or a single circuit can control all cat qubits in a given circuit. In this example, a single command circuit 8 controls all the data cat qubits 4 and ancilla cat qubits 6. Furthermore, cat qubits are sometimes referred to as "modes" when referring to the resonators in which they are encoded.

[0047] Figure 2 shows a block diagram of the operation of a CNOT gate (also called a CX gate). In the first operation 200, the feedforward Hamiltonian (element 2) is turned on and the stabilization of data cat qubit 4 is turned off. Element 1 is always applied because the ancilla cat qubit 6 needs to be stabilized. Alternatively, operation 200 may include turning on the time-dependent dissipation of the feedforward Hamiltonian (element 2) and the data cat qubit (element 3), or enabling only the time-dependent dissipation of the data cat qubit (element 3). After a selected period of operation 200 (the gate period described below), in operation 210 the feedforward Hamiltonian (or its variation of operation 200) is turned off and the stabilization of data cat qubit 4 is restored.

[0048] Stabilization of an ancilla qubit and data cat qubit can be performed using one of the five stabilizations described above.

[0049] The applicant's initial research revealed that the means used to construct the feedforward Hamiltonian also introduce very strong deterministic spurious effects. Therefore, eliminating these deterministic spurious effects is essential to obtaining any results.

[0050] More precisely, when the term of the feedforward Hamiltonian is advanced, the term -2α(a2 † a2) is the detuning of the data cat qubit, which does not need to be physically implemented and can be done in software by redefining α of the data cat qubit. On the other hand, term -α 2 g CX (a1+a1 † ) is the linear displacement of an Ancillane cat qubit that is typically achieved.

[0051] The entangled part of the interaction, that is, the most difficult to realize, is g CX (a1+a1 † )a2 †This is a2. Typically, modes a1 and a2 are adjacent on the chip and both are incorporated into a nonlinear circuit element (typically an asymmetric threaded superconducting quantum interference device or "ATS" as described in the paper "Exponential suppression of bit-flips in a qubit encoded in an oscillator", Lescanne R. et. Al., Nature Physics, 2020), and this term is pump ε p (t) = ε p It is directly driven by cos(ω1t). However, applying a pump at this frequency induces a linear drive in the Ancillane cat qubit a1, which needs to be precisely compensated for.

[0052] Figure 3 illustrates this effect on the Wigner function of an ancilla cat qubit. When stabilized and without a feedforward Hamiltonian, the Wigner distribution of the ancilla cat qubit is shown on the horizontal axis. When a feedforward Hamiltonian is applied, a displacement occurs as shown by the solid arrow, which, combined with stabilization, causes decoherence and / or leakage from the cat qubit code space. Therefore, compensation is necessary, as shown by the dotted line. This can be achieved, for example, by turning on resonant driving for the ancilla cat qubit to precisely compensate for the effect of the pump that realizes the entangled portion of the feedforward Hamiltonian.

[0053] While this compensation is feasible, it becomes burdensome when multiple CNOT gates are executed, as it requires configuring multiple gates almost simultaneously without causing adverse effects.

[0054] In their search for a suitable solution to implement the feedforward Hamiltonian, the applicant unexpectedly discovered that they could improve upon the "N-1 interaction between two quantum entities" paradigm and that they could directly implement the CNOTNOT gate without relying on a series of CNOT steps.

[0055] More specifically, the applicant discovered a way to reduce the error correction cycle time when using its gate to implement the measurement of quantum operators in the context of quantum error correction codes, thereby improving its performance (threshold) as demonstrated below. As a further advantage, the applicant found that this can be achieved while solving the compensation problem of the feedforward Hamiltonian.

[0056] Figure 4 shows a schematic diagram of a quantum system for executing the CNOTNOT quantum gate 2 according to the present invention. Hereafter, the terms "CNOTNOT gate," "CNOTNOT quantum gate," "CXX gate," or "CXX quantum gate" are interchangeable. Similarly, the terms "MNOTNOT operation," "MNOTNOT quantum operation," "MXX operation," or "MXX quantum operation" are also interchangeable.

