Quantum operation evaluation method and information processing apparatus

Abstract

To efficiently evaluate an error of a quantum gate.SOLUTION: Acquiring measurement data indicating an execution result of a quantum computer for a quantum circuit in which a quantum gate array including a first quantum gate is repeated N times (N is an integer of 1 or more); determining, from a first matrix indicating an ideal value of the quantum gate array, a period k (k is an integer of 1 or more) at which the first matrix raised to the power of k becomes a unit matrix; Processing for generating a linear approximation function for linearly approximating the influence of an error on measurement data by approximating the repetition of a quantum gate array and processing for estimating the error by using the linear approximation function and the measurement data are executed.SELECTED DRAWING: Figure 1

Description

[Technical Field]

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

[0002] Quantum gate-based quantum computers perform various quantum operations on qubits. They 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 that cause quantum operations to deviate from ideal operations. Users may evaluate the quantum operation errors and, based on the evaluation results, perform calibration to adjust the values ​​of the control parameters of the quantum computer. For example, quantum computers may have control parameters for adjusting the time width, amplitude, waveform, etc. of the drive pulses emitted to quantum bits. Adjusting the values ​​of the control parameters can improve the accuracy of quantum operations.

[0004] Here, because the measurement values ​​of qubits are generated through multiple types of quantum operations, it is not easy to directly evaluate the errors of individual quantum operations. Therefore, quantum tomography is one technique for estimating quantum operation errors. Quantum tomography involves conducting experiments in which measurements are repeatedly obtained from a quantum computer while changing the combination of quantum operations. Quantum tomography analyzes the experimental data to estimate the errors of individual quantum operations.

[0005] In quantum process tomography, which evaluates errors in quantum gates, there is a technology that uses an error amplifier circuit to repeat the same quantum gate to amplify the minute errors in the quantum gate. There is also a technology that performs spectral quantum process tomography synchronously for each of two quantum bits to estimate the crosstalk strength between the two quantum bits. There is also a technology that repeatedly measures the quantum state after applying a phase gate while changing the phase, and analyzes this experimental data to estimate the errors in the measuring device. [Prior art documents] [Patent documents]

[0006] [Patent Document 1] International Publication No. 2021 / 131695 [Patent Document 2] International Publication No. 2022 / 151737 [Patent Document 3] US Patent Application Publication No. 2023 / 0306294 Summary of the Invention [Problem to be solved by the invention]

[0007] The error amplifier circuit improves the reliability of the evaluation results by making the error components contained in the measurement values ​​more visible. However, because the error amplifier circuit has a nonlinear amplification effect on errors, data analysis may become more complicated. As a result, the burden of data analysis may increase and the numerical stability of the evaluation results may decrease.

[0008] For example, when the amplification effect of an error amplifier circuit is approximated by a linear function, a singularity that calculates an abnormal value such as infinity may appear during calculation of the linear function, depending on the quantum gate. In this case, it is difficult to evaluate the error of the quantum gate using a simple approximation method. Therefore, in one aspect, the present invention aims to improve the efficiency of error evaluation of quantum gates. [Means for solving the problem]

[0009] In one aspect, a quantum operation evaluation program is provided that causes a computer to execute the following steps: acquiring measurement data indicating the execution results of a quantum computer for a quantum circuit that repeats a quantum gate array including a first quantum gate N times (N is an integer greater than or equal to 1); determining, from a first matrix indicating the ideal value of the quantum gate array, a period of k (k is an integer greater than or equal to 1) such that raising the first matrix to the kth power results in an identity matrix; generating a linear approximation function that linearly approximates the effect of the error on the measurement data by approximating the repetition of the quantum gate array using the matrix exponent of the first conversion result obtained by converting the error that the quantum computer has for the first quantum gate using the first matrix, N, and the period; and estimating the error using the linear approximation function and the measurement data. [Effects of the Invention]

[0010] One aspect is that it makes it more efficient to evaluate errors in quantum gates. [Brief explanation of the drawings]

[0011] [Figure 1] FIG. 1 is a diagram illustrating an information processing apparatus according to a first embodiment. [Figure 2] FIG. 10 illustrates an example of hardware of an information processing system according to a second embodiment. [Figure 3] FIG. 10 is a diagram illustrating an example of an evaluation quantum circuit including an error amplifier circuit. [Figure 4] FIG. 10 is a diagram illustrating an example of combining quantum gates included in a quantum gate array. [Figure 5] FIG. 10 is a diagram illustrating an example of the action of the first quantum gate array on a generator error. [Figure 6] FIG. 10 is a diagram illustrating an example of the effect of the second quantum gate array on a generator error. [Figure 7] FIG. 10 is a diagram illustrating an example of correct answers and estimated values ​​of generator errors. [Figure 8] FIG. 1 is a diagram illustrating an example of a quantum gate having a singularity. [Figure 9]FIG. 10 is a diagram illustrating an example of a generator of a quantum gate having a singularity. [Figure 10] FIG. 2 is a block diagram illustrating an example of functions of the information processing device. [Figure 11] FIG. 10 is a diagram illustrating an example of setting data. [Figure 12] FIG. 10 is a diagram showing examples of experimental data and evaluation data. [Figure 13] 1 is a first flowchart showing an example of a procedure for evaluating a quantum operation. [Figure 14] 10 is a second flowchart showing an example of the procedure for evaluating quantum operations. [Figure 15] 10 is a second flowchart (continued) showing an example of the procedure for evaluating quantum operations. [Figure 16] FIG. 10 is a diagram illustrating an example of a pseudo program for determining the period of a quantum gate. DETAILED DESCRIPTION OF THE INVENTION

[0012] Hereinafter, this embodiment will be described with reference to the drawings. (a) First embodiment FIG. 1 is a diagram illustrating an information processing device according to a first embodiment. The information processing device 10 according to the first embodiment evaluates the implementation accuracy of quantum operations executed by a quantum computer using quantum tomography. The information processing device 10 may be a so-called classical computer. The information processing device 10 may also be a client device or a server device. The information processing device 10 may also be called a quantum operation evaluation device.

[0013] The information processing device 10 includes a storage unit 11 and a processing unit 12. The storage unit 11 may be a volatile semiconductor memory such as a random access memory (RAM), or may be a non-volatile storage such as a hard disk drive (HDD) or a flash memory.

[0014] The processing unit 12 is, for example, a processor such as a central processing unit (CPU), a graphics processing unit (GPU), or a digital signal processor (DSP). However, the processing unit 12 may also include an electronic circuit such as an application specific integrated circuit (ASIC) or a field programmable gate array (FPGA). The processor executes a program stored in a memory such as a RAM. A set of processors may be called a multiprocessor or simply a "processor." Furthermore, multiple processing steps described below may be executed by different processors.

[0015] The storage unit 11 stores measurement data 24. The measurement data 24 indicates the results of execution of a quantum computer on the quantum circuit 21. The quantum computer that executes the quantum circuit 21 is the quantum computer whose implementation accuracy is to be evaluated. The information processing device 10 may be connected to the quantum computer and may read measurement values ​​from the quantum computer. The information processing device 10 may also receive the measurement data 24 from another information processing device connected to the quantum computer. The information processing device 10 may also sequentially read measurement values ​​from the quantum computer while proceeding with the data analysis described below.

[0016] The quantum circuit 21 is a quantum computation model that describes quantum operations on one or more quantum bits using quantum gates. The quantum circuit 21 repeats a quantum gate array 22 including a quantum gate 23 N times. N is an integer equal to or greater than 1 and indicates the number of iterations 28. This quantum circuit 21 is sometimes called an error amplifier circuit because it amplifies the error component. Typically, N may be an integer equal to or greater than 2. The information processing device 10 may use the execution results of different quantum circuits corresponding to different Ns including N=1, such as N=1, 3, 5, ...

