Quantum gate calibration program, quantum gate calibration method, and information processing device.

Abstract

To improve the calibration accuracy of quantum gates. [Solution] The information processing device 10 acquires error data 14 indicating an error in the quantum gate 13a. The information processing device 10 uses an approximation function 15 that linearly approximates the effect that an additional quantum gate attached to the quantum gate 13a has on the operation of the quantum gate 13a, to generate a linear equation 16 that includes a variable corresponding to the additional quantum gate and shows a relationship in which the error is canceled by the additional quantum gate. The information processing device 10 determines the quantum gate 13b corresponding to the additional quantum gate by solving the linear equation 16 for the variable.

Description

[Technical Field]

[0001] The present invention relates to a quantum gate calibration program, a quantum gate calibration method, and an information processing device. [Background technology]

[0002] Quantum gate-type quantum computers perform various quantum operations on qubits. Quantum computers initialize qubits, apply quantum gates to qubits, and measure the values ​​of qubits. Quantum computers are implemented using physical platforms such as superconducting quantum circuits, semiconductor quantum dots, diamond nitrogen vacancy (NV) centers, and nuclear magnetic resonance (NMR) molecules.

[0003] Implemented quantum computers typically have errors (errors) in their quantum operations, causing them to deviate from ideal operation. Users may perform calibration to adjust the values ​​of the quantum computer's control parameters to minimize these errors. For example, a quantum computer may have control parameters to change the waveform of the microwave pulse signal irradiated onto the qubits.

[0004] Furthermore, there are techniques to reduce errors in quantum computers by generating multiple logically equivalent but different quantum circuits and performing measurements on these multiple quantum circuits. There are also techniques to make a quantum gate sequence equivalent to an identity quantum gate by inserting the same type of quantum gate between multiple unitary quantum gates included in the quantum gate sequence. In addition, there are techniques to reduce qubit readout errors in quantum circuits by inserting a random Pauli quantum gate immediately before measuring a qubit. There are also techniques to insert a certain quantum gate and another quantum gate with the opposite effect into a quantum circuit. [Prior art documents] [Patent Documents]

[0005] [Patent Document 1] International Publication No. 2021 / 101829 [Patent Document 2] U.S. Patent No. 11348027 [Patent Document 3] International Publication No. 2022 / 129204 [Patent Document 4] U.S. Patent Application Publication No. 2023 / 0176935 [Overview of the project] [Problems that the invention aims to solve]

[0006] The errors in a quantum gate being calibrated may include error components that are difficult to cancel simply by changing the control parameter values ​​of that quantum gate. Therefore, errors may remain in the quantum gate after calibration. Thus, in one aspect, the present invention aims to improve the calibration accuracy of quantum gates. [Means for solving the problem]

[0007] In one aspect, a quantum gate calibration program is provided that allows a computer to perform the following steps: acquire error data indicating an error in the first quantum gate; use an approximation function that linearly approximates the effect of an additional quantum gate attached to the first quantum gate on the operation of the first quantum gate; generate a linear equation that includes a variable corresponding to the additional quantum gate and shows a relationship in which the error is canceled by the additional quantum gate; and determine the second quantum gate corresponding to the additional quantum gate by solving the linear equation for the variable. [Effects of the Invention]

[0008] One aspect of this is an improvement in the calibration accuracy of quantum gates. [Brief explanation of the drawing]

[0009] [Figure 1] This is a diagram illustrating the information processing device of the first embodiment. [Figure 2] This figure shows an example of the hardware of the information processing system according to the second embodiment. [Figure 3] This figure shows an example of a quantum circuit in which a calibration quantum gate is added to the target quantum gate. [Figure 4] This figure shows an example of adding a calibration quantum gate to an X90 gate. [Figure 5] This figure shows an example of adding a calibration quantum gate to an X180 gate. [Figure 6] This graph shows an example of X90 gate calibration using a first-order approximation. [Figure 7] This graph shows an example of X180 gate calibration using a first-order approximation. [Figure 8] This figure shows an example of error component classification for the ZX90 gate. [Figure 9] This figure shows an example of a transformation coefficient vector that takes non-commutativity into account. [Figure 10] This figure shows an example of adding a calibration quantum gate to a ZX90 gate. [Figure 11] This figure shows an example of the deployment of the first calibration quantum gate. [Figure 12] This figure shows an example of the deployment of the second calibration quantum gate. [Figure 13] This figure shows an example of the deployment of the third calibration quantum gate. [Figure 14] This figure shows an example of the deployment of the fourth calibration quantum gate. [Figure 15] This graph shows an example of the relationship between pre-calibration error and post-calibration error. [Figure 16] This graph shows an example of error reduction after calibration. [Figure 17] This is a block diagram showing examples of functions of an information processing device. [Figure 18] This flowchart shows an example of the procedure for calibrating a quantum gate. [Modes for carrying out the invention]

[0010] This embodiment will be described below with reference to the drawings. (a) First embodiment Figure 1 is a diagram illustrating an information processing device of a first embodiment. The information processing device 10 of the first embodiment calibrates quantum gates executed by a quantum computer. The information processing device 10 is, for example, a von Neumann type classical computer. The information processing device 10 may be a client device or a server device. The information processing device 10 may be called a computer or a quantum gate calibration device.

[0011] The information processing device 10 includes a storage unit 11 and a processing unit 12. The storage unit 11 may be a volatile memory such as RAM (Random Access Memory). Alternatively, the storage unit 11 may be a non-volatile storage such as an HDD (Hard Disk Drive) or SSD (Solid State Drive).

[0012] The processing unit 12 is a processor, such as a CPU (Central Processing Unit), a GPU (Graphics Processing Unit), or a DSP (Digital Signal Processor). However, the processing unit 12 may also include electronic circuits such as an ASIC (Application Specific Integrated Circuit) or an FPGA (Field Programmable Gate Array). The processor executes a program stored in memory, such as RAM. The processor is sometimes called a processor circuit. A collection of processors is sometimes called a multiprocessor or simply a "processor." Different processes among the multiple processes described later may be executed by different processors.

[0013] The memory unit 11 stores error data 14 indicating an error in the quantum gate 13a implemented in the quantum computer. The quantum gate 13a is the quantum gate to be calibrated. The error in the quantum gate 13a indicates the degree to which the operation of the quantum gate 13a deviates from the ideal value. The quantum gate 13a may be a one-input quantum gate that acts on one qubit, or a multi-input quantum gate that acts on two or more qubits.

[0014] The ideal value of quantum gate 13a may be expressed as a generator (Lindbladian). The generator corresponds to the matrix logarithm of the unitary matrix that represents the transformation of the quantum state. The error of quantum gate 13a may be expressed as a generator error that represents the divergence on the generator.

[0015] The information processing device 10 may generate error data 14 using a quantum computer, or it may obtain error data 14 from another information processing device. For example, the information processing device 10 has a quantum computer execute a quantum gate 13a and obtains experimental data showing the result of executing the quantum gate 13a. The information processing device 10 evaluates the error by analyzing the experimental data and generates error data 14. Examples of error evaluation methods include quantum process tomography, GST (Gate Set Tomography), HEAT (Hamiltonian Error Amplifying Tomography), and RB (Randomized Benchmarking).

[0016] The processing unit 12 performs calibration to reduce the error of the quantum gate 13a according to the error data 14. In the first embodiment, the processing unit 12 reduces the error by adding an additional quantum gate to the quantum gate 13a. The processing unit 12 determines a quantum gate 13b corresponding to the additional quantum gate and adds quantum gate 13b to quantum gate 13a. This generates a quantum circuit 13 including quantum gates 13a and 13b. Quantum gate 13b may include a pre-quantum gate added before quantum gate 13a, or a post-quantum gate added after quantum gate 13a.

[0017] In determining quantum gate 13b, the processing unit 12 generates a linear equation 16 using an approximation function 15. The approximation function 15 provides a linear approximation of the effect of the additional quantum gate on the operation of quantum gate 13a. A linear approximation is performed because, typically, the two matrices corresponding to two quantum gates connected in series do not commutate.

[0018] For example, matrices M1 and M2 may have the property of being non-commutative, meaning that their product M1×M2 and product M2×M1 are not identical. If matrices M1 and M2 are non-commutative, then the matrix exponent e of matrix M1 is... M1 and the matrix index e of matrix M2 M2 Regarding the product e M1 ×e M2 is e M1+M2 This does not match. Therefore, even if an additional quantum gate with generators opposite to those of quantum gate 13a is added to quantum gate 13a, the composite quantum gate formed by combining quantum gate 13a and the additional quantum gate may still have uncancellable generator errors.