[0057] In the example described here, quantum system 2 executing a CXX gate concatenates two data cat qubits 4 and one ancilla cat qubit 6. In this invention, "data cat qubit" and "target qubit" are interchangeable and refer to physical qubits whose data is to be known, controlled, or modified. Similarly, "control cat qubit" and "ancilla cat qubit" are interchangeable and refer to physical qubits used to read the photon parity of the data cat qubit for the purposes described above. In other words, a data cat qubit means a physical qubit containing the quantum information that the QECC seeks to protect. An ancilla cat qubit means the complement of the data cat qubit in the QECC; that is, this physical qubit is used to detect errors in the data cat qubit.

[0058] To theoretically implement a CXX gate between three cat qubits stabilized in modes a1, a2, and a3, the following three elements are used. 1) Jump operator for control qubit a1 Dissipative stabilization in TIFF2026510324000010.tif6150. Here, a1 is the photon annihilation operator for mode a1, and α is the complex number defining the Ancillane cat qubit.

[0059] 2) In the circuit, Add the “feedforward” Hamiltonian (also called the “CXX” Hamiltonian) represented by the formula in TIFF2026510324000011.tif8150. Here, T is the duration of the CXX gate, a1 is the photon annihilation operator for the ancilla cat qubit 6, a2 and a3 are the photon annihilation operators for the data cat qubit in mode a2, and α is the complex number defining the cat qubit. For simplicity, it is assumed that the photon distributions of the ancilla qubit and the data qubit are equal, but these may be different. That is, a1 may be α1 2 a2 has α2 2 a3 has α3 2 This corresponds to the feedforward Hamiltonian, which has a frequency of ω a1 It consists of a pump that acts as an attraction to data cat qubit 4. While this feedforward Hamiltonian is applied, the stabilization of data cat qubit 4 is turned off.

[0060] 3) Depending on the state of the Ancilla cat qubit a1, one or more dissipative stabilizations may be used to act as resistances on one or both data cat qubits 4. The jump operator is, TIFF2026510324000012.tif11150 The filename is TIFF2026510324000013.tif11150. Jump operator L t2 and L t3 teeth, To provide buffer modes b2 and b3 to the data cat qubit 4, Hamiltonian To compose TIFF2026510324000014.tif6150, frequency TIFF2026510324000015.tif6150 and frequency Pumping with TIFF2026510324000016.tif9150, frequency This can be achieved by driving TIFF2026510324000017.tif6150. By replacing a2 and b2 with a3 and b3 in the above formula, the jump operator L t3 The same thing can be done with respect to this as well.

[0061] The best implementation theoretically uses these three elements. However, the applicant's experiments have shown that the first two elements offer a good compromise between ease of implementation and quality of results. As with the CNOT gate mentioned above, stabilization schemes 1) and 3) can be implemented using any of the five stabilization schemes mentioned above.

[0062] As shown in this diagram, the two data cat qubits 4 are connected to the ancilla cat qubit 6 via a CXX gate 11. However, unlike conventional techniques that use two consecutive CNOT gates, there is a single CXX gate 11, and as shown in the diagram, the two data cat qubits 4 are connected to the CXX gate 11 simultaneously, so the CXX gate is executed simultaneously for the two data cat qubits 4.

[0063] Figure 5 shows a block diagram of the operation of the CXX gate in Figure 4. In the first operation 500, the feedforward Hamiltonian (element 2) is turned on for both data cat qubit 4 and ancilla cat qubit 6. Element 1 is always applied because the ancilla cat qubit needs to be stabilized, but the stabilization of data cat qubit 4 is turned off. Alternatively, operation 500 is performed on the time-dependent dissipation L of the feedforward Hamiltonian (element 2) and the data cat qubit (element 3). t2 and L t3 Turning on, or the time-dependent dissipation L of the data cat qubit (element 3) t2 and L t3This may include enabling only that. After the selected period of operation 500 (the gate period described below), the feedforward Hamiltonian (or its variation of operation 500) is turned off in operation 510, and the stabilization of data cat qubit 4 is restored.

[0064] To demonstrate that the CXX strategy leads to an overall performance improvement, the analytical error of the CXX gate is calculated and compared to the performance of two consecutive CX gates, and this analytical calculation is verified using the numerical simulation shown in Figure 6.