[0017] The quantum gate array 22 may include quantum gates other than the quantum gate 23. The quantum gate array 22 may execute two or more quantum gates in series. The quantum gate 23 is a quantum gate for which the implementation accuracy is evaluated. Examples of the quantum gate 23 include a 180-degree rotation gate, a 90-degree rotation gate, and a 45-degree rotation gate. The quantum computer has an error 26 for the quantum gate 23 that indicates a deviation from the ideal value. The ideal value indicates an ideal effect on the quantum state represented by the quantum bit. The error 26 may be referred to as an implementation error.

[0018] The error 26 is unknown and is determined by a variable. The quantum circuit 21 amplifies the error 26 by repeating the quantum gate array 22. The quantum gate array 22 may include two or more quantum gates to be evaluated whose errors are unknown. The quantum gate array 22 may also include one or more quantum gates not to be evaluated whose errors are known. The information processing device 10 may consider the errors of the quantum gates not to be evaluated to be zero or a constant value.

[0019] In a quantum computer, even if the quantum bit initialization method, quantum gate type, and measurement method are the same, measured values ​​are obtained probabilistically. Therefore, the measurement data 24 represents, for example, a probability distribution listing the occurrence probability of each of multiple values ​​that the quantum bit can take. The information processing device 10 may cause the quantum computer to perform multiple attempts to initialize the quantum bit, repeat the quantum gate array 22 N times, and measure the value of the quantum bit. The information processing device 10 may calculate the occurrence probability of each value by dividing the number of occurrences of each value by the number of attempts.

[0020] The measurement data 24 may include multiple measurement results corresponding to different initialization methods or different measurement methods, or may include multiple measurement results corresponding to different quantum gate arrays or multiple measurement results corresponding to different numbers of iterations 28.

[0021] The processing unit 12 analyzes the measurement data 24 to estimate the error 26. Strictly speaking, the effect of the error 26 on the measurement data 24 is nonlinear. In response to this, to facilitate data analysis, the processing unit 12 generates a linear approximation function 25 that linearly approximates the effect of the error 26 on the measurement data 24. The linear approximation function 25 is expressed, for example, by a matrix.

[0022] First, the processing unit 12 determines a period 29 from a matrix 27 indicating the ideal value of the quantum gate array 22. The period 29 is the period of k such that the matrix 27 becomes an identity matrix when raised to the kth power. The period 29 can also be said to be the smallest value of k such that the matrix 27 becomes an identity matrix when raised to the kth power. k is an integer greater than or equal to 1. The property of a matrix becoming an identity matrix when raised to the kth power is sometimes called cyclicity. Many quantum gates, such as rotation gates, are cyclic. Typically, k may be an integer greater than or equal to 2. The information processing device 10 may use the execution results of different quantum circuits corresponding to different k including k=1, such as k=1, 2, 4, 8, ...

[0023] An ideal 180-degree rotation gate restores the quantum state in two turns and is therefore represented by a matrix with a period of 2. An ideal 90-degree rotation gate is represented by a matrix with a period of 4. An ideal 45-degree rotation gate is represented by a matrix with a period of 8. When quantum gate array 22 includes two or more quantum gates, matrix 27 may be calculated by sequentially combining the ideal values ​​of those two or more quantum gates. Matrix 27 may also be the product of matrices that indicate the ideal values ​​of the quantum gates.

[0024] The processing unit 12 generates a linear approximation function 25. At this time, the processing unit 12 linearly approximates the amplification of error 26 due to the repetition of the quantum gate array 22. This linear approximation indicates a component of the amplification of error 26 that is proportional to the number of repetitions 28. The processing unit 12 approximates the repetition of the quantum gate array 22 by the matrix exponent of the transformation result obtained by transforming error 26 using a matrix 27, the number of repetitions 28, and a period 29. The matrix exponent of matrix X is defined as a square matrix obtained by adding up the a-th power of matrix X divided by the factorial of a, from a=0 to infinity.

[0025] For example, the iteration action of the quantum gate array 22 is approximately represented as the matrix exponent of a matrix described below. The processing unit 12 sums the products of the mth power of the matrix 27 and the mth power of the adjoint of the matrix 27, from m=0 to k-1. The adjoint of the matrix X is a matrix obtained by transposing the matrix X and taking its complex conjugate. The processing unit 12 multiplies the above sum by the reciprocal of the iteration number 28 and the period 29. The matrix exponent of this matrix is ​​an approximation of the iteration action of the quantum gate array 22.

[0026] This linear approximation takes advantage of the fact that matrix 27 is a unitary matrix and has cyclicity. This linear approximation also expresses quantum gate 23 with error 26 as the product of a unitary matrix representing the ideal value of quantum gate 23 and the matrix exponent of error 26. The matrix logarithm of the matrix representing the quantum gate is sometimes called the generator or Lindbladian. Error 26 may also be called the generator error.

[0027] If the quantum gate array 22 includes two or more quantum gates, the processing unit 12 may use a matrix indicating ideal values ​​of the two or more quantum gates to linearly approximate the amplification effect of the error 26 during one execution of the quantum gate array 22. The processing unit 12 may generate the linear approximation function 25 by multiplying a matrix that linearly approximates the combined effect of the two or more quantum gates by a matrix that linearly approximates the repeated effect of the quantum gate array 22.

[0028] The processing unit 12 estimates the error 26 using a linear approximation function 25 and the measurement data 24. For example, the processing unit 12 extracts a first-order component proportional to the number of iterations 28 from the measurement data 24. The linear approximation function 25 has the error 26 as an argument and outputs a predicted value of the first-order component included in the measurement data 24. The processing unit 12 optimizes the error 26 to be input to the linear approximation function 25 so as to minimize the difference between the extracted first-order component and the output of the linear approximation function 25. For example, the processing unit 12 defines this optimization problem as a quadratic programming problem with a semidefinite constraint and calculates the error 26 using a mathematical programming solver.

[0029] Here, the space to which the matrix representing the quantum operation of the quantum circuit 21 belongs may contain a singular point. A singular point is a point representing a matrix at which an eigenvalue that satisfies a certain condition is calculated. A singular point is sometimes called a critical point. For example, a singular point corresponds to a matrix having different eigenvalues ​​such that the value of an exponential function with Napier's constant e as the base is the same. When the eigenvalues ​​are complex numbers, the values ​​of the exponential function calculated from different eigenvalues ​​may be the same.

[0030] Depending on the type of quantum gates included in the quantum gate array 22, a matrix corresponding to a singularity may appear. Any quantum gate included in the quantum gate array 22 may correspond to a singularity by itself, or a composite quantum gate may correspond to a singularity. An example of a quantum gate corresponding to a singularity is a rotation gate. Examples of such rotation gates include a 180-degree rotation gate and a quantum gate that includes a 45-degree rotation component in addition to a 180-degree rotation component (for example, the product of a CX gate and a 90-degree rotation gate around the X axis). At a singularity, an abnormal value such as division by zero may be calculated.

[0031] In response to this, the processing unit 12 performs linear approximation as described above for the iterations of the quantum gate array 22 so as to suppress the influence of the singularity. In the second embodiment, the processing unit 12 does not need to perform spectral decomposition, which decomposes the matrix 27 into the sum of products of eigenvalues ​​and projection matrices, and can avoid abnormal calculations related to eigenvalues ​​even if there is a quantum gate corresponding to a singularity. Note that the processing unit 12 may generate the linear approximation function 25 before acquiring the measurement data 24.