[0019] The effect of an additional quantum gate on the operation of quantum gate 13a is generally nonlinear. Therefore, it is not easy to exactly calculate an additional quantum gate that can cancel the error of quantum gate 13a. Thus, the processing unit 12 uses an approximation function 15. The processing unit 12 may decompose the matrix representing quantum gate 13a (e.g., generators) into a plurality of eigenvalues ​​and a plurality of projection matrices by eigenvalue decomposition or spectral decomposition. The processing unit 12 may generate an approximation function 15 using the plurality of eigenvalues ​​and a plurality of projection matrices.

[0020] The linear equation 16 includes a variable corresponding to the additional quantum gate and shows a relationship in which the error of quantum gate 13a is canceled by the additional quantum gate. For example, the linear equation 16 shows that the action of the additional quantum gate transformed by the approximation function 15 matches the error shown in the error data 14 with the sign reversed.

[0021] Candidate additional quantum gates may also be rotation gates that rotate the quantum state by a fixed angle around a fixed rotation axis. The variables included in linear equation 16 may indicate the additional quantum gate to be used, or they may specify the rotation angle of the additional quantum gate. Linear equation 16 may also be expressed using a coefficient matrix and a right-hand side vector.

[0022] The processing unit 12 solves the linear equation 16 for the variables. From the solution values ​​of the variables, the processing unit 12 determines the quantum gate 13b corresponding to the additional quantum gate. The processing unit 12 may also solve the linear equation 16 analytically by calculating the inverse of the coefficient matrix. Alternatively, the processing unit 12 may calculate an approximate solution to the linear equation 16 using an iterative linear solver.

[0023] The processing unit 12 may determine the values ​​of control parameters that control the operation of the quantum gate 13b from the solution of the linear equation 16. For example, if the quantum gate is implemented using a microwave pulse signal, the processing unit 12 may determine the values ​​of control parameters related to the waveform, such as the time width and amplitude of the microwave pulse signal, from the rotation angle of the quantum gate 13b. Generally, the waveform area of ​​the microwave pulse signal corresponds to the rotation angle.

[0024] After calibration in the first embodiment, the quantum computer will execute the pair of quantum gate 13a and quantum gate 13b instead of executing quantum gate 13a alone. The addition of quantum gate 13b to quantum gate 13a may be performed automatically by the quantum computer, or the information processing device 10 may instruct the quantum computer to do so each time, or another information processing device may instruct the quantum computer each time. The processing unit 12 outputs a calibration result that includes information about quantum gate 13b. The processing unit 12 may store the calibration result in non-volatile storage, display it on a display device, or transmit it to another information processing device.

[0025] As described above, the information processing device 10 of the first embodiment acquires error data 14 indicating an error in the quantum gate 13a. The information processing device 10 generates a linear equation 16 using an approximation function 15. The approximation function 15 linearly approximates the effect that an additional quantum gate attached to the quantum gate 13a has on the operation of the quantum gate 13a. The linear equation 16 includes a variable corresponding to the additional quantum gate and shows a relationship in which errors are canceled out by the additional quantum gate. The information processing device 10 determines the quantum gate 13b corresponding to the additional quantum gate by solving the linear equation 16 for the variable.

[0026] This reduces the error of the quantum gate 13a implemented in the quantum computer. Furthermore, it reduces error components that are difficult to cancel simply by changing the control parameter values ​​of the quantum gate 13a, thereby improving calibration accuracy. Additionally, by using the approximation function 15, the relationship in which additional quantum gates cancel errors can be concisely expressed by the linear equation 16. Therefore, the information processing device 10 can efficiently determine the quantum gate 13b.

[0027] (b) Second embodiment Figure 2 shows an example of the hardware of the information processing system according to the second embodiment. The information processing system according to the second embodiment includes an information processing device 100 and a quantum computer 20. The information processing device 100 is a von Neumann type classical computer. The information processing device 100 calibrates the quantum gates implemented in the quantum computer 20. The information processing device 100 corresponds to the information processing device 10 of the first embodiment.

[0028] The information processing device 100 includes a CPU 101, RAM 102, HDD 103, GPU 104, input interface 105, media reader 106, communication interface 107, and interface 108. The CPU 101 corresponds to the processing unit 12 of the first embodiment. The RAM 102 or HDD 103 corresponds to the storage unit 11 of the first embodiment.

[0029] The CPU 101 is a processor that executes program instructions. The CPU 101 loads the program and data from the HDD 103 into the RAM 102 and executes the program. The information processing device 100 may have multiple processors.

[0030] RAM 102 is a volatile semiconductor memory that temporarily stores programs executed by CPU 101 and data used for calculations by CPU 101. The information processing device 100 may have a type of volatile memory other than RAM.

[0031] The HDD 103 is a non-volatile storage device that stores software programs such as the operating system, middleware, and application software, as well as other data. The information processing device 100 may have other types of non-volatile storage, such as an SSD or flash memory.

[0032] The GPU 104 works in conjunction with the CPU 101 to perform image processing and outputs the image to the display device 111 connected to the information processing device 100. The display device 111 is, for example, a CRT (Cathode Ray Tube) display, a liquid crystal display, an organic EL (Electro Luminescence) display, or a projector. The GPU 104 may also be used as a GPGPU (General Purpose Computing on Graphics Processing Unit). The GPU 104 can execute programs in response to instructions from the CPU 101. The information processing device 100 may have volatile semiconductor memory other than RAM 102 as GPU memory.

[0033] The input interface 105 receives input signals from an input device 112 connected to the information processing device 100. The input device 112 is, for example, a mouse, a touch panel, or a keyboard. Multiple input devices may be connected to the information processing device 100.

[0034] The media reader 106 is a reading device that reads programs and data recorded on the recording medium 113. The recording medium 113 is, for example, a magnetic disk, an optical disk, or semiconductor memory. Magnetic disks include flexible disks (FD) and HDDs. Optical disks include CDs (Compact Discs) and DVDs (Digital Versatile Discs). The media reader 106 copies the programs and data read from the recording medium 113 to other recording media such as RAM 102 or HDD 103. The read programs may be executed by the CPU 101.

[0035] The recording medium 113 may be a portable recording medium. The recording medium 113 may be used for distributing programs and data. The recording medium 113 and the HDD 103 may also be referred to as computer-readable recording media.

[0036] The communication interface 107 communicates with other information processing devices via the network 114. The communication interface 107 may be a wired communication interface connected to a wired communication device such as a switch or router, or a wireless communication interface connected to a wireless communication device such as a base station or access point.

[0037] Interface 108 is connected to the quantum computer 20. Interface 108 sends commands to the quantum computer 20 in response to instructions from the CPU 101. Interface 108 receives the execution results of the commands from the quantum computer 20 and stores the received execution results in RAM 102.

[0038] The quantum computer 20 has a quantum computing unit 21 and a control unit 22. The quantum computing unit 21 includes a plurality of qubits. The quantum computing unit 21 performs quantum operations such as initializing qubits, executing quantum gates on qubits, and measuring qubits, in response to instructions from the control unit 22. Quantum operations change the quantum state represented by the qubits. The behavior of quantum operations is adjusted by control parameter values. For example, the quantum computing unit 21 irradiates the qubits with microwave pulse signals having waveforms corresponding to the control parameter values.

[0039] The control unit 22 receives commands from the information processing device 100. Calibration commands include the names and values ​​of control parameters. The control unit 22 holds the control parameter values ​​included in the command and controls the quantum operation performed by the quantum computation unit 21. Examples of control parameters include the time width, amplitude, and phase of the microwave pulse signal. The control unit 22 also instructs the quantum computation unit 21 to perform quantum operations in response to quantum computation commands. Furthermore, the control unit 22 reads out the measured values ​​generated by the measurement of the qubits in response to a command to acquire measured values ​​and transmits them to the information processing device 100.

[0040] Generally, quantum information processing includes quantum computing, quantum simulation, quantum communication, quantum cryptography, and quantum sensing. Examples of physical platforms for quantum information processing include superconducting quantum circuits, semiconductor quantum dots, diamond NV centers, NMR molecules, neutral atoms, trapped ions, and light. Typical quantum information processing protocols based on quantum circuits utilize three types of quantum operations: initialization, quantum gates, and measurement.

[0041] The quantum operations implemented in the quantum computer 20 have errors that indicate a deviation from ideal quantum operations. The information processing device 100 performs evaluation and calibration on the quantum computer 20 to improve the accuracy of the quantum operations. Evaluation estimates the errors of the quantum operations. Calibration changes the control parameter values ​​based on the error data to reduce the errors. The information processing device 100 may repeat the evaluation and calibration process.