[0065] In the following demonstration, we again consider that the CXX gate is realized between three cat qubits, each having a mode given by 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 is represented by the master equation This is implemented by TIFF2026510324000018.tif15150. Here, a1, a2, and a3 are the control qubit and the annihilation operators for the two target qubits, respectively. TIFF2026510324000019.tif9150 TIFF2026510324000020.tif9150 The filename is TIFF2026510324000021.tif8150.

[0066] In the cat qubit architecture, hardware quality is measured by the ratio of two time scales: time 1 / κ2 and time 1 / κ1. Here, κ2 is the two-photon dissipation rate that stabilizes the qubit, and κ1 is the one-photon loss rate. Phase shift (rate κ) Φ ), there are other sources of error such as thermal excitation, self-Kerr effect, cross-Kerr effect, or undesirable coupling with other quantum systems present in the memory vicinity.

[0067] However, since single-photon loss is the dominant error mechanism, for clarity, we will consider only this physical error mechanism below. In general, below, it is acceptable to replace the ratio κ1 / κ2 with the sum of the error mechanism rates divided by κ2.

[0068] A hybrid basis is used to analyze the master equation. In the hybrid basis, the control qubit is described by a shifted Fock basis, and the two target qubits are described by a normal Fock basis. The Hamiltonian is expressed as follows: TIFF2026510324000022.tif8150

[0069] In a rotating system, under a fully shifted Fock basis, the master equation can be expressed as follows: TIFF2026510324000023.tif39154 Here, the jump operator for the target qubit is given by the following equation: TIFF2026510324000024.tif9150 TIFF2026510324000025.tif9150

[0070] If we consider only the first excited state for each of the three qubits, ignore internal coupling, and ignore the last term of both target transition operators, the master equation becomes as follows: TIFF2026510324000026.tif11150

[0071] By adiabatically removing the excited states, the following master equation is obtained. TIFF2026510324000027.tif21150

[0072] Integrating from 0 to time T and ignoring higher-order terms, the equation becomes as follows: TIFF2026510324000028.tif28118

[0073] By returning to the initial frame and removing the unital perfect CXX, the error channel of CXX is obtained. TIFF2026510324000029.tif23123

[0074] Performing integration and focusing on the diagonal terms, the Z error rate of CXX is given as follows: TIFF2026510324000030.tif11150 (Here, the second term corresponds to the non-adiabatic error) TIFF2026510324000031.tif9150

[0075] Figure 6 shows the simulation results of the quantum circuit using the quantum gate from Figure 4, compared with the non-adiabatic error mentioned above. The small dashed line represents α 2 For various values ​​of κ², as a function of the reciprocal κ² The values ​​for TIFF2026510324000032.tif6150 are plotted. α 2 For each value, the simulation error level at a specific κ2 value is plotted (as circles, triangles, and quadrilaterals). This figure shows that the theoretically determined non-adiabatic error is expected to be met in the actual implementation, and the error calculation below demonstrates the aforementioned advantages. In this simulation, the effect of κ1 is well understood and does not need to be verified, so κ1 is set to 0.

[0076] To specifically evaluate the gain associated with using the CXX gate according to the present invention, we compare the error amount with that of two CNOT gates connected in series. The total time required to realize the CXX gate, or two CNOT gates connected in series, is fixed at T. Below, TIFF2026510324000033.tif6150 represents the average number of photons in the cat state.

[0077] *Error amount of CNOT gates connected in series The error of an ancilla qubit is given by the following formula: The error of two data qubits in TIFF2026510324000034.tif11150 is given by the following formula: The correlation error between data qubits and ancilla qubits is given by the following formula: TIFF2026510324000035.tif9150 Therefore, the total error is given by the following formula: TIFF2026510324000036.tif9150 TIFF2026510324000037.tif11150

[0078] *Error amount of CXX gate The error of an ancilla qubit is given by the following formula: The error of two data qubits in TIFF2026510324000038.tif11150 is given by the following formula: The correlation error between data qubits and ancilla qubits in TIFF2026510324000039.tif6150 is given by the following formula: Therefore, the total error is given by the following formula: TIFF2026510324000040.tif6150 TIFF2026510324000041.tif11150

[0079] This means that the CXX gate according to the present invention can be executed at high speed with the same errors as when two CNOT gates are executed at low speed, or it can be executed with better fidelity in the same amount of time as when two CNOT gates are executed.