[0032] As described above, the information processing device 10 of the first embodiment acquires measurement data 24 indicating the results of quantum computer execution for the quantum circuit 21 that repeats N times the quantum gate array 22 including the quantum gate 23. From the matrix 27 indicating the ideal values ​​of the quantum gate array 22, the information processing device 10 determines the period 29 of k, at which the matrix 27 raised to the kth power becomes an identity matrix.

[0033] The information processing device 10 approximates the repetition of the quantum gate array 22 with the matrix exponent of the transformation result obtained by transforming an error 26 that the quantum computer has for the quantum gate 23 using a matrix 27, an iteration count 28, and a period 29, thereby generating a linear approximation function 25 that linearly approximates the effect of the error 26 on the measurement data 24. The information processing device 10 estimates the error 26 using the linear approximation function 25 and the measurement data 24.

[0034] Quantum tomography allows the information processing device 10 to efficiently estimate the error 26. Furthermore, by using the quantum circuit 21 that repeats the quantum gate array 22, the information processing device 10 can amplify the minute error 26 of the quantum gate 23, improving the accuracy of estimating the error 26 from the measurement data 24. Furthermore, based on the estimated error 26, the user can adjust the values ​​of the control parameters of the quantum computer, thereby improving the accuracy of the quantum operation of the quantum computer.

[0035] Furthermore, because the linear approximation function 25 is used, the information processing device 10 does not need to perform highly complex nonlinear data analysis, reducing the load of data analysis and improving the numerical stability of the estimation result of the error 26. Furthermore, by utilizing the fact that the matrix 27 is a unitary matrix and has cyclicity, the information processing device 10 can linearly approximate the repetitive action without using spectral decomposition. Therefore, it is possible to complete the approximation calculation while avoiding the influence of singularities, and the number of types of quantum gates to which linear approximation can be applied increases.

[0036] In particular, the information processing device 10 can efficiently estimate errors in various types of quantum gates with cyclicity, such as a 180-degree rotation gate, a 90-degree rotation gate, and a 45-degree rotation gate. Furthermore, the information processing device 10 does not need to distinguish between approximation methods depending on whether the matrix 27 corresponds to a singularity, thereby simplifying the approximation calculation. Furthermore, the information processing device 10 does not need to perform spectral decomposition, which is a heavy load, and can reduce the amount of calculation.

[0037] (b) Second embodiment 2 is a diagram illustrating an example of hardware of an 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 115. The information processing device 100 evaluates the accuracy of quantum operations performed by the quantum computer 115 using quantum tomography. The information processing device 100 performs calibration to adjust the values ​​of control parameters of the quantum computer 115 so as to improve the accuracy. The information processing device 100 corresponds to the information processing device 10 according to the first embodiment.

[0038] The information processing device 100 has a CPU 101, a RAM 102, a HDD 103, a GPU 104, an input interface 105, a medium reader 106, a communication interface 107, and an interface 108. These units are connected to a bus. The CPU 101 corresponds to the processing unit 12 in the first embodiment. The RAM 102 or the HDD 103 corresponds to the storage unit 11 in the first embodiment.

[0039] The CPU 101 is a processor that executes program instructions. The CPU 101 loads programs and data stored in the HDD 103 into the RAM 102 and executes the programs. The information processing device 100 may have multiple processors.

[0040] The RAM 102 is a volatile semiconductor memory that temporarily stores programs executed by the CPU 101 and data used in calculations by the CPU 101. The information processing device 100 may have a type of volatile memory other than a RAM.

[0041] The HDD 103 is a nonvolatile storage that stores software programs such as an operating system (OS), middleware, and application software, as well as data. The information processing device 100 may also have other types of nonvolatile storage, such as a flash memory or an SSD (Solid State Drive).

[0042] The GPU 104 performs image processing in cooperation with the CPU 101 and outputs an image to a 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. Other types of output devices, such as a printer, may be connected to the information processing device 100.

[0043] The GPU 104 may also be used as a general purpose computing on graphics processing unit (GPGPU). The GPU 104 may execute a program in response to an instruction from the CPU 101. The information processing device 100 may include a volatile semiconductor memory other than the RAM 102 as a GPU memory.

[0044] The input interface 105 receives an input signal 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. A plurality of input devices may be connected to the information processing device 100.

[0045] The medium 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 a semiconductor memory. Magnetic disks include flexible disks (FDs) and HDDs. Optical disks include compact discs (CDs) and digital versatile discs (DVDs). The medium reader 106 copies the programs and data read from the recording medium 113 to other recording media such as the RAM 102 or the HDD 103. The read programs may be executed by the CPU 101.

[0046] The recording medium 113 may be a portable recording medium. The recording medium 113 may be used to distribute programs and data. The recording medium 113 and the HDD 103 may also be referred to as computer-readable recording media.

[0047] 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 a router, or may be a wireless communication interface connected to a wireless communication device such as a base station or an access point.

[0048] The interface 108 is connected to the quantum computer 115. The interface 108 transmits commands to the quantum computer 115 in response to instructions from the CPU 101. The interface 108 also receives data from the quantum computer 115 and writes the received data to the RAM 102 or the HDD 103.

[0049] The quantum computer 115 has a quantum operation unit 116 and a control unit 117. The quantum operation unit 116 includes a plurality of quantum bits. In response to instructions from the control unit 117, the quantum operation unit 116 executes quantum operations such as initializing the quantum bits, applying quantum gates to the quantum bits, and measuring the quantum bits. The quantum operations change the quantum state represented by the quantum bits. The behavior of the quantum operations is adjusted by the values ​​of control parameters provided by the control unit 117.

[0050] The control unit 117 receives a command from the information processing device 100. The calibration command includes the name and value of a control parameter. In response to the calibration command, the control unit 117 inputs the value of the control parameter to the quantum operation unit 116. Examples of the control parameter include the time width, amplitude, and waveform of a drive pulse issued to a quantum bit. Changing the value of the control parameter changes the accuracy of the quantum operation.

[0051] Furthermore, in response to a command for quantum operation, the control unit 117 instructs the quantum operation unit 116 to perform a quantum operation. Furthermore, in response to a command for obtaining a measurement value, the control unit 117 reads out a measurement value generated by measuring the quantum bit and transmits the measurement value to the information processing device 100.

[0052] Generally, quantum information processing includes quantum computing, quantum simulation, quantum communication, quantum cryptography, quantum sensing, etc. Examples of physical platforms for quantum information processing include superconducting quantum circuits, semiconductor quantum dots, diamond NV centers, NMR molecules, neutral atoms, trapped ions, light, etc. Typical quantum information processing protocols based on quantum circuits use three types of quantum operations: initialization, quantum gates, and measurement.

[0053] Quantum operations implemented in the quantum computer 115 have errors that indicate deviations from ideal quantum operations. The information processing device 100 performs evaluation and calibration on the quantum computer 115 to improve the accuracy of the quantum operations. Evaluation estimates the errors of the quantum operations. Calibration changes the values ​​of control parameters based on the error information so as to reduce the errors. The information processing device 100 may repeat evaluation and calibration.

[0054] The information processing device 100 evaluates quantum operations using quantum tomography. The information processing device 100 collects measurements of qubits from the quantum computer 115 while changing combinations of initialization, quantum gates, and measurements. The information processing device 100 analyzes experimental data that associates the tried combinations with the measurements, and estimates errors in the quantum operations. Quantum tomography makes it possible to estimate errors in multiple quantum gates at once.