[0042] Next, we will explain the errors of quantum gates. The action of a quantum gate on a quantum state is described by a unitary matrix. If we denote the Hamiltonian that describes the time evolution of the quantum system in question as H(t), then the unitary matrix U that represents the action of the quantum gate realized by the time evolution from time 0 to time t is defined as shown in equation (1). In equation (1), T is Dyson's time order operator and e is the matrix exponential function.

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[0044] A unitary matrix is ​​a matrix whose product with its adjoint matrix is ​​the identity matrix. The adjoint matrix is ​​the matrix obtained by transposing the original matrix and taking its complex conjugate. For a unitary matrix U, there exists a Hermitian matrix A that satisfies the second equality in equation (1). A Hermitian matrix is ​​a matrix whose adjoint matrix is ​​equal to the original matrix. In the second embodiment, this Hermitian matrix A is sometimes called a quantum gate generator. The generator corresponds to the matrix logarithm of the unitary matrix U that represents the quantum gate.

[0045] A quantum circuit is a quantum computing model that describes the execution procedure of multiple quantum gates. Typically, the multiple quantum gates included in a quantum circuit are executed in an order from left to right. A composite quantum gate, which represents the overall action of multiple quantum gates executed in series, is represented by the product of multiple unitary matrices corresponding to those quantum gates. In this case, the unitary matrices of the quantum gates executed first are placed on the right, and the unitary matrices of the quantum gates executed later are placed on the left. Therefore, the execution order is reversed between quantum circuits and matrix operations.

[0046] A quantum gate has the effect of rotating a quantum state by a specific angle around a rotation axis specified by a Hermitian matrix A. If we denote the matrix specifying the rotation axis as P and the rotation angle as θ, then the quantum gate is P θIt may be expressed as. When the matrix P is a Pauli matrix or a tensor product of two or more Pauli matrices, the quantum gate P θ is expressed as in Equation (2).

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[0048] For example, the X90 gate (X π / 2 ) that rotates 90 degrees around the X-axis is expressed as in Equation (3). The X90 gate is a one-input quantum gate that acts on one qubit. Also, the ZX90 gate (ZX π / 2 ) that rotates 90 degrees around the ZX-axis is expressed as in Equation (4). The ZX90 gate is a two-input quantum gate that acts on two qubits.

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[0051] Let A be the ideal value of the generator and ε be the generator error indicating the deviation from the ideal value. Also, let ΔA be the difference in the generator that can be changed by adjusting the control parameter of the quantum gate itself. Then, the unitary matrix U of the quantum gate implemented in the quantum computer 20 is expressed as in Equation (5). In the second embodiment, it is assumed that the influence of the change in ΔA on the generator error ε is sufficiently small and that ε and ΔA are independent.

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[0053] Ideally, the generator error ε should be completely canceled, such as ΔA = -ε, by adjusting the control parameters of the quantum gate itself. However, for some quantum gates, the control parameters may only affect a portion of the angular components of the generator, and some error components may remain that cannot be canceled even if the control parameter values ​​are changed. Therefore, the information processing device 100 attempts to reduce the generator error by adding calibration quantum gates before and after the target quantum gate during the calibration of the target quantum gate.

[0054] Figure 3 shows an example of a quantum circuit in which calibration quantum gates are added to a target quantum gate. Quantum gate 130 is the target quantum gate to be calibrated. Quantum gate 131 is a calibration quantum gate added before quantum gate 130. Quantum gate 132 is a calibration quantum gate added after quantum gate 130. Quantum gate 133 is a composite quantum gate in which quantum gates 131, 132, and 133 are considered as a single quantum gate.

[0055] The unitary matrix U' of quantum gate 133 is expressed as shown in equation (6). By adding quantum gates 131 and 132 to quantum gate 130, the unitary matrix U in equation (5) changes to the unitary matrix U' in equation (6). In Figure 3 and equation (6), ΔB is a generator of quantum gate 131, and ΔC is a generator of quantum gate 132.

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[0057] Here, due to the non-commutativity of the generators, the generators of quantum gate 133 do not coincide with A+ε+ΔA+ΔB+ΔC, but rather become A+ε+ΔA+ΔB'+ΔC'. In general, the product of the matrix exponents of matrix A and matrix exponents of matrix B exhibits non-commutativity as shown in equation (7). Therefore, the generator ΔB of quantum gate 131 changes to ΔB' through composition, and the generator ΔC of quantum gate 132 changes to ΔC' through composition. The unitary matrix U' of quantum gate 133 can also be expressed as shown in equation (8). Δ' indicates the influence that quantum gates 131 and 132 have on the generators of quantum gate 130, and strictly speaking depends on A, ε, ΔA, ΔB, and ΔC.

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[0060] The information processing device 100 prefers to select ΔA, ΔB, and ΔC such that ε' in equation (8) becomes zero, and it is preferable to select ΔA, ΔB, and ΔC such that ΔA + ΔB' + ΔC' = -ε. However, Δ' is nonlinear with respect to A, ε, ΔA, ΔB, and ΔC, and it is not easy to find an exact solution for ΔA, ΔB, and ΔC. Therefore, the information processing device 100 introduces an approximation function that expresses Δ' as a first-order approximation with respect to ε.

[0061] Regarding the approximation of the product of matrix indices, there is the BCH (Baker-Campbell-Hausdorff) formula. Equation (9) represents the BCH formula. However, the BCH formula assumes that the sum of the norm of matrix A (e.g., the Frobenius norm) and the norm of matrix B is less than ln² (where ln is the natural logarithm). On the other hand, many quantum gate generators do not satisfy this assumption. For this reason, it is difficult to accurately approximate the composition of quantum gates with the BCH formula.

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[0063] First, the information processing device 100 performs eigenvalue decomposition of the matrix A relating to the target quantum gate, as shown in equation (10). In equation (10), a j P is the j-th eigenvalue (j=1,2,...). j is the j-th projection matrix. The information processing device 100 processes the pair of eigenvalues ​​a of matrix A. j ,a k For each, the coefficient l shown in formula (11) jk Calculate.

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[0066] Then, the information processing device 100 generates the projection matrix P j ,P k and coefficient l jk Using this, the function cml shown in equation (12) A And the function cmr shown in equation (13) A The function cml defines the following. A ,cmr A This is a linear function with respect to matrix A.

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[0069] Matrix index e of matrix B B and the matrix index e of matrix A A Product of e B ×e A is the function cmlA Using (B), it can be expressed as in equation (14). In equation (14), ||B|| 2 is a term of order two or higher with respect to matrix B. Therefore, the product e B ×e A In the range of a first-order approximation, A + cml A (B) is approximated by the matrix exponents. Also, the product e A ×e B The function cmr A Using (B), it can be expressed as in equation (15). Therefore, the product e A ×e B In the range of a first-order approximation, A+cmr A This is approximated by the matrix exponents in (B).

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[0072] The information processing device 100 uses the unitary matrix U' of the quantum gate 133 to perform the function cml -iA ,cmr -iA We approximate it linearly using equation (16): ΔB'=cmr -iA (ΔB) and ΔC' = cml -iA (ΔC) Therefore, the information processing device 100 selects ΔA, ΔB, and ΔC to cancel the generator error ε so as to satisfy equation (17).

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[0075] Furthermore, the information processing device 100 does not need to adjust the control parameters of the target quantum gate. In that case, ΔA = O. Also, the information processing device 100 may place a calibration quantum gate either before or after the target quantum gate. If a calibration quantum gate is not placed before the target quantum gate, ΔB = O. If a calibration quantum gate is not placed after the target quantum gate, ΔC = O.

[0076] Furthermore, the calibration quantum gate preceding the target quantum gate may be expanded into two or more quantum gates. The order of the expanded two or more quantum gates is usually interchangeable as it does not affect the first-order approximation. Similarly, the calibration quantum gate following the target quantum gate may be expanded into two or more quantum gates. The order of the expanded two or more quantum gates is usually interchangeable as it does not affect the first-order approximation.

[0077] The information processing device 100 selects ΔA, ΔB, and ΔC as follows: {ΔA} is the set of generators available for use as ΔA. α The set of generators available as ΔB is {ΔB}. α The set of generators available as ΔC is {ΔC}. α This is written as}. Multiple ΔA α This corresponds, for example, to the multiple control parameters of the target quantum gate. α This corresponds to multiple rotation gates such as X-axis rotation gates, Y-axis rotation gates, and Z-axis rotation gates. Similarly, multiple ΔC α It supports multiple rotary gates.