[0080] Another advantage of implementing CXX in one step instead of using two consecutive CNOT gates is that the compensation problem mentioned above can be solved by selecting a specific phase configuration for the feedforward Hamiltonian. More precisely, The Hamiltonian of the form TIFF2026510324000042.tif6150 all produce the desired effect. The applicant is H CXFor the specific case where the last term of the equation is a "-" sign, each term of the feedforward Hamiltonian (a1+a1) is (as explained in Figure 3). † )(a2 † a2) and (a1+a1 † )(a3 † We found that the undesirable displacement caused by a3) is precisely offset, as shown in Figure 7. This means that compensation design is no longer necessary. In practice, due to experimental imperfections, a very small amount of compensation may still be needed, but since most of it has already been offset, the design becomes much easier.

[0081] In alternative embodiments, compensation can be constructed in a manner known as prior art.

[0082] Figure 8 shows a schematic diagram of the XX syndrome measurement using the CXX quantum gate of Figure 4. The system in Figure 8 is very similar to the system in Figure 4, except that it includes a control qubit preparation 12 and a control qubit measurement 14. By providing these elements, it becomes possible to measure the coupled photon parity between the data cat qubits 4 (which are stabilizers for the repeating code on the two-component cat qubit) by measuring the ancilla cat qubit 6. It should be noted that the measurement operation of the present invention is not limited to the measurement of "XX" as with the CXX gate. In other words, other gates that operate with the same structure (simultaneous CNOT gate) but are labeled differently by certain conventions are still within the scope of the present invention. Therefore, although the expression CXX is for ease of understanding in light of existing standards, the true aim of the present invention is a gate that inverts the state of the control when the coupled photon parity of the target is odd. The same applies to MXX for any measurement operation with the CXX gate or its equivalent.

[0083] Figure 9 shows a block diagram of the execution of the measurement operation in Figure 8. This is similar to Figure 5, except that operation 900, which prepares the Ancilan cat qubit 6 to the "+" state, precedes the feedforward Hamiltonian in operation 910, followed by operation 920, which measures the Ancilan cat qubit 6. In operation 910, if this embodiment utilizes them, conditional dissipative stabilization L t2 and L t3 This is also turned on at the same time. Operation 900 also allows you to prepare control qubit 6 to the |-> state.

[0084] As a result, Figure 9 shows that the XX syndrome measurement procedure according to the present invention is performed in three time steps, whereas conventional XX syndrome measurement requires four time steps due to consecutive CNOTs. The advantage gained by saving one time step has been demonstrated above.

[0085] Figure 10 shows a typical timeline of QECC using the measurement operation in Figure 8. For simplicity, only the Ancilla cat qubit 6, its preparation 12, and its measurement 14 are shown. The command circuit 8 and data cat qubit 4 are intentionally omitted, and the data cat qubit is indicated by a line showing the MXX operation.

[0086] If two operations 11 are aligned vertically, this means that the operations are simultaneous. If they are offset vertically, this means that they are offset in time, with one performing before the other. As described above, the execution of the MXX operation is done by turning on the corresponding feedforward Hamiltonian while turning off the normal stabilization (and possibly the corresponding conditional dissipative stabilization) for data cat qubit 4.

[0087] In the present invention, the QECC is configured such that each data cat qubit 4, except for the data cat qubits 4 at both ends of the QECC, is connected to two different ancilla cat qubits 6 via their respective MXX operations. The symbols shown in Figure 10 have a distance of 3. In another embodiment, the data cat qubits 4 may be arranged in a circle, and all data cat qubits 4 may be connected to two different ancilla cat qubits 6.

[0088] Figure 10 shows an example of a QECC with a distance of 3, but QECCs may have longer distances of 5 or more. In this case, the MXX operation is performed simultaneously in an alternating scheme. That is, in the first time step, an MXX operation is performed for each data cat qubit, but it is associated with a single non-linearly coupled ancilla cat qubit. After this first time step, the MXX operation is performed for the remaining ancilla cat qubits.