[0055] The information processing device 100 generates a quantum circuit including an error amplifier circuit that repeats the same quantum gate array multiple times, and causes the quantum computer 115 to execute this quantum circuit. The number of iterations may be, for example, 16, 128, or 1024. Errors in the quantum gates included in the quantum gate array are amplified through the error amplifier circuit. This increases the error components included in the measured values, improving evaluation accuracy. Examples of quantum tomography techniques that use error amplifier circuits include GST (Gate-Set Tomography), IT (Idle Tomography), and HEAT (Hamiltonian Error Amplifying Tomography).

[0056] GST is described, for example, in the following non-patent document: 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.

[0057] HEAT is described, for example, in the following non-patent 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.

[0058] Error amplifier circuits are also described in the following non-patent document: Gabriel O. Samach, Ami Greene, Johannes Borregaard, Matthias Christandl, Joseph Barreto, David K. Kim, Christopher M. McNally, Alexander Melville, Bethany M. Niedzielski, Youngkyu Sung, Danna Rosenberg, Mollie E. Schwartz, Jonilyn L. Yoder, Terry P. Orlando, Joel I-Jan Wang, Simon Gustavsson, Morten Kjaergaard and William D. Oliver, "Lindblad Tomography of a Superconducting Quantum Processor," Physical Review Applied of the American Physical Society, Vol. 18, page 064056, December 2022.

[0059] Yanwu Gu, Rajesh Mishra, Berthold-Georg Englert and Hui Khoon Ng, "Randomized Linear Gate-Set Tomography", PRX Quantum of the American Physical Society, Volume 2, page 030328, August 2021.

[0060] 3 is a diagram showing an example of an evaluation quantum circuit including an error amplifier circuit. This quantum circuit includes an initialization circuit 141, a quantum gate array 142, and a measurement circuit 143. The initialization circuit 141 initializes one or more quantum bits to generate a desired quantum state. The quantum gate array 142 includes one or more quantum gates. The error amplifier circuit repeats the quantum gate array 142 N times in series. The measurement circuit 143 measures the value of the quantum bit.

[0061] A non-repeating quantum gate may be included between the initialization circuit 141 and the quantum gate array 142. Also, a non-repeating quantum gate may be included between the quantum gate array 142 and the measurement circuit 143. In FIG. 3, an X gate is a rotation gate that rotates around the X axis, a Y gate is a rotation gate that rotates around the Y axis, and a Z gate is a rotation gate that rotates around the Z axis. A CR gate is a cross-resonance gate. An R gate is a rotation gate that rotates by a fixed amount.

[0062] Quantum tomography collects experimental data while changing the combination of quantum gate array 142, number of iterations N, initialization circuit 141, and measurement circuit 143. Quantum gate array 142 includes one or more quantum gates to be evaluated that evaluate errors. Quantum gate array 142 may also include one or more quantum gates not to be evaluated that do not evaluate errors.

[0063] The ideal behavior of the quantum gate to be evaluated and the quantum gate not to be evaluated is known. The error of the quantum gate to be evaluated in quantum computer 115 is unknown. The error of the quantum gate not to be evaluated in quantum computer 115 may be known or unknown. In the latter case, quantum tomography may consider the error of the quantum gate not to be evaluated to be zero or may assume a constant value.

[0064] Here, if we try to strictly define the effect of the error amplifier circuit on quantum gate errors, we will have to solve a highly nonlinear numerical optimization problem. This may increase the burden of experimental data analysis and reduce the stability of solution search. Therefore, the information processing device 100 linearly approximates the effect of the error amplifier circuit and defines the search for quantum gate errors as a quadratic programming problem with a semidefinite constraint. A quadratic programming problem with a semidefinite constraint is a numerical optimization problem that optimizes a quadratic function under linear constraints, and the eigenvalues ​​of the matrix are nonnegative. The method of the second embodiment may be called RLT (Robust Lindbladian Tomography).

[0065] The quantum tomography calculation of the second embodiment will be explained below. The action of a quantum gate acting on a d-dimensional quantum system is described by the linear mapping shown in Equation (1). This linear mapping is a mapping from a d×d-dimensional complex space to a d×d-dimensional complex space, and is a trace-preserving and completely positive mapping. Here, the representation matrix of the linear mapping of Equation (1) under the orthonormal basis shown in Equation (2) is denoted as G. This orthonormal basis is a matrix of d x d complex spaces whose norm is 1 and which are orthogonal to each other. 2 Individual Source B α The element G in the αth row and βth column of the matrix G is αβ is an element B of the orthonormal basis corresponding to the α-row, as shown in equation (3). α and the element B of the orthonormal basis corresponding to the β column β is calculated using the trace.

[0066]

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[0067]

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[0068]

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[0069] There exists a linear map that satisfies the relationship of Equation (4) between the quantum gate defined in Equation (1) via the exponential map exp. This linear map is sometimes called a generator (Lindbrandian). If the representation matrix of the linear map in Equation (4) is expressed as L, then the relationship of Equation (5) holds between the representation matrix G of the quantum gate and the representation matrix L of the generator using the matrix exponential function e. If the representation matrix of the ideal value of the generator is L, then ideal , the representation matrix of the error of the generator is denoted as δL, and then G=e L is expanded as in equation (5).

[0070]

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[0071]

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[0072] When errors are amplified through an error amplifier circuit, it is easier to analyze the generator error, which corresponds to the matrix logarithm, than the quantum gate error itself. This is because, when errors accumulate, a rotation occurs in the quantum bit, and the angular deviation may not be exactly proportional to the number of iterations N. Therefore, in the second embodiment, the information processing device 100 estimates the generator error from experimental data. Since the ideal value of the generator is known, if the generator error can be known, the actual G in the quantum computer 115 can be known. Note that, for simplicity of explanation, the following will not distinguish between a linear map and its representation matrix. For example, the quantum gate may be denoted as G, the generator as L, and the generator error as δL.

[0073] To prepare the error amplifier circuit, we use a set of quantum gates I g is given by the user. Set I g Each quantum gate in n is identified by a unique number. g Among the different quantum gates, the first n g,1 quantum gates are quantum gates to be evaluated, and the remaining quantum gates are quantum gates not to be evaluated.

[0074]

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[0075] The quantum gate array that forms the error amplifier circuit is set I g A quantum circuit is a circuit in which one or more (typically two or more) quantum gates included in are arranged in series. The quantum gate array a is defined as in Equation (7). In Equation (7), the length of the quantum gate array is m. The elements on the right side of Equation (7) are in the set I gThe same type of quantum gate may appear more than once in the quantum gate sequence a.

[0076]

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[0077] If the composite quantum gate corresponding to the overall action of the quantum gate sequence a is represented as G(a), G(a) can be expanded as shown in Equation (8). Here, the product of three or more matrices is calculated in order from right to left. If the number of iterations of the error amplifier circuit is represented as N, the quantum gate corresponding to the overall action of the error amplifier circuit is G(a). N It is expressed as:

[0078]

number

[0079] n shown in Equation (9) g,1 The generator errors δL are to be estimated. In order to collect measurements taken under various conditions, a set of quantum gate arrays a shown in Equation (10) and a set of iteration numbers N are provided by the user. For the quantum gate array, a n different quantum gate sequences are given. The number of iterations is N A different number of iterations is given.

[0080]

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[0081]

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[0082] Different iteration counts N and N' given to the same quantum gate array satisfy the condition of equation (11). The effect of an ideal error-free quantum gate array when iterated N times is the same as the effect of that quantum gate array when iterated N' times. This makes it easy to extract amplified error components from measurements with different iteration counts.