[0078] ΔA, ΔB, and ΔC can be decomposed into linear sums of available generators, as shown in equation (18). ΔA α The coefficient parameter ν that acts on it α This corresponds to the amount of adjustment to the rotation angle by the target quantum gate itself. ΔB α The coefficient parameter ν that acts on it α This corresponds to the rotation angle of the preceding calibration quantum gate. ΔC α The coefficient parameter ν that acts on it αThis corresponds to the rotation angle of the subsequent calibration quantum gate. Furthermore, to handle rotation angles that are linked between ΔA, ΔB, and ΔC, such as when the preceding and subsequent calibration quantum gates are linked, the coefficient parameter ν is used. α is ΔA α ,ΔB α ,ΔC α It acts in common with them.

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[0080] Substituting equation (18) into equation (17) and rearranging, we obtain equation (19). Equation (19) is the coefficient parameter ν α This is a linear equation with variable ν. The information processing device 100 uses this linear equation with coefficient parameter ν α We will solve for this. For example, the information processing device 100 solves the linear equation of formula (19) by the following method.

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[0082] The function that vectorizes a matrix is ​​denoted as vec, and the coefficient parameter ν α The vector representing the enumerated elements is denoted as ν. The information processing device 100 also generates a matrix Λ whose elements in row β and column α are defined as shown in equation (20). Then, equation (19) can be expressed as equation (21) using the matrix Λ, the vector ν, and the generator error ε. If the matrix Λ is a square and invertible matrix, the information processing device 100 can analytically solve the linear equation as shown in equation (22) using the inverse matrix of matrix Λ. The information processing device 100 can also calculate an approximate solution for ν in equation (21) using an iterative linear solver.

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[0086] In this way, the information processing device 100 determines the coefficient parameter ν α Calculate the value of the coefficient parameter ν. α From the value of ν, the information processing device 100 determines the quantum gate to be used as the calibration quantum gate in the preceding stage and the quantum gate to be used as the calibration quantum gate in the subsequent stage. For example, the information processing device 100 determines the coefficient parameter ν α We decide not to use quantum gates whose absolute value is zero or less than the threshold.

[0087] Furthermore, the information processing device 100 uses the coefficient parameter ν α From the value of ν, the amount of adjustment of the rotation angle by the target quantum gate is determined, and the control parameter value to realize that adjustment amount is determined. In addition, the information processing device 100 determines the coefficient parameter ν α From the value of ν, the rotation angle of the preceding calibration quantum gate is determined, and the control parameter values ​​(e.g., the time width and amplitude of the microwave pulse signal) to realize that rotation angle are determined. Similarly, the information processing device 100 determines the coefficient parameter ν α From this value, the rotation angle of the subsequent calibration quantum gate is determined, and the control parameter values ​​are determined.

[0088] Next, we will explain examples of calibration for one-input quantum gates and two-input quantum gates. Below, as examples of one-input quantum gates, we will use the X90 gate and the X180 gate (which rotates 180 degrees around the X-axis). As an example of a two-input quantum gate, we will use the ZX90 gate.

[0089] First, a quantum gate X rotates the quantum state, represented by one qubit, around the X axis by a rotation angle θ. θ Let's consider X θIn implementing this, the quantum computer 20 may generate microwave pulse signals using a method called DRAG (Derivative Removal by Adiabatic Gate). DRAG inserts the original waveform signal into the in-phase channel (I channel) and the derivative signal of the I channel into the quadrature-phase channel (Q channel). DRAG suppresses leakage transitions, which are unintentional transitions in the energy of a qubit to the third energy level |f>.

[0090] DRAG is also described in the following publication: F. Motzoi, JM Gambetta, P. Rebentrost, and FK Wilhelm, "Simple Pulses for Elimination of Leakage in Weakly Nonlinear Qubits", Physics Review Letters, Volume 103, Issue 11, September 2009.

[0091] However, DRAG can, as a side effect, generate a generator error in the Z component. Here, we consider canceling out the generator error in the Z component caused by DRAG with the generator error in the Y component originating from higher-order terms in the Magnus expansion.

[0092] Quantum gate X θ The function corresponding to cmr -iθ / 2X It is calculated as shown in equation (23). Also, quantum gate X θ The function corresponding to cml -iθ / 2X It is calculated as shown in formula (24). When 0 < θ < π, that is, when the rotation angle θ is greater than 0 degrees and less than 180 degrees, X θ By placing one Z-axis rotation gate before and one after the function, both the Y-component generator error and the Z-component generator error are canceled out.

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[0095] In this case, ΔA, ΔB, and ΔC are defined as shown in equation (25). Quantum gate X θ No adjustments are made using its own control parameters. The rotation angle of the preceding Z-axis rotation gate is determined by the angle parameter θ used to cancel out the Y component. Y And the angle parameter θ for canceling out the Z component. Z Using θ, Z +θ Y It is expressed as follows. Also, the rotation angle of the subsequent Z-axis rotation gate is θ Z -θ Y It is expressed as follows: Angular parameter θ Y Regarding this, the Z-axis rotation gate in the preceding stage and the Z-axis rotation gate in the subsequent stage have symmetrical rotation angles.

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[0097] Figure 4 shows an example of adding calibration quantum gates to an X90 gate. Quantum gate 134 is an X90 gate with a rotation angle θ of 90 degrees. Quantum gate 135 is added before quantum gate 134, and quantum gate 136 is added after quantum gate 135. Quantum gate 135 is a Z-axis rotation gate used as a calibration quantum gate. The rotation angle of quantum gate 135 is θ. Z +θ Y Quantum gate 136 is a Z-axis rotation gate used as a calibration quantum gate. The rotation angle of quantum gate 136 is θ. Z -θ Y That is the case.

[0098] However, the information processing device 100 uses the quantum gate X θ By using a virtual Z gate as an implementation, the Y and Z components of the generator error can be canceled without increasing the number of quantum gates. The virtual Z gate is a quantum gate X θThis quantum gate can achieve the same effect as when two Z-axis rotation gates are added before and after it, by adjusting the microwave pulse signal for that purpose. The effect of the two Z-axis rotation gates on the generator is expressed as shown in equation (26).

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[0100] Next, consider the case where θ=π, that is, the case where the target quantum gate is an X180 gate. The function cmr corresponding to the X180 gate is... -i(π / 2)X It is calculated as shown in formula (27). Also, the function cml corresponds to the X180 gate. -i(π / 2)X It is calculated as shown in formula (28).

[0101]

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[0102]

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[0103] The generator error of an X180 gate can be canceled by adding a calibration quantum gate either before or after the X180 gate. In the following explanation, the calibration quantum gate is added after the X180 gate. However, a Z-axis rotation gate acts on the Y component of the generator error, and a Y-axis rotation gate acts on the Z component of the generator error.

[0104] ΔA, ΔB, and ΔC are defined as shown in equation (29). No adjustment is made by the control parameters of the X180 gate itself. Also, no calibration quantum gate is added before the X180. The calibration quantum gate after the X180 includes a Y-axis rotation gate and a Z-axis rotation gate. The rotation angle of the Y-axis rotation gate is θ to cancel the Z component. Z Therefore, the rotation angle of the Z-axis rotation gate is θ to cancel out the Y component.Y is as follows.

[0105]

Number

[0106] FIG. 5 is a diagram showing an example of adding a calibration quantum gate to an X180 gate. Quantum gate 137 is an X180 gate. A quantum gate 138 is added to the subsequent stage of the quantum gate 137, and a further quantum gate 139 is added to the subsequent stage of the quantum gate 138. However, the order of the quantum gates 138 and 139 may be reversed. The quantum gates 138 and 139 are calibration quantum gates. The quantum gate 138 is a Y-axis rotation gate with a rotation angle of θ Z and the quantum gate 139 is a Z-axis rotation gate with a rotation angle of θ Y .

[0107] Here, a numerical example of the calibration of the X90 gate will be described. The Pauli matrices I, X, Y, Z of a single qubit system are defined as in Equation (30). When a calibration quantum gate of the generator ΔB is added to the previous stage of the X90 gate and a calibration quantum gate of the generator ΔC is added to the subsequent stage of the X90 gate, the unitary matrix U' of the composite quantum gate is approximated as in Equation (31).

[0108]

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[0109]

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[0110] Assume that ΔB and ΔC are defined as in Equation (32). The calibration quantum gate in the previous stage is a Z-axis rotation gate with a rotation angle of θ Z -θ Y and the calibration quantum gate in the subsequent stage is a Z-axis rotation gate with a rotation angle of θ Z +θ Y . For convenience of explanation in Equation (32), the angular parameter θ YThe sign of the function is reversed compared to Figure 4 and equation (25).