[0089] Starting from Figure 10, in a QECC at distance 5, the next sequence of Ancilan cat qubits aligns with the first sequence of Ancilan cat qubits, and the last sequence of Ancilan cat qubits aligns with the second sequence of Ancilan cat qubits.

[0090] Figure 11 shows a block diagram of the QECC operation in Figure 10 at distance J. Operation 1100 performs an MXX operation on each ancillane cat qubit with an odd row number (2i+1). Then, operation 1110 performs an MXX operation on each ancillane cat qubit with an even row number 2i.

[0091] Figure 12 shows two charts comparing the performance of conventional QECC (left) and QECC according to the present invention (right). In the design evaluation of QECC, the error tolerance threshold can be considered as the horizontal axis where error tolerance improves as the code distance increases.

[0092] In Figure 12, we assume that an 8-photon cat qubit is used and that all operations take the same time 1 / κ2. The logic Z error is plotted as a function of κ1 / κ2 with different code distances. On the left, two CNOT gates are used to detect the Z error, and on the right, a CXX gate is used. The threshold has improved significantly by approximately 25%, from 0.289 to 0.366.

[0093] Figure 13 shows a less desirable alternative to the QECC code in Figure 10. In this embodiment, all CXX operations 11 are performed simultaneously. While this may seem desirable at first glance, when actually implemented, it introduces potential drawbacks compared to the embodiment in Figure 10.

[0094] This can be proven by starting by distinguishing between two implementation cases. In the first implementation case, the CXX gate is realized using elements 1 and 2. In the second implementation case, CXX is realized using element 3 as well.

[0095] In the first implementation case, the sequence shown in Figure 13 is technically feasible. However, the fact that the data cat qubit functions as the target qubit for two ancilla qubits presents several problems. This is because, in this first implementation case, the drives need to be turned on to obtain feedforward Hamiltonians at two different frequencies (the frequencies of two different ancilla cat qubits). Turning on multiple drives at different frequencies simultaneously can lead to crosstalk, frequency congestion, and each feedforward Hamiltonian needs to be driven with a small amplitude to avoid supplying excess energy that could raise the temperature of the circuit.

[0096] The applicant determined that, at this stage, it is best to use the offset feedforward Hamiltonian with the maximum output. Therefore, the embodiment shown in Figure 10 is preferred.

[0097] In the second implementation case (adding element 3), the current implementation of CXX described above makes it impossible to execute two CXX gates simultaneously on the same data qubit. Doing so would require a more difficult dissipation, a configuration the applicant has not yet figured out. Therefore, the embodiment in Figure 10 is still preferred. While the embodiments in Figures 4-6 consist of only two data cat qubits, the applicant has discovered that CXX can be made into CX^N by connecting N data cat qubits, each to the same ancilla cat qubit. Like CXX, CX^N can also be realized in a single time step. Figure 14 shows a schematic diagram of the CX^N gate. Data cat qubits 4(1)-4(N) are connected to an ancilla cat qubit 6 via the CX^N gate 16.

[0098] Figure 15 shows a block diagram of the operation of the CX^N gate in Figure 14. The first operation 1500 consists of turning on N feedforward Hamiltonians (element 2) for each data cat qubit 4(1) to 4(N) and an ancilla cat qubit 6. Element 1 is always applied because the ancilla cat qubits need to be stabilized. Alternatively, operation 1500 is the feedforward Hamiltonian (element 2) and the time-dependent dissipation L of the data cat qubits. t1 From L tN (Element 3) is turned on, or the time-dependent dissipation L of the data cat qubit is turned on. t1 and L tN It may also include enabling only (element 3). After the selected period of operation 1500 (the gate period described below), the feedforward Hamiltonian (or its variation of operation 1500) is turned off in operation 1510.