[0083]

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[0084] For each quantum gate array, a set of pairs of initialization ρ and measurement Π shown in Equation (12) is given by the user. This set is divided into n pairs of initialization ρ and measurement Π. t,i Therefore, the quantum computer 115 generates an initial quantum state by initialization ρ, executes the quantum gate sequence a repeatedly for N iterations, and obtains a measurement value by measurement Π. The quantum computer 115 executes this experiment for all combinations of a, N, (ρ, Π).

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[0086] If we express the vectorization of matrix X with respect to the base B as |X>>, the probability of obtaining a measured value x in each experiment can be calculated as shown in Equation (13). Matrix X is an element of the d×d-dimensional complex space, and vector |X>> is a d 2 In the following, i may be used as an identifier of the quantum gate sequence a, j as an identifier of the number of iterations N, and k as an identifier of the set of initialization and measurement (ρ, Π). Due to the nature of quantum computing, measurements are obtained probabilistically, so the quantum computer 115 calculates n for each of (i, j, k). i,j,k A sample of measurements is taken.

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[0088] The frequency at which measurement value x is obtained is f i,j,k,x Then, the frequency distribution of Equation (14) is calculated for each (i, j, k). i,j,k,x is the number of samples for which the measurement value x was obtained, n i,j,k It is calculated by dividing by.

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[0090] Following the experimental phase of collecting frequency distributions as described above, the information processing device 100 executes a data processing phase of analyzing experimental data indicating the frequency distributions. The data processing phase includes step #1 of extracting amplification components from the experimental data, step #2 of linearly approximating the function of the error amplifier circuit, and step #3 of solving a numerical optimization problem defined using the results of steps #1 and #2. However, steps #1 and #2 may be executed in reverse order, or step #2 may be executed before the experimental phase. Furthermore, the information processing device 100 may omit step #2 by reusing the function of the error amplifier circuit that was previously calculated.

[0091] In step #1, the information processing device 100 extracts a first-order component proportional to the number of iterations N from the experimental data. The experimental data includes a constant component unrelated to the number of iterations N, a first-order component proportional to the number of iterations N, and higher-order components that are quadratic or higher with respect to the number of iterations N. The ideal value of the generator corresponds to the constant component. The generator error amplified by the error amplifier circuit corresponds to the first-order component. A method for extracting the first-order component by extrapolation will be described below.

[0092] The information processing device 100 selects and fixes one combination of the quantum gate array a, the set of initialization and measurement (ρ, Π), and the measurement value x. The information processing device 100 calculates the probability p x Expand (i,j,k) as a series for the number of iterations N. Probability p x(i,j,k) is the probability that the measured value x is obtained under the experiment (i,j,k). The νth order expansion coefficient is h ν Due to the resolution accuracy, the information processing device 100 performs the series expansion up to the infinite order shown in Equation (15) as follows: N Approximate by a finite series sum up to order -1.

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[0094] Equation (16) is the probability p x (i,j,k) to n N The vector summarizing the number of iterations is shown below. T represents the transpose. Using the Vandermonde matrix V for the number of iterations N in equation (17), the sum of the finite series is expressed as in equation (18). The Vandermonde matrix V is a matrix that ranges from 0 to n N A row vector of powers of the iteration number N up to the power -1, where n N n corresponding to the number of iterations N The coefficient vector h in Equation (18) is a matrix containing row vectors. v from 0th to nth N It is a vector arranged up to degree -1.

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[0098] n NWhen the numbers of iterations are different from each other, the Vandermonde matrix V is an invertible matrix. The coefficient vector h is calculated as shown in Equation (19) using the inverse matrix of the Vandermonde matrix V. In an experiment using a quantum computer 115, the probability p x The frequency f corresponding to the estimate of (i,j,k) i,j,k,x is measured. Then, the information processing device 100 measures the probability p x Instead of (i,j,k), frequency f i,j,k,x Formula (20) uses the frequency f i,j,k,x n N The vector summarizing the iterations is shown below. The estimated coefficient vector h est is calculated as shown in Equation (21).

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[0102] In this way, the information processing device 100 calculates the measured frequency f i,j,k,x From 0th to nth N The information processing device 100 calculates expansion coefficients up to the -1st order. However, the information processing device 100 may calculate the expansion coefficients by other methods instead of the method using the Vandermonde matrix V. For example, the information processing device 100 may perform polynomial approximation using the least squares method.

[0103] The second element of the coefficient vector h is a primary component proportional to the number of iterations N. The information processing device 100 extracts the primary component while changing the measurement value x. The information processing device 100 extracts the coefficient vector h by grouping primary components with the same (i, k) but different measurement values ​​x. est1(i, k). The information processing device 100 calculates the coefficient vector h est 1(i,k) is calculated for all (i,k).

[0104] In step #2, the information processing device 100 g Among the errors of the generators, n are evaluated. g,1 The errors of the n generators are parameterized together by the parameter v, which is a vector. g,1 The transformation from the generator errors to the parameter v may be a predefined affine transformation, and may be defined using a matrix. g,1 To estimate the generator errors, this parameterization is defined to be invertible.

[0105] The information processing device 100 linearly approximates the action of the error amplifier circuit for each (i, k) and calculates a matrix C i,k Generate the matrix C i,k The product of and the parameter v indicates the effect of the error of the generator under evaluation on the first-order component of the experimental data.

[0106] In addition, the information processing device 100 g Among the errors of the generators, the errors of the generators that are not the object of evaluation are vectorized in the same way as the parameter v. For this vectorization, the same conversion as for the errors of the generators that are the object of evaluation is used. The information processing device 100 applies a matrix C i,k Applying i,k Generate a vector b i,k indicates the influence of the error of the unevaluated generator on the first component of the experimental data. Since the error of the unevaluated generator is given by the user, the vector b i,k The elements of are calculated as specific numerical values. A linear approximation of the operation of the error amplifier circuit will be described later.

[0107] In step #3, the information processing device 100 calculates the coefficient vector h est1(i,k) and the matrix C generated in step #2 i,k and vector b i,k The objective function F(v) shown in Equation (22) is generated using W i,k is a weighting matrix set by the user for (i, k). The information processing device 100 calculates the parameter v that minimizes the value of the objective function F(v), as shown in Equation (23). est The information processing device 100 calculates the parameter v est returns the error of the generator being evaluated.

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[0110] The numerical optimization problem to solve Equation (23) is a physically constrained least-squares fitting. est is searched for to satisfy the physical constraints that the generator error has. Since the parameter v is an affine parameterization of the generator error, the physical constraints that the parameter v has are expressed by linear semidefinite constraint conditions. In addition, the objective function F(v) is a quadratic function of the parameter v. Therefore, this numerical optimization problem is defined as a quadratic programming problem with semidefinite constraints, and is solved using a semidefinite programming solver.

[0111] Next, we will explain the linear approximation in step #2. Consider n×n complex square matrices A, B, and P as in equation (24). Matrix A is diagonalizable and corresponds to the ideal value of the generator. Matrix A is decomposed by spectral decomposition as in equation (25). In equation (25), V is a matrix containing eigenvectors as column vectors, and Λ is a diagonal matrix with eigenvalues ​​arranged on the diagonal line. a j is a complex number that is an eigenvalue, P j is the eigenvalue a jThe matrix A is the projection matrix corresponding to the eigenvalue a j and the projection matrix P j It is decomposed into a sum of products.

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[0114] The projection matrix P satisfies the formula (26). jk is the Kronecker delta. For j=k, δ jk = 1, and if j and k are different, δ jk = 0. Therefore, the product of different projection matrices corresponding to different eigenvalues ​​is a zero matrix, while the square of the same projection matrix is ​​the projection matrix itself.