[0111]

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[0112] The influence that the two Z-axis rotation gate generators ΔB and ΔC have on the X90 gate generator is calculated as shown in the first equation of equation (33). Therefore, the θ that satisfies the second equation of equation (33) is... Y ,θ Z This cancels the generator error ε. The Y component included in the generator error ε is ε Y , the Z component included in the generator error ε Z Then, θ Y ,θ Z It is calculated as shown in formula (34). ε Y =π / 2×0.01,ε Z If = -π / 2 × 0.02, then as shown in formula (35), θ Y =-0.02,θ Z = 0.04.

[0113]

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[0114]

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[0115]

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[0116] Next, we will explain the approximation accuracy of the combination of the target quantum gate and the calibration quantum gate. Here, we consider the case where a Z-axis rotation gate with a rotation angle θ is added after the target quantum gate having a generator H0. As shown in equation (36), the generator of the combined quantum gate is H0 + Δ(θ). Δ(θ) indicates the effect of the calibration quantum gate and depends on the rotation angle θ.

[0117]

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[0118] Figure 6 is a graph showing an example of calibration of an X90 gate using a first-order approximation. Graph 141 shows the relationship between θ and Δ(θ) when the target quantum gate is an X90 gate. Curve 142 plots the X component included in Δ(θ) up to the second-order term. Line 143 shows the Y and Z components included in Δ(θ). In the case of the X90 gate, the first-order approximation of the Y component matches the exact value, and the first-order approximation of the Z component matches the exact value.

[0119] Figure 7 is a graph showing an example of calibration of an X180 gate using a first-order approximation. Graph 144 shows the relationship between θ and Δ(θ) when the target quantum gate is an X180 gate. Curve 145 plots the X component included in Δ(θ) up to the second-order term. Curve 146 shows the exact value of the Y component included in Δ(θ). Line 147 shows the Z component included in Δ(θ). Line 148 shows the analytical solution corresponding to the first-order approximation of the Y component included in Δ(θ).

[0120] In the case of the X180 gate, the first-order approximation of the Z component matches the exact value. On the other hand, the first-order approximation of the Y component does not match the exact value. However, in the region where θ is close to 0, the line 148 approximates the curve 146 with high accuracy, and the Y component is approximated with sufficient accuracy for practical purposes.

[0121] Next, the calibration of the ZX90 gate implemented by cross resonance (CR) will be described. The calibration quantum gate described below is just one example, and the information processing device 100 may use other types of quantum gates as calibration quantum gates. The combination of quantum gates to be used may be determined through optimization by the information processing device 100, or it may be specified by the user of the information processing device 100.

[0122] The generator error of a two-input quantum gate is expanded into 15 error components, excluding component II. It is preferable that these 15 error components be calibrated as independently as possible. Furthermore, it is preferable that each error component be calibrated using a one-input quantum gate as much as possible. The 15 error components are classified into five categories based on whether they can be calibrated by the ZX90 gate itself, their non-commutativity with the ideal value A of the generator, and the number of qubits involved.

[0123] Figure 8 shows an example of the classification of error components in a ZX90 gate. Table 126 shows the classification of generator errors in a ZX90 gate. Category 0 represents error components that can be directly calibrated by the control parameters of the ZX90 gate. Category 0 includes the ZX and ZY components. Category 1 represents error components that are commutative with the ideal value A of the generator and act on one qubit. Category 1 includes the ZI and IX components.

[0124] Category 2 represents error components that are commutative with the ideal value A of the generator and act on two qubits. Category 2 includes the XY, XZ, YZ, and YY components. Category 3 represents error components that are noncommutative with the ideal value A of the generator and act on one qubit. Category 3 includes the XI, YI, IY, and IZ components. Category 4 represents error components that are noncommutative with the ideal value A of the generator and act on two qubits. Category 4 includes the XX, YX, and ZZ components.

[0125] Error components belonging to Category 0 are calibrated through adjustment of the control parameters of the ZX90 gate. Error components belonging to Category 1 or Category 2 are calibrated by a calibration quantum gate placed either before or after the ZX90 gate. However, the calibration quantum gate for Category 2 error components acting on two qubits is implemented by combining a ZX gate or ZY gate with a one-input quantum gate.

[0126] Error components belonging to Category 3 or Category 4 cannot be easily calibrated using only a calibration quantum gate placed either before or after the ZX90 gate. This is because secondary error components are generated due to non-commutativity. Therefore, the information processing device 100 places two calibration quantum gates of the same type with opposite rotation angles before and after the ZX90 gate. However, the calibration quantum gates for Category 3 or Category 4 error components can be implemented using a single-input quantum gate. Thus, a two-input quantum gate is not required for Category 1, 3, and 4 error components.

[0127] The following describes an example of the structure of a calibration quantum gate. Here, we assume that the ZX gate and ZY gate are available as two-input quantum gates, and the X-axis rotation gate, Y-axis rotation gate, and Z-axis rotation gate are available as one-input quantum gates. Note that the relationship between the angle parameter θ and the coefficient parameter ν is θ = 2ν.

[0128] ΔA is the matrix ZX, ZY and coefficient parameter ν ZX ν ZY It is defined as shown in equation (37) using the matrix IY, IZ, XI, YI and coefficient parameter ν. IY ν ZZ ν IZ ν XI ν YX ν YI ν XX It is defined as shown in equation (38) using the matrix IX, ZI, IY, IZ, XI, YI, XY, XZ, YY, YZ and coefficient parameter ν IX ν ZI ν IY ν ZZ ν IZ ν XI ν YX ν YI ν XX ν XY ν XZ ν YY ν YZ It is defined as shown in formula (39) using [the given formula].

[0129]

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[0130]

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[0131]

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[0132] Let's assume that we select a normalized Pauli matrix basis for a 2-qubit system as the representation basis for the vectorization function vec. Also, as shown in equation (40), let the vector λ be a vector obtained by arranging the diagonal elements of the matrix 1 / 2Λ. Then, λ is calculated as follows.

[0133]

number

[0134] Figure 9 shows an example of a transformation coefficient vector considering non-commutativity. Vector 127 represents vector λ. The first dimension of vector 127 is "1", which corresponds to the IX component of category 1. The second dimension of vector 127 is "π / 2", which corresponds to the IY component of category 3. The third dimension of vector 127 is "π / 2", which corresponds to the IZ component of category 3. The fourth dimension of vector 127 is "π / 2", which corresponds to the XI component of category 3.

[0135] The fifth dimension of vector 127 is "π / 2", which corresponds to the XX component of category 4. The sixth dimension of vector 127 is "1", which corresponds to the XY component of category 2. The seventh dimension of vector 127 is "1", which corresponds to the XZ component of category 2. The eighth dimension of vector 127 is "π / 2", which corresponds to the YI component of category 3. The ninth dimension of vector 127 is "π / 2", which corresponds to the YX component of category 4.

[0136] The 10th dimension of vector 127 is "1", which corresponds to the YY component of category 2. The 11th dimension of vector 127 is "1", which corresponds to the YZ component of category 2. The 12th dimension of vector 127 is "1", which corresponds to the ZI component of category 1. The 13th dimension of vector 127 is "1", which corresponds to the ZX component of category 0. The 14th dimension of vector 127 is "1", which corresponds to the ZY component of category 0. The 15th dimension of vector 127 is "π / 2", which corresponds to the ZZ component of category 4.

[0137] Therefore, among the elements of vector λ, the elements corresponding to the error components of categories 0, 1, and 2 are "1", and the elements corresponding to the error components of categories 3 and 4 are "π / 2". The generators for the error components of categories 0, 1, and 2 are composed as is, so the conversion coefficient is "1". On the other hand, the generators for the error components of categories 3 and 4 are affected by non-commutativity, so the conversion coefficient is "π / 2".

[0138] The generator error ε is the Pauli matrix P α and the Pauli procession P α The corresponding rotation angle θ ε,α Using these, it is expanded as shown in equation (41). α represents 15 different rotation axes. Since the matrix Λ is diagonalized, the angular parameter θ of the calibration quantum gate is used to reduce the error component of the rotation axis α. α This is calculated as shown in equation (42). This indicates that the 15 error components that make up the generator error can be calibrated independently of each other.

[0139]

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[0140]

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[0141] Therefore, the information processing device 100 can reduce the cost of error evaluation and streamline calibration. Furthermore, the information processing device 100 may employ a sweep method that repeatedly evaluates and calibrates generator errors while gradually narrowing the range of rotation angles of the calibration quantum gate. The independence of the error components is well-suited to the sweep method and is a practically advantageous characteristic.

[0142] Figure 10 shows an example of adding calibration quantum gates to a ZX90 gate. Quantum gate 201 is a ZX90 gate. Quantum gate 201 has the generator π / 4ZX+ε+ΔA. Quantum gate 202 is added before quantum gate 201, and quantum gate 203 is added after quantum gate 201. Quantum gate 202 is a calibration quantum gate with the generator ΔB. Quantum gate 203 is a calibration quantum gate with the generator ΔC.