[0099] The phases of the pumps that turn on the N feedforward Hamiltonians can be selected so that no compensation is needed. Figure 16 shows an example of a compensation scheme. Here, the pumps for the N feedforward Hamiltonians are selected so that the drive is precisely compensated for the Ancilan cat qubit, as indicated by the arrows in the figure. If all the feedforward Hamiltonians have the same amplitude, referring to the explanation in Figure 7, this reduces to the number of feedforward Hamiltonians in the "+" case being equal to the number of feedforward Hamiltonians in the "-" case. Possible examples are: The filename is TIFF2026510324000043.tif6150, where compensation is ensured by alternating signs. Even when the amplitudes are not exactly equal, the phase compensation scheme yields excellent results in first-order analysis. When N is odd, the solution is for even feedforward Hamiltonians to cancel each other out, resulting in one (or more) conventional compensations.

[0100] The CZ^N gate may be implemented in one time step instead of using N CZ gates in series. The dissipation of the data and Ancilan cat qubits remains unchanged, and the frequency ω a -ω b In this pump, the so-called beam splitter Hamiltonian H=a † b+b † It is worth remembering that the CZ gate can be easily realized by turning on 'a'. Unlike the CNOT gate, this pump does not induce deterministic spurious effects on the control qubit, so no compensation is required.

[0101] While the above focuses on examples of repetitive codes, the CX^N gate and MX^N operations of the present invention can be generalized to all codes that implement X or Z stabilizer measurements. Below, we show how they can be applied to surface codes and XZZX codes.

[0102] Surface coding involves sequentially measuring an X-stabilizer with a weight of 4 and a Z-stabilizer with a weight of 4. In state-of-the-art implementations, both the X-stabilizer and Z-stabilizer measurements are performed in parallel using four gates over four time steps.

[0103] Figure 17 shows CX 4 (and Frontier is CX 2 This demonstrates the implementation of an X-stabilizer measurement using a CNOT gate. If the CNOT gate implementation uses only the feedforward Hamiltonian and the dissipation of the Ancilla cat qubit (i.e., elements 1 and 2), the X-stabilizer measurement can be implemented in one time step. If the CNOT gate is implemented with the time-dependent dissipation of the data cat qubit (element 3), two time steps are required.

[0104] In this diagram, the data cat qubit is represented by an intermediate-sized circle connected to an ancilla cat qubit, indicated by a smaller circle with the CNOT symbol (the circle with a cross indicates that the data cat qubit acts as the target) (the larger circle surrounding the smaller circle indicates that the ancilla cat qubit acts as the control). This indicates that half of the ancilla cat qubit remains inactive as it is used for the Z stabilizer.

[0105] Figure 18 shows the first implementation of a Z-stabilizer measurement using a CXX gate (and a CX gate in the Frontier). If the CNOT gate embodiment uses only the feedforward Hamiltonian and the dissipation of the Ancilla cat qubit (i.e., elements 1 and 2), the Z-stabilizer measurement can be implemented in one time step. If the CNOT gate is implemented with the time-dependent dissipation of the data cat qubit (element 3), two time steps are required.

[0106] Figure 19 shows CZ 4This demonstrates an alternative implementation of Z-stabilizer measurement using a gate (and a CZZ gate in the Frontier). Regardless of which of the three elements is used to perform this gate, the gate can be implemented in one time step. In this figure, the CZ operation is shown as a black dot.

[0107] Therefore, by using conditionally stabilized data cat qubits (element 1, element 3, and optionally element 2), the X stabilizer can be measured in 2 time steps, and the Z stabilizer can be measured in CZ 4 It can be measured in a single time step using a gate. This results in a total of 3 time steps. When using unstabilized data cat qubits (elements 1 and 2 only), both the X stabilizer and Z stabilizer can be measured in a single time step (the latter being CZ). 4 Alternatively, use a CXX gate. This results in a total of two time steps.

[0108] The CX^N and CZ^N gates can also be used in XZZX codes. Here, each ancilla qubit measures the XZZX stabilizer. State-of-the-art implementations achieve this in four time steps.