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[0116] Eigenvalue a of matrix A j and the projection matrix P j Using the above, the six linear mappings of Equations (27) to (32) are defined as linear mappings for the matrix B. A denotes decomposition to the left with respect to matrix A. A indicates decomposition to the right with respect to matrix A. A denotes Composition to the Left with respect to matrix A.

[0117] cmr Adenotes Composition to the Right with respect to matrix A. ssp A denotes the sum of spectral projections with respect to matrix A. sspc A denotes the sum of the complement of the spectral projection with respect to matrix A.

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[0124] The coefficients used in the equations (27) to (30) are defined as in the equation (33). j ,P k If are the same, l jk = 1. Projection matrix P j ,P k If different, l jkis the projection matrix P j ,P k The eigenvalue a corresponding to j ,a k These six linear mappings can be expressed as matrices using an orthonormal basis in the n×n-dimensional complex space. The representation matrices of these six linear mappings are 2 ×n 2 is a matrix of , and is expressed as equation (34).

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[0127] Using dcl in equation (27), the matrix exponent of the sum of matrices A and B is calculated as in equation (35). The second term on the right-hand side is a quadratic or higher order component of matrix B. Therefore, the matrix exponent of the sum of matrices A and B is linearly approximated as in the first term on the right-hand side. Furthermore, using dcr in equation (28), the matrix exponent of the sum of matrices A and B is calculated as in equation (36). Therefore, the matrix exponent of the sum of matrices A and B is also linearly approximated as in the first term on the right-hand side.

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[0130] Furthermore, when cml in Equation (29) is used, the product of the matrix exponent of matrix A and the matrix exponent of matrix B is calculated as shown in Equation (37). Therefore, the product of the matrix exponent of matrix A and the matrix exponent of matrix B is linearly approximated as shown in the first term on the right-hand side. Furthermore, when cmr in Equation (30) is used, the product of the matrix exponent of matrix B and the matrix exponent of matrix A is calculated as shown in Equation (38). Therefore, the product of the matrix exponent of matrix B and the matrix exponent of matrix A is linearly approximated as shown in the first term on the right-hand side.

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[0133] Using the linear mapping of Equations (27) to (30), the action of the composite quantum gate obtained by combining quantum gates G1 and G2 is approximated as in Equation (39). ideal is a composite generator corresponding to the matrix logarithm of the ideal value of the composite quantum gate, and is defined as Equation (40).

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[0136] Therefore, the generator error δL1 for the quantum gate G1 changes through the composite quantum gate as shown in Equation (41). In Equation (41), the linear map dcr L ,cmr L The change in δL1 is linearly approximated using the above equation. The generator error δL2 for the quantum gate G2 changes through the composite quantum gate as shown in equation (42). In equation (42), the linear mapping dcl L ,cml LThe change in δL2 is linearly approximated using Equation (43). Equation (44) is a representation matrix of the change in δL1 shown in Equation (41). Equation (42) is a representation matrix of the change in δL2 shown in Equation (42).

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[0141] When a quantum gate array includes three or more quantum gates, the information processing device 100 combines the quantum gates one by one from the beginning to the end. The information processing device 100 first combines the first quantum gate with the second quantum gate. Next, the information processing device 100 combines this combined quantum gate with the third quantum gate. By repeating this combination, the information processing device 100 generates a combined quantum gate equivalent to the entire quantum gate array. This allows a linear approximation of the change in generator error through the quantum gate array.

[0142] Next, the information processing device 100 calculates the effect of repeating the quantum gate array. Repeating the quantum gate G N times is defined as in Equation (45). The composite quantum gate obtained by combining the quantum gate arrays is e A+BAssuming that there exists a c such that mod(nA)=cA, the N iterations of this composite quantum gate are defined as in Equation (46) using ssp and sspc in Equations (31) and (32). c is a constant that does not depend on the number of iterations N. The first term on the right-hand side is a first-order component of matrix B. The second term on the right-hand side is a second-order or higher-order component of matrix B.

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[0145] Therefore, the generator error δL for quantum gate G changes as shown in equation (47) through N iterations. In equation (47), the change in δL is linearly approximated using linear mappings ssp, sspc. The first term on the right-hand side indicates the component that is not amplified by N iterations. The second term on the right-hand side indicates the linear component that is proportional to the number of iterations N. Equation (48) shows the representation matrix of the action that does not amplify δL during N iterations. Equation (49) shows the representation matrix of the action that amplifies δL during N iterations.

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[0149] Finally, for an error amplification circuit that repeats a quantum gate array N times, the information processing device 100 calculates the effect on the generator error by integrating the quantum gate synthesis effect and the quantum gate array repetition effect. For example, the information processing device 100 generates a synthesis effect representation matrix and a repetition effect representation matrix as described above, and pre-multiplies the synthesis effect representation matrix by the repetition effect representation matrix. For example, the matrix of Equation (49), which is proportional to the number of iterations N, is used as the repetition effect representation matrix.

[0150] In this way, the information processing device 100 linearly approximates the amplification effect of the error amplifier circuit amplifying the generator error. The amplification effect calculated here is an effect in the generator space. In performing numerical optimization using the objective function F(v), the information processing device 100 converts the effect in the generator space into an effect in the space of the probability distribution of the measurement values.

[0151] Here, the information processing device 100 generates a transformation matrix by rearranging the elements of the representation matrices of the initialization ρ and measurement Π used in the experiment, and multiplies the representation matrix of the error amplifier circuit by the transformation matrix from the left. The representation matrices of the initialization ρ and measurement Π used here include known or expected errors related to ρ and Π, and are sometimes called model values. As a result, the matrix C included in the objective function F(v) i,k is calculated. Also, for errors of generators not evaluated, the matrix C i,k By applying i,k is calculated.

[0152] Next, the improvement of the linear approximation in step #2 will be explained. For the linear approximation of the action of the quantum gate array, the linear mapping cml shown in Equation (29) and the linear mapping cmr shown in Equation (30) are used. The linear mappings cml and cmr are calculated by the coefficient l shown in Equation (33). jk Includes the reciprocal of

[0153] Here, the ideal value of the generator is different for the eigenvalue a j ,a k and e's a j a to the power of ek There exists a quantum gate such that the eigenvalues ​​a and b are the same. j ,a k If is a complex number, then a of e, as in equation (50), j a to the power of e k The power and the squared power may be the same value.

[0154] Examples of such quantum gates include rotation gates with a 180-degree rotation component, such as the 180-degree rotation gate. The X180 gate, Y180 gate, Z180 gate, CNOT gate, CZ gate, and SWAP gate have a 180-degree rotation component. The CNOT·XI90 gate and CNOT·YI90 gate have a 45-degree rotation component in addition to the 180-degree rotation component. The CNOT·XI90 gate is a quantum gate that performs a CNOT gate followed by an XI90 gate. The CNOT·YI90 gate is a quantum gate that performs a CNOT gate followed by a YI90 gate.

[0155] In this case, l jk = 0, and its inverse is infinity, so the values ​​of the linear maps cml and cmr cannot be calculated. This indicates that the differential map of the exponential map has a zero eigenvalue at the point representing the above generator, which is included in the generator space. Such points on the generator space are sometimes called singular points or critical points. At singular points, it can also be said that the linear approximation of the action of the quantum gate array breaks down. If the quantum gate array contains a quantum gate equivalent to a singular point, or if a composite quantum gate equivalent to a singular point is generated during synthesis, the action of the error amplifier circuit cannot be calculated using the linear approximation method described above.