[0143] Quantum gate 202 expands into quantum gates 211, 212, 213, and 214. Quantum gate 211 is an X-axis rotation gate acting on the first qubit, with a rotation angle θ. XI +θ YX It has. Quantum gate 212 is a Y-axis rotation gate that acts on the second qubit, with a rotation angle θ IY +θ ZZ It has. Quantum gate 213 is a Y-axis rotation gate that follows quantum gate 211, with a rotation angle θ YI -θ XX It has. Quantum gate 214 is a Z-axis rotation gate that follows quantum gate 212, with a rotation angle θ IZ It has.

[0144] Quantum gate 203 expands into quantum gates 221, 222, 223, 224, 225, 226, 230, 240, 250, and 260. Quantum gate 221 is an X-axis rotation gate acting on the first qubit, with a rotation angle θ. XI -θ YX It has the following characteristics. Quantum gates 211 and 221 are a pair and have symmetrical angular parameters. Quantum gate 222 is an X-axis rotation gate acting on the second qubit, with a rotation angle θ IX It has.

[0145] Quantum gate 223 is a Y-axis rotation gate that follows quantum gate 221, with a rotation angle θ. YI +θ XX It has the following characteristics. Quantum gate 213 and quantum gate 223 are a pair and have symmetrical angular parameters. Quantum gate 224 is a Y-axis rotation gate that follows quantum gate 222, with a rotation angle θ IY -θ ZZ It has the following characteristics. Quantum gate 212 and quantum gate 224 are a pair and have symmetrical angular parameters. Quantum gate 225 is a Z-axis rotation gate following quantum gate 223, with a rotation angle θ ZI It has. Quantum gate 226 is a Z-axis rotation gate that follows quantum gate 224, with a rotation angle θ IZ It has. Quantum gates 214 and 226 are a pair.

[0146] Quantum gate 230 is a two-input quantum gate that follows quantum gates 225 and 226. Quantum gate 230 has a rotation angle θ XY It is an XY gate with the following characteristics. Quantum gate 240 is located after quantum gate 230, with a rotation angle θ. XZ This is an XZ gate with [a specific characteristic]. Quantum gate 250 is located after quantum gate 240, with a rotation angle θ. YY This is a YY gate. Quantum gate 260 is located after quantum gate 250, with a rotation angle θ. YZ It is a YZ gate.

[0147] Note that changes in the order of quantum gates only affect higher-order terms of order two or higher with respect to the generator error ε, and do not affect the first-order approximation. Therefore, the order of quantum gates 211 and 213 may be reversed. The order of quantum gates 212 and 214 may also be reversed. The order of quantum gates 221, 223, and 225 may also be changed. The order of quantum gates 222, 224, and 226 may also be changed. The order of quantum gates 230, 240, 250, and 260 may also be changed.

[0148] Figure 11 shows an example of the deployment of the first calibration quantum gate. Quantum gate 230 is deployed into quantum gates 231, 232, 233, 234, and 235. Quantum gate 231 is a Y-axis rotation gate acting on the first qubit, with a rotation angle θ -π / 2 It has. Quantum gate 232 is a Z-axis rotation gate that acts on the second qubit, with a rotation angle θ -π / 2 It has.

[0149] Quantum gate 233 is a two-input quantum gate that follows quantum gates 231 and 232. Quantum gate 233 has a rotation angle θ XY This is a ZX gate. Quantum gate 234 is a single-input quantum gate that follows quantum gate 233. Quantum gate 234 is a Y-axis rotation gate that acts on the first qubit, with a rotation angle θ +π / 2 It has. Quantum gate 235 is a single-input quantum gate that follows quantum gate 233. Quantum gate 235 is a Z-axis rotation gate that acts on the second qubit, with a rotation angle θ +π / 2 It has.

[0150] Quantum gate 230 is equivalent to quantum gate 230a. The information processing device 100 may use quantum gate 230a instead of quantum gate 230. Quantum gate 230a is expanded into quantum gates 231, 234, and 236. Quantum gate 236 is a two-input quantum gate that follows quantum gate 231. Quantum gate 236 has a rotation angle θ XY This is a ZY gate. Quantum gate 234 is located after quantum gate 236.

[0151] Figure 12 shows an example of the deployment of the second calibration quantum gate. Quantum gate 240 is deployed into quantum gates 241, 242, 243, 244, and 245. Quantum gate 241 is a Y-axis rotation gate acting on the first qubit, with a rotation angle θ -π / 2 It has. Quantum gate 242 is a Y-axis rotation gate that acts on the second qubit, with a rotation angle θ +π / 2 It has.

[0152] Quantum gate 243 is a two-input quantum gate that follows quantum gates 241 and 242. Quantum gate 243 has a rotation angle θ XZ This is a ZX gate. Quantum gate 244 is a single-input quantum gate that follows quantum gate 243. Quantum gate 244 is a Y-axis rotation gate that acts on the first qubit, with a rotation angle θ +π / 2 It has. Quantum gate 245 is a single-input quantum gate that follows quantum gate 243. Quantum gate 245 is a Y-axis rotation gate that acts on the second qubit, with a rotation angle θ -π / 2 It has.

[0153] Quantum gate 240 is equivalent to quantum gate 240a. The information processing device 100 may use quantum gate 240a instead of quantum gate 240. Quantum gate 240a expands into quantum gates 241, 244, 246, 247, and 248. Quantum gate 246 is an X-axis rotation gate acting on the second qubit, with a rotation angle θ -π / 2 It has.

[0154] Quantum gate 247 is a two-input quantum gate that follows quantum gates 241 and 246. Quantum gate 247 has a rotation angle θ XZ This is a ZY gate. Quantum gate 244 is located after quantum gate 247. Quantum gate 248 is a single-input quantum gate located after quantum gate 247. Quantum gate 248 is an X-axis rotation gate acting on the second qubit, with a rotation angle θ +π / 2 It has.

[0155] Figure 13 shows an example of the deployment of the third calibration quantum gate. Quantum gate 250 is deployed into quantum gates 251, 252, 253, 254, and 255. Quantum gate 251 is an X-axis rotation gate acting on the first qubit, with a rotation angle θ +π / 2 It has. Quantum gate 252 is a Z-axis rotation gate that acts on the second qubit, with a rotation angle θ -π / 2 It has.

[0156] Quantum gate 253 is a two-input quantum gate that follows quantum gates 251 and 252. Quantum gate 253 has a rotation angle θYY It is a ZX gate having. Quantum gate 254 is a one-input quantum gate at the subsequent stage of quantum gate 253. Quantum gate 254 is an X-axis rotation gate acting on the first qubit, with a rotation angle θ -π / 2 having. Quantum gate 255 is a one-input quantum gate at the subsequent stage of quantum gate 253. Quantum gate 255 is a Z-axis rotation gate acting on the second qubit, with a rotation angle θ +π / 2 having.

[0157] Quantum gate 250 is equivalent to quantum gate 250a. Information processing apparatus 100 may use quantum gate 250a instead of quantum gate 250. Quantum gate 250a is expanded into quantum gates 251, 254, 256. Quantum gate 256 is a two-input quantum gate at the subsequent stage of quantum gate 251. Quantum gate 256 is a ZY gate having a rotation angle θ YY having. Quantum gate 254 is located at the subsequent stage of quantum gate 256.

[0158] FIG. 14 is a diagram showing an expansion example of a fourth calibration quantum gate. Quantum gate 260 is expanded into quantum gates 261, 262, 263, 264, 265. Quantum gate 261 is an X-axis rotation gate acting on the first qubit, with a rotation angle θ +π / 2 having. Quantum gate 262 is a Y-axis rotation gate acting on the second qubit, with a rotation angle θ +π / 2 having.

[0159] Quantum gate 263 is a two-input quantum gate at the subsequent stage of quantum gates 261, 262. Quantum gate 263 is a ZX gate having a rotation angle θ YZ having. Quantum gate 264 is a one-input quantum gate at the subsequent stage of quantum gate 263. Quantum gate 264 is an X-axis rotation gate acting on the first qubit, with a rotation angle θ -π / 2 having. Quantum gate 265 is a one-input quantum gate at the subsequent stage of quantum gate 263. Quantum gate 265 is a Y-axis rotation gate acting on the second qubit, with a rotation angle θ -π / 2 having.