[0109] Figures 20 and 21 show two sequences that enable XZZX measurements. The first sequence shown in Figure 20 can be constructed using conditionally stabilized data cat qubits (element 1, element 3, and optionally element 2). This requires splitting the CXX operation into two time steps. If unstabilized data cat qubits (elements 1 and 2 only) are used, the CXX operation can be performed in a single time step. The second sequence shown in Figure 21 uses the CZZ operation and can be performed in a single time step regardless of which of the three elements is used. This results in a maximum total of 3 time steps when using stabilized cat qubits (elements 1, 3, and optionally element 2). By using unstabilized data cat qubits (elements 1 and 2 only), both CXX and CZZ can be performed in a single time step, for a total of 2 time steps.

[0110] Similar to the CXX gate, the present invention can save at least one time step in MXX operations and QECC as well.

[0111] The above only considered CXX gates between two cat qubits. However, the applicant has found that, in principle, the invention works even if only the target qubit is a cat qubit and the control qubit is generally a normal qubit, or any two-level system having two quantum states |0> and |1>. In some embodiments, this normal control qubit may be a transmon qubit or a fraxonium qubit.

[0112] In this example, the CXX gate is Hamiltonian This is implemented using TIFF2026510324000044.tif8150, where a2 and a3 are annihilation operators for the two target cat qubits. Since the control is a normal qubit, there is no stabilization for the control qubit. In some embodiments, the two target cat qubits are jump operators. TIFF2026510324000045.tif10150 and It is described in TIFF2026510324000046.tif10150.

[0113] Alternatively, any of the stabilization schemes a) through e) described above may be used in a form that is conditionally applied depending on the state of the ancilla qubit.

[0114] In this example, there is no compensation applied to the control qubit, but executing this multi-qubit quantum gate instead of a series of two-qubit gates is still beneficial for reducing gate time.

[0115] A similar, simple generalization applies to the case of a CX^N quantum gate between a normal level of control qubit and N target cat qubits.

[0116] A CX^N gate between a regular qubit and N cat qubits can be used in quantum error correction codes, similar to the example of the two target cat qubits mentioned earlier.

Claims

1. A quantum system for executing quantum gates, A command circuit (8) for selectively applying radiation, N data resonators (4) of which are 2 or more, The system comprises an ancillary resonator (6) having an ancillary resonant frequency, Each data resonator (4) has its own resonant frequency and is connected to the command circuit (8) to stabilize each data cat qubit. The ansila resonator (6) is connected to the command circuit (8) to stabilize the ansila cat qubit, and is nonlinearly connected to the data resonator (4) via the command circuit (8), The command circuit (8) is, a) While stabilizing the ancilla cat qubit, apply radiation having the ancilla resonant frequency so that the data resonator (4) and the ancilla resonator (6) substantially simultaneously follow the Hamiltonian resulting from the radiation having the ancilla resonant frequency. b) Stop the radiation having the ancilla resonant frequency after a predetermined period of time. Configured to execute quantum gates, Quantum systems.

2. The quantum system according to claim 1, wherein the command circuit (8) is further configured to apply dissipative stabilization to at least one of the data resonators (4) in operation a) depending on time and the state of the Ansila cat qubit.

3. The quantum system according to claim 1 or 2, wherein the command circuit (8) is further configured to prepare the ancilla cat qubit to the state |+> or state |-> of the X operator before operation a), and after stopping the emission having the ancilla resonant frequency, apply the measurement operation of the operator X to the ancilla resonator (6).

4. The command circuit (8) is configured to apply N radiations having the ancilla resonant frequency in operation a), The data resonator (4) follows substantially simultaneously the Hamiltonian resulting from one of the N radiations. Of the N radiations mentioned above, the even-numbered radiations are selected to have opposite amplitudes. A quantum system according to any one of claims 1 to 3.