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[0157] Therefore, the information processing device 100 of the second embodiment changes the definition of the generator error and suppresses linear approximation using linear maps cml and cmr. The information processing device 100 uses Equation (51) instead of Equation (5) to define the quantum gate G. The second matrix exponent on the right-hand side is the matrix exponent of the ideal value of the generator and is a unitary matrix. A unitary matrix is ​​a matrix U having the properties of Equation (52). The product of a unitary matrix and its adjoint matrix is ​​an identity matrix. The first matrix exponent on the right-hand side is the matrix exponent of the generator error δL'.

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[0160] The information processing device 100 estimates the generator error δL' defined in Equation (51) for the quantum gate to be evaluated. The generator error δL' is different from the generator error δL defined in Equation (5). According to the definition of Equation (51), an estimated value different from that in the case of Equation (5) is calculated. However, the generator errors δL and δL' are common in that they are indicators of the error possessed by the quantum gate G. The user may directly use the calculated generator error δL' to calibrate the quantum computer. Alternatively, the user may convert the generator error δL' to the generator error δL and use it to calibrate the quantum computer.

[0161] The information processing device 100 modifies Equation (39) to Equation (53) for the composition of quantum gates G1 and G2 included in the quantum gate array. In the modified composition, the linear maps dcl, dcr, cml, and cmr are not used. The matrix U1 is a unitary matrix indicating the ideal value of the quantum gate G1, and corresponds to the matrix exponent of the ideal value of the generator L1. The matrix U2 is a unitary matrix indicating the ideal value of the quantum gate G2, and corresponds to the matrix exponent of the ideal value of the generator L2. The modification of Equation (53) utilizes the fact that the matrices U1 and U2 are unitary matrices.

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[0163] The composite quantum gate shown in Equation (53) is approximated as shown in Equation (54). The approximation in Equation (54) utilizes the fact that the generator errors δL1', δL2' are small compared to the matrices U1, U2. The ideal value of the composite quantum gate is the product of the matrices U1, U2. The error of the composite quantum gate is expressed using the generator errors δL1', δL2' and the matrix U2. The error of the composite quantum gate is the matrix exponent of the generator error δL1' transformed using the matrix U2 and its adjoint matrix, plus the generator error δL2'.

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[0165] The synthesis of quantum gates included in a quantum gate array is performed regardless of whether a matrix corresponding to a singularity appears in the calculation process. Because linear mappings cml and cmr are not used, quantum gate synthesis is possible even if a matrix corresponding to a singularity appears. The information processing device 100 obtains the entire synthesized quantum gate of the quantum gate array by successively applying the linear approximation of Equations (53) to (54) from the beginning to the end of the quantum gate array.

[0166] 4 is a diagram showing an example of combining quantum gates included in a quantum gate array. Quantum gate array 144 includes, from the top, quantum gates 145, 146, 147, and 148. First, information processing device 100 performs combining by multiplying quantum gate 145 by quantum gate 146. At this time, information processing device 100 multiplies the representation matrix of quantum gate 145 to the left by the representation matrix of quantum gate 146.

[0167] Next, the information processing device 100 performs composition by multiplying the composite quantum gate of quantum gates 145 and 146 by quantum gate 147. At this time, the information processing device 100 multiplies the representation matrix of the composite quantum gate by the representation matrix of quantum gate 147 on the left side. Next, the information processing device 100 performs composition by multiplying the composite quantum gate of quantum gates 145, 146, and 147 by quantum gate 148. At this time, the information processing device 100 multiplies the representation matrix of the composite quantum gate by the representation matrix of quantum gate 148 on the left side. In this way, the information processing device 100 obtains a composite quantum gate that represents the entire quantum gate array 144.

[0168] Next, the information processing device 100 obtains a matrix representing the action when the composite quantum gate representing the entire quantum gate array is repeated N times from the composite quantum gate. The composite result of the quantum gate array is e δL’ It is in the form of U, where U is a unitary matrix that represents the ideal value of the composite quantum gate. δL' is the generator error of the composite quantum gate.

[0169] Here, we assume that the unitary matrix U has the cyclic property shown in equation (55). Multiplying a cyclic unitary matrix U k times results in an identity matrix. Circularity indicates that applying an ideal quantum gate to a qubit k times in succession will return the qubit to its original quantum state. Many quantum gates are cyclic, such as a rotation gate with a fixed rotation amount. A composite quantum gate, which is a combination of two or more cyclic quantum gates, also has cyclicity. Therefore, the cyclicity of the error amplifier circuit is a reasonable assumption.

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[0171] The smallest positive integer k that satisfies Equation (55) is the period of the unitary matrix U. The period of an ideal 180-degree rotation gate is 2. The period of an ideal 90-degree rotation gate is 4. The period of an ideal 45-degree rotation gate is 8. The number of iterations N is assumed to be a multiple of k. For example, if the period of the composite quantum gate is expected to be 8, such as when a quantum gate array includes a 45-degree rotation gate, N is set to 8, 16, 24, 32, ... Note that the period k is different from the identifier that identifies the initialization and measurement pair (ρ, Π) described above.

[0172] The information processing device 100 of the second embodiment performs an approximate calculation for the iteration of the combined quantum gate by utilizing the fact that the ideal value of the combined quantum gate is a unitary matrix and has cyclicity. In this approximate calculation, as in the composition of quantum gates, the linear maps dcl, dcr, cml, and cmr are not used. The iteration of the combined quantum gate is performed regardless of whether the combined quantum gate has a singularity or not. Because the linear maps cml and cmr are not used, it is possible to calculate the action of the iteration even if the combined quantum gate has a singularity.

[0173] The information processing device 100 uses an approximation formula shown in Formula (56) instead of Formula (46). U is a linear mapping of equation (31) for the unitary matrix U. Based on the assumption that the unitary matrix U is cyclic, equation (56) can be expanded to equation (57). The matrix logarithm of the repeated action is proportional to the number of iterations N and inversely proportional to the period k. Furthermore, the matrix logarithm of the repeated action is proportional to the sum of the mth-th (m=1,...,k) term for the unitary matrix. Due to the cyclicity, this equation remains valid even if the range of m is replaced by 0 to k-1.

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[0176] Here, we explain the derivation of Equation (57). From the unitarity of U, the generator L ideal is (L ideal ) -1 =-L ideal By choosing a Hermitian matrix as the representation basis, the representation matrix of a quantum gate is reduced to a real matrix. In group theory, the period k is sometimes called the order of the unitary matrix U.

[0177] A unitary matrix, a Hermitian matrix, and an anti-Hermitian matrix are diagonalizable matrices, and can be spectrally decomposed as shown in Equation (25). j satisfies uniqueness, and the projection matrix P j satisfies orthogonality and decomposability. Uniqueness is the property that different projection matrices P j The eigenvalue a corresponding to j The orthogonality is the property shown in Equation (26). The decomposability is that all projection matrices P j The sum of these is the identity matrix.

[0178] If a matrix A is normal, then it is unitarily diagonalizable. Unitary, Hermitian, and anti-Hermitian matrices are normal matrices. As a consequence, the projection matrix P j satisfies the Hermitic property shown in Equation (58). Hermiticity is the property that a matrix and its adjoint matrix are identical. A projection that satisfies the Hermitic property is called an orthogonal projection, which indicates that the space before the projection and the space after the projection are orthogonal. The projection matrix obtained by spectrally decomposing a unitary matrix U satisfies the Hermitic property. In addition, the ideal generator L ideal The projection matrix obtained by spectrally decomposing also satisfies the Hermitian property. To explain the derivation of the approximation formula, we introduce the linear mappings shown in Equation (59) and Equation (60).