[0160] Quantum gate 260 is equivalent to quantum gate 260a. The information processing device 100 may use quantum gate 260a instead of quantum gate 260. Quantum gate 260a expands into quantum gates 261, 264, 266, 267, and 268. Quantum gate 266 is an X-axis rotation gate acting on the second qubit, with a rotation angle θ -π / 2 It has.

[0161] Quantum gate 267 is a two-input quantum gate that follows quantum gates 261 and 266. Quantum gate 267 has a rotation angle θ YZ This is a ZY gate. Quantum gate 264 is located after quantum gate 267. Quantum gate 268 is a single-input quantum gate located after quantum gate 267. Quantum gate 268 is an X-axis rotation gate acting on the second qubit, with a rotation angle θ +π / 2 It has the following characteristics. Next, we will describe an example of the calibration results for the ZX90 gate.

[0162] Figure 15 is a graph showing an example of the relationship between pre-calibration error and post-calibration error. Graph 151 shows the relationship between the norm of the generator error ε before calibration (e.g., the Frobenius norm) and the norm of the generator error ε' after calibration according to the second embodiment.

[0163] In generating graph 151, the information processing device 100 randomly generates a generator error ε for the ZX90 gate. This generator error ε contains 15 random error components. The information processing device 100 solves a linear equation using the generated generator error ε to determine the calibration quantum gate. The information processing device 100 numerically calculates the remaining generator error ε' in the quantum circuit including the ZX90 gate and the calibration quantum gate. The information processing device 100 repeats the above process to generate a large number of samples. Equation (43) shows the unitary matrix U before calibration and the unitary matrix U' after calibration of the ZX90 gate.

[0164]

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[0165] Graph 151 plots a number of generated samples. In Graph 151, the horizontal and vertical axes are logarithmic scales. Line 152 indicates the point where the norm of the generator error ε is equal to the norm of the generator error ε'. Line 153 indicates the point where the square of the norm of the generator error ε is equal to the norm of the generator error ε'. As shown in Graph 151, all samples are plotted below line 153. Thus, Graph 151 shows that the linear component is eliminated from the generator error ε by calibration.

[0166] The information processing device 100 may repeatedly perform evaluation and calibration of the generator error for the same target quantum gate. The first calibration eliminates the first-order component from the generator error ε. When the target quantum gate with the calibration quantum gate attached is executed and evaluated by the quantum computer 20, a generator error ε'1 is calculated that includes higher-order components of the original generator error ε. The information processing device 100 performs a second calibration using this generator error ε'1. Theoretically, the second calibration eliminates the second-order component of the original generator error ε from the generator error ε'1.

[0167] In the second calibration, the information processing device 100 does not need to increase the number of calibration quantum gates; it only needs to update the coefficient parameter values ​​of the calibration quantum gates that have already been added. When the target quantum gate is executed on the quantum computer 20 using the updated calibration quantum gates and evaluated, a generator error ε'2 is calculated that includes higher-order components of the original generator error ε, of order three or higher. The information processing device 100 then performs a third calibration using this generator error ε'2. Theoretically, the third calibration eliminates the third-order component of the original generator error ε from the generator error ε'2.

[0168] Thus, by repeatedly evaluating and calibrating the generator error, theoretically, by the time the Nth calibration (N=1,2,3,...) is completed, the lower-order components up to the Nth order of the original generator error ε are eliminated. Therefore, the calibrated generator error ε' decreases stepwise.

[0169] Figure 16 is a graph showing an example of error reduction after calibration. Graph 154 shows the norm of the generator error ε before calibration and the generator error ε' after the Nth calibration. N The relationship with the norm is shown. Similar to Graph 151, the information processing device 100 randomly generates a generator error ε for the ZX90 gate. The information processing device 100 repeats the calibration of the second embodiment. Graph 154 plots the numerous samples that have been generated.

[0170] The points in Graph 154 are divided into four layers vertically. The top layer shows the relationship between the norm of the generator error ε and the norm of the generator error ε'1 after the first comparison. The second layer from the top shows the relationship between the norm of the generator error ε and the norm of the generator error ε'2 after the second comparison. The third layer from the top shows the relationship between the norm of the generator error ε and the norm of the generator error ε'3 after the third comparison. The bottom layer shows the relationship between the norm of the generator error ε and the norm of the generator error ε'4 after the fourth comparison.

[0171] Similar to Graph 151, the horizontal and vertical axes of Graph 154 are on a logarithmic scale. Line 155 indicates the point where the norm of generator error ε is equal to the norm of generator error ε'. Line 156 indicates the point where the square of the norm of generator error ε is equal to the norm of generator error ε'. Line 157 indicates the point where the cube of the norm of generator error ε is equal to the norm of generator error ε'. Line 158 indicates the point where the fourth-powered norm of generator error ε is equal to the norm of generator error ε'. Line 159 indicates the point where the fifth-powered norm of generator error ε is equal to the norm of generator error ε'.

[0172] As shown in Graph 154, points representing one calibration are plotted below line 156. Therefore, the first-order component is eliminated from the generator error ε by one calibration. Also, points representing two calibrations are plotted below line 157. Therefore, the second-order component is eliminated from the generator error ε by two calibrations.

[0173] Furthermore, points representing three calibrations are plotted below line 158. Thus, the cubic component is eliminated from the generator error ε by three calibrations. Also, points representing four calibrations are plotted below line 159. Thus, the quartic component is eliminated from the generator error ε by four calibrations.

[0174] Next, we will provide further explanation regarding the evaluation of the generator error ε. The information processing device 100 collects experimental data by having the quantum computer 20 execute a quantum circuit including the target quantum gate and reading out the measured values ​​of the qubits from the quantum computer 20. The information processing device 100 estimates the generator error ε by analyzing the experimental data.

[0175] Examples of evaluation methods include quantum process tomography, GST, IT (Idle Tomography), HEAT, and Randomized Benchmarking. The information processing device 100 may use any of these evaluation methods as long as it can ultimately obtain the generator error ε.

[0176] GST is also described in the following publication: Erik Nielsen, John King Gamble, Kenneth Rudinger, Travis Scholten, Kevin Young, and Robin Blume-Kohout, "Gate Set Tomography", the Open Journal for Quantum Science, Volume 5, Page 557, October 2021.

[0177] When a certain evaluation method outputs evaluation information other than the generator error ε, the information processing apparatus 100 may convert the output evaluation information into the generator error ε using a data analysis method as described in the following document. Takanori Sugiyama, Shinpei Imori, and Fuyuhiko Tanaka, "Reliable Characterization for Improving and Validating Accurate Quantum Operations", arXiv:1806.02696, December 2020.

[0178] HEAT is also described in the following document. Neereja Sundaresan, Isaac Lauer, Emily Pritchett, Easwar Magesan, Petar Jurcevic, and Jay M. Gambetta, "Reducing Unitary and Spectator Errors in Cross Resonance with Optimized Rotary Echoes", PRX Quantum of the American Physical Society, Volume 1, Page 020318, December 2020.

[0179] Next, an explanation of the determination of the control parameter value will be supplemented. In an experiment using the quantum computer 20, due to the limit of the performance of the control unit 22, the control parameter value θ specified by the information processing apparatus 100 in and the control parameter value θ realized by the quantum computer 20 may deviate slightly. In this case, the information processing apparatus 100 may try a control parameter value θ in that is larger than θ by a small amount δ, and a control parameter value θ in,1 = θ in + δ, and a control parameter value θ in that is smaller than θ by a small amount δ. in,2 = θ in - δ. By doing so, the information processing apparatus 100 can determine the control parameter value θ in,1The corresponding generator error ε1 and the control parameter value θ in,2 The corresponding generator error ε2 is obtained.

[0180] The information processing device 100 performs a linear interpolation as shown in equation (43) to obtain the control parameter value θ. in,1 ,θ in,2 And from the generator errors ε1 and ε2, the control parameter value θ in The information processing device 100 corrects the corrected control parameter value θ. in This is specified to the control unit 22.

[0181]

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[0182] In the second embodiment, the realized control parameter value θ is linearly parameterized with respect to the generator, and the control parameter value θ behaves affinely with respect to the input to the control unit 22. For this reason, the calibration method and first-order interpolation described above are compatible.

[0183] Furthermore, the calibration method described above can calibrate multiple error components included in the generator error independently of each other. In this respect, the calibration method of the second embodiment and first-order interpolation are well compatible. This is because if multiple control parameters corresponding to multiple error components are not independent of each other, modifying the control parameter value to reduce one error component will affect another error component. Next, the functions and processing procedures of the information processing device 100 will be described.