5. A quantum system for performing quantum error correction coding, A command circuit (8) for selectively applying radiation, Two or more J data resonators (4), It comprises J-1 ancillary resonators (6), Each data resonator (4) has its own resonant frequency and is connected to the command circuit (8) to stabilize each data cat qubit. Each ansila resonator (6) has an ansila resonant frequency and is connected to the command circuit (8) to stabilize the ansila cat qubit. Each of the J-1 ansila resonators (6) is nonlinearly connected to two data resonators of each of the J data resonators, and each data resonator of the J data resonators is connected to at most two auxiliary resonators of the J-1 ansila resonators. The command circuit (8) is, 1) For each of the J-1 ansila resonators, prepare the ansila cat qubit in each of the J-1 ansila resonators to state + or state - of the X operator, 2) For each of the J-1 ancilla resonators, while stabilizing the ancilla cat qubit, the radiation having the ancilla resonant frequency of each ancilla resonator is applied to the command circuit (8) so that the data resonator (4) and the ancilla resonator (6) connected to each ancilla resonator follow substantially simultaneously the Hamiltonian resulting from the radiation having the ancilla resonant frequency of each ancilla resonator. 3) Stopping the radiation from each of the ansila resonators having the ansila resonant frequency, 4) Applying the measurement to the operator X of each of the ancillary resonators, Configured to perform J-1 quantum operations, Quantum systems.

6. The quantum system according to claim 5, wherein the quantum system is configured to simultaneously perform the first half of the J-1 quantum operations, the quantum operations being performed on a single nonlinearly coupled ancillane cat qubit for each data cat qubit, and then simultaneously performing the second half of the J-1 quantum operations for the remaining ancillane cat qubits.

7. The quantum system according to claim 5, configured to perform J-1 quantum operations substantially simultaneously.

8. The quantum system according to any one of claims 5 to 7, wherein the command circuit (8) is configured to periodically perform the J-1 quantum operations.

9. A method for executing a quantum gate between two or more N data resonators (4) and an ancilla resonator (6), Each data resonator (4) has its own resonant frequency and holds its own data cat qubit. The ansila resonator (6) has an ansila resonant frequency, holds an ansila cat qubit, and is nonlinearly connected to the data resonator (4). The aforementioned method, Operation a) While stabilizing the ancilla cat qubit, apply radiation having the ancilla resonant frequency so that the data resonator (4) and the ancilla resonator (6) substantially simultaneously follow the Hamiltonian resulting from the radiation having the ancilla resonant frequency. Operation b) Stop the radiation having the ancilla resonant frequency after a predetermined period of time. Methods that include...

10. The method according to claim 9, wherein operation a) includes applying dissipative stabilization to at least one of the data resonators (4) depending on time and the state of the Ancilan cat qubit.

11. The method according to claim 9 or 10, further comprising preparing the ansila cat qubit to the state + or state - of the X operator before the operation a), and further comprising the operation c) applying a measurement operation of the operator X to the ansila resonator (6) after stopping the emission having the ansila resonant frequency.

12. The operation a) includes applying N radiations having the ancilla resonant frequency, The data resonator (4) follows substantially simultaneously the Hamiltonian resulting from one of the N radiations. Of the N radiations mentioned above, the even-numbered radiations are selected to have opposite amplitudes. The method according to any one of claims 9 to 11.

13. A method for performing quantum error correction coding between two or more J data resonators (4) and J-1 ancilla resonators (6), Each data resonator (4) has its own resonant frequency and holds its own data cat qubit. Each Ansila resonator has an Ansila resonant frequency and holds an Ansila cat qubit. Each of the J-1 ansila resonators (6) is nonlinearly connected to two data resonators of each of the J data resonators, and each data resonator of the J data resonators is connected to at most two auxiliary resonators of the J-1 ansila resonators. The aforementioned method, 1) For each of the J-1 ansila resonators, prepare the ansila cat qubit in each of the J-1 ansila resonators to state + or state - of the X operator, 2) For each of the J-1 ancilla resonators, while stabilizing the ancilla cat qubit, radiation having the ancilla resonant frequency of each ancilla resonator (6) is applied so that the data resonator (4) and the ancilla resonator (6) connected to each ancilla resonator (6) substantially simultaneously follow the Hamiltonian resulting from the radiation having the ancilla resonant frequency of each ancilla resonator (6), 3) Stopping the radiation from each of the ansila resonators having the ansila resonant frequency, 4) Applying the measurement of the operator X of each ancillary resonator (6) and This involves performing J-1 quantum operations, method.

14. The method according to claim 13, wherein the first half of the J-1 quantum operations is performed simultaneously for each data cat qubit on a nonlinearly coupled single ancilla cat qubit, and then the second half of the J-1 quantum operations is performed simultaneously on the remaining ancilla cat qubits.

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

Images