[0184] Figure 17 is a block diagram showing an example of the functions of an information processing device. The information processing device 100 includes a setting storage unit 121, an error storage unit 122, a parameter storage unit 123, an evaluation unit 124, and a calibration unit 125. The setting storage unit 121, the error storage unit 122, and the parameter storage unit 123 are implemented using, for example, RAM 102 or HDD 103. The evaluation unit 124 and the calibration unit 125 are implemented using, for example, CPU 101 and a program.

[0185] The setting memory unit 121 stores setting data related to the quantum gate. The setting data includes the ideal values ​​of the generators of the target quantum gate and the generators of each candidate calibration quantum gate. The setting data also includes the control parameters of each quantum gate and data showing the relationship between the rotation angle of each quantum gate and the control parameter values. The error memory unit 122 stores error data indicating generator errors of the target quantum gate.

[0186] The parameter storage unit 123 stores parameter data indicating the calibration results. The parameter data includes the type of calibration quantum gate selected. The parameter data also includes coefficient parameter values ​​and angle parameter values ​​for each quantum gate. The parameter data may also include control parameter values ​​indicating the waveform of the microwave pulse signal.

[0187] The evaluation unit 124 uses the setting data stored in the setting storage unit 121 to perform an experiment in which the quantum computer 20 executes the target quantum gate. At this time, the evaluation unit 124 may also specify control parameter values ​​to the quantum computer 20 using the parameter data stored in the parameter storage unit 123. The evaluation unit 124 acquires and analyzes the experimental data and calculates the generator error of the target quantum gate. The evaluation unit 124 generates error data indicating the generator error and stores it in the error storage unit 122.

[0188] The calibration unit 125 uses the setting data stored in the setting storage unit 121 and the error data stored in the error storage unit 122 to calibrate the target quantum gate. The calibration unit 125 generates parameter data showing the calibration result and stores it in the parameter storage unit 123.

[0189] At this time, the calibration unit 125 performs eigenvalue decomposition of the ideal values ​​of the generators of the target quantum gate and generates functions cml and cmr for a first-order approximation. Using the functions cml and cmr, the calibration unit 125 generates a linear equation that shows the relationship in which generator errors are canceled by adjusting the target quantum gate itself and adding a calibration quantum gate. By solving the linear equation, the calibration unit 125 determines the adjustment amount of the target quantum gate, the type of calibration quantum gate, and the rotation angle of the calibration quantum gate. A linear solver may be used to solve the linear equation.

[0190] The information processing device 100 outputs the calibration result. The information processing device 100 may also transmit control parameter values ​​corresponding to the calibration result to the quantum computer 20. Alternatively, the information processing device 100 may display the calibration result on the display device 111, or transmit the calibration result to another information processing device, such as a classical computer.

[0191] Figure 18 is a flowchart illustrating an example of the quantum gate calibration procedure. In step S10, the evaluation unit 124 reads the quantum gate configuration data. In step S11, the evaluation unit 124 generates a quantum circuit including the target quantum gate to be calibrated according to the configuration data and causes the quantum computer 20 to execute the quantum circuit. The evaluation unit 124 acquires experimental data by reading the measured values ​​of the qubits from the quantum computer 20.

[0192] In step S12, the evaluation unit 124 analyzes the experimental data and evaluates the generator error of the target quantum gate. In step S13, the calibration unit 125 determines whether the magnitude of the generator error is less than a threshold. If the magnitude of the generator error is less than the threshold, the quantum gate calibration is completed. If the magnitude of the generator error is greater than or equal to the threshold, the process proceeds to step S14. The calibration unit 125 may also determine whether the accuracy of the target quantum gate is sufficient using other methods.

[0193] In step S14, the calibration unit 125 generates a linear equation from the generator error and the available calibration quantum gates that shows the relationship in which the generator error cancels out. At this time, the calibration unit 125 decomposes the ideal values ​​of the generators of the target quantum gate into eigenvalues ​​and generates an approximation function for a linear approximation using the eigenvalues ​​and projection matrix. In step S15, the calibration unit 125 solves the linear equation using a linear solver. However, the calibration unit 125 may also solve the linear equation analytically using the inverse of the coefficient matrix.

[0194] In step S16, the calibration unit 125 determines a calibration quantum gate to be added to the target quantum gate from the solution of the linear equation. The calibration unit 125 also determines the control parameter values ​​of the target quantum gate and the calibration quantum gate, respectively, from the solution of the linear equation, so that a specific rotation angle is realized. The calibration unit 125 transmits the control parameter values ​​to the quantum computer 20. Then, the process returns to step S11.

[0195] As described above, the information processing device 100 of the second embodiment evaluates the error of the target quantum gate and calibrates the target quantum gate to reduce the error. This improves the accuracy of the quantum operations of the quantum computer 20. Furthermore, the information processing device 100 adds a calibration quantum gate to the target quantum gate and has the quantum computer 20 execute the target quantum gate and the calibration quantum gate as a single unit. This reduces error components that are difficult to cancel by optimizing the control parameters of the target quantum gate alone, thereby improving the calibration accuracy.

[0196] Furthermore, the information processing device 100 generates functions cml and cmr for a first-order approximation by eigenvalue decomposition of the generators of the target quantum gate, and generates a first-order equation using the functions cml and cmr. As a result, the information processing device 100 can concisely approximate the effect of the calibration quantum gate even under the non-commutativity of the generators, and can efficiently determine the calibration quantum gate.

[0197] Furthermore, the information processing device 100 calibrates multiple error components included in the generator error independently using different parameters. Therefore, the information processing device 100 can calibrate the target quantum gate more efficiently than when there are dependencies between the error components. In addition, the information processing device 100 may add calibration quantum gates with symmetric parameters before and after the target quantum gate to address specific error components. This allows the information processing device 100 to cancel specific error components with high precision.

[0198] Furthermore, the information processing device 100 gradually reduces the order of the remaining generator error by repeatedly evaluating and calibrating the target quantum gate. This improves the calibration accuracy and makes it easier to determine when to stop the calibration process. [Explanation of symbols]

[0199] 10 Information Processing Devices 11 Storage section 12 Processing Units 13 Quantum circuit 13a, 13b Quantum Gates 14 Error Data 15 Approximation Functions 16 Linear equations

Claims

1. We obtain error data indicating an error in the first quantum gate, Using an approximation function that linearly approximates the effect of an additional quantum gate added to the first quantum gate on the operation of the first quantum gate, a linear equation is generated that includes a variable corresponding to the additional quantum gate and shows a relationship in which the error is canceled by the additional quantum gate. By solving the aforementioned linear equation for the aforementioned variables, the second quantum gate corresponding to the aforementioned additional quantum gate is determined. A quantum gate calibration program that causes a computer to perform a process.

2. The generation process includes decomposing the matrix representing the first quantum gate into a plurality of eigenvalues ​​and a plurality of projection matrices, and generating the approximate function using the plurality of eigenvalues ​​and the plurality of projection matrices. The quantum gate calibration program according to claim 1.

3. The aforementioned additional quantum gate is a rotation gate that rotates the quantum state by a certain angle around a certain axis of rotation, The aforementioned variable indicates the constant angle, The quantum gate calibration program according to claim 1.

4. The second quantum gate includes a third quantum gate added before the first quantum gate and a fourth quantum gate added after the first quantum gate. The third quantum gate and the fourth quantum gate are quantum gates of the same type with symmetrical parameter values. The quantum gate calibration program according to claim 1.

5. The linear equation further includes other variables corresponding to control parameters that control the operation of the first quantum gate, The process of determining the above includes a process of determining the value of the control parameter by solving the linear equation for the other variables. The quantum gate calibration program according to claim 1.

6. The computer is further instructed to perform the following processes: acquire other error data indicating an error in the quantum circuit including the first quantum gate and the second quantum gate; update the linear equation using the other error data; and modify the second quantum gate by solving the updated linear equation. The quantum gate calibration program according to claim 1.

7. We obtain error data indicating an error in the first quantum gate, Using an approximation function that linearly approximates the effect of an additional quantum gate added to the first quantum gate on the operation of the first quantum gate, a linear equation is generated that includes a variable corresponding to the additional quantum gate and shows a relationship in which the error is canceled by the additional quantum gate. By solving the aforementioned linear equation for the aforementioned variables, the second quantum gate corresponding to the aforementioned additional quantum gate is determined. A quantum gate calibration method in which a computer performs the processing.

8. A memory unit that stores error data indicating an error in the first quantum gate, A processing unit that determines a second quantum gate corresponding to the additional quantum gate by using an approximation function that linearly approximates the effect of an additional quantum gate attached to the first quantum gate on the operation of the first quantum gate, generates a linear equation that includes a variable corresponding to the additional quantum gate and shows a relationship in which the error is canceled by the additional quantum gate, and solves the linear equation for the variable, An information processing device having

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