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[0182] For the sake of explanation, we will generalize the number of iterations N to any positive integer. Then, the number of iterations N is expressed as in formula (61). q is the quotient of N divided by k, and r is the remainder. Substituting U1=U2=U and δL1'=δL2'=δL into formulas (53) and (54), we obtain formula (62). Similarly, G2=G, G1=G 2 Substituting and combining with the equation for N=2, we obtain equation (63) for N=3.

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[0186] By repeating the above substitution, an approximation to formula (64) is obtained for the number of iterations N. m is a non-negative integer. Due to the cyclicity of U, formulas (65) and (66) hold. Therefore, formula (64) is expanded as formula (67), and then further expanded as formula (68). However, in formula (67), when r = 0, the second term in the matrix exponential function on the right-hand side is considered to be 0. Substituting r = 0 into formula (68) gives formula (69).

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[0193] Due to the cyclicity of U, equation (70) holds, and equation (71) holds for any integer m. If m'=km, then the linear mapping sum U,k Therefore, from equations (69) and (72), equation (57) holds.

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[0197] Next, we will explain the derivation of Equation (56). Below, we will explain the linear mapping sum in Equation (68). U,k and the linear map rsd U,r,k The goal is to rewrite the spectral decomposition of U as shown in equation (73). j are the eigenvalues, and Q j is the projection matrix. Since U is a unitary matrix, the projection matrix Q j is a Hermitian matrix, and the eigenvalues ​​ω j The absolute value of is 1. If the cyclicity of Equation (55) holds, the projection matrix Q j From the orthogonality and decomposition of all projection matrices Q j The sum of these is expanded as in equation (74). Therefore, the eigenvalue ω of the unitary matrix U j is the k-th root of 1, as shown in Equation (75).

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[0201] Linear mapping sum U,k and the linear map rsd U,r,k are all linear mappings sum U,n It contains the linear map sum U,n is defined as in equation (76) and expanded as in equation (77). In order to calculate the series sum in equation (77), we use S n The eigenvalue ω j The absolute value of is 1 and the eigenvalue ω j Due to the uniqueness of (77), the series in (77) can be calculated as (79).

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[0206] When n=k, the formula (80) holds, and the formula (79) is simplified to the formula (81). jj’ is the Kronecker delta. For general n, equation (77) is expressed as equation (82) and expanded as equation (83). Therefore, the linear mapping sum U,k is a linear mapping ssp U It is expressed as equation (84) using the linear mapping sum U,r is a linear mapping ssp U This is expressed as equation (85) using

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[0213] Linear mapping rsd U,r,k is defined as in Equation (86) and expanded as in Equation (87). Therefore, Equation (68) can be rewritten as Equation (88) using the spectral decomposition of the unitary matrix U.

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[0217] When the generator error δL' is expressed in a basis that diagonalizes the unitary matrix U, ssp U The non-zero elements of (δL') are only in the diagonal blocks, and rsd U,r,k The non-zero components of (δL') exist only in the off-diagonal blocks. Therefore, the components of the diagonal blocks of the generator error δL' are amplified in proportion to the number of iterations N, while the components of the off-diagonal blocks are not amplified. Therefore, the iteration of the composite quantum gate is linearly approximated as shown in Equation (56).

[0218] Linear mapping rsd U,r,k By definition, it can be calculated without spectral decomposition. Ucan be calculated without spectral decomposition from Equation (84). In the second embodiment, spectral decomposition is not required for the synthesis of quantum gates or the repetition of synthesized quantum gates. Therefore, the linear approximation is made more efficient and the numerical stability is improved.

[0219] In the iteration of the composite quantum gate, the information processing device 100 does not need to determine whether the unitary matrix U has a singularity or not, and does not need to distinguish between cases depending on whether a singularity exists or not. This is because the approximation formula of Equation (57) can be used for various composite quantum gates with period k, and is not limited to a specific type of quantum gate such as a 180-degree rotation gate.

[0220] This also gives the linear map ssp with respect to the unitary matrix U. U If the unitary matrix U has no singular points, then the generator L ideal Linear mapping ssp with respect to L and the linear map ssp with respect to the unitary matrix U U On the other hand, if the unitary matrix U has a singular point, then the linear mapping ssp L ,ssp U are not equivalent. The linear map ssp with respect to the unitary matrix U U By using this, approximate calculations can be performed that are independent of the presence or absence of singularities.

[0221] Here, the linear map ssp L ,ssp U The difference between generator L and ideal The spectral decomposition of is expressed as in Equation (89). ideal The unitary matrix U, which is the matrix exponent of the matrix, is expressed as in Equation (90).

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[0224] generator L ideal If the set of eigenvalues ​​of satisfies equation (91), then the eigenvalues ​​of the unitary matrix U and the projection matrix and generator L ideal There is a one-to-one correspondence between the eigenvalues ​​and projection matrices of the unitary matrix U and the generator L. ideal The set of projection matrices is the same as that of the matrix (93), and the equality of equation (93) holds.

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[0228] On the other hand, the generator L ideal If there is at least one pair in the set of eigenvalues ​​of that does not satisfy equation (91), then the generator L ideal Since different eigenvalues ​​of correspond to the same eigenvalue of the unitary matrix U, equation (90) does not undergo spectral decomposition. ideal If the set of projection matrices between and the unitary matrix U is different, then the generator L ideal is a singular point. At a singular point, the equality of equation (93) does not hold.

[0229] Next, we will explain numerical examples of the matrices used in the above calculations. Here, we consider a one-qubit system with d=2. First, we define 2 × 2 matrices σ0, σ1, σ2, σ3, called Pauli matrices, as shown in Equation (94). Using these matrices σ0, σ1, σ2, σ3, we select the Pauli matrix basis shown in Equation (95) as the basis for the matrix representation of quantum gates and generators.

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[0232] In addition, as a basis for the vector representation of quantum gates and generators, S1 to S2 shown in Eqs. (96) to (109) are used. 16 In Equations (96) to (109), the overline on B indicates a complex conjugate, and the product operator indicates a tensor product.

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[0247] Now, consider two quantum gates. Quantum gate G1 is a rotation gate that rotates 180 degrees around the X axis and can be denoted as an X180 gate. Quantum gate G2 is a rotation gate that rotates 90 degrees around the Z axis and can be denoted as a Z90 gate. Quantum gate G1 is the quantum gate to be evaluated, and its generator error is unknown. Quantum gate G2 is a quantum gate not to be evaluated, and its generator error is assumed to be 0.

[0248] The information processing device 100 sets, as quantum gate arrays to be tested, a quantum gate array including only quantum gate G1 and a quantum gate array including quantum gates G1 and G2 in that order. The ideal action of quantum gate G1 on quantum state ρ is given by equation (110). The ideal action of quantum gate G2 on quantum state ρ is given by equation (111). The representation matrix of quantum gates G1 and G2 under the above Pauli matrix basis B is given by equation (112). Quantum gate G1 is a singular point. Quantum gate G2 is not a singular point. The composite quantum gate G2G1 is a singular point.

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[0252] 5 is a diagram showing an example of the effect on generator errors by the first quantum gate array. Matrix 151 shows the effect on generator errors through quantum gate synthesis and quantum gate array iteration for a quantum gate array including only quantum gate G1, and indicates components proportional to the number of iterations N. Matrix 151 is a representation matrix based on the above-mentioned basis S.

[0253] 6 is a diagram showing an example of the effect on generator errors by the second quantum gate array. Matrix 152 shows the effect on generator errors through quantum gate composition and quantum gate array iteration for a quantum gate array including quantum gates G1 and G2, and indicates components proportional to the number of iterations N. Matrix 152 is a representation matrix based on the above-mentioned basis S.

[0254] Furthermore, the information processing device 100 selects five iteration numbers N to be tried: 2, 4, 8, 16, and 32. The information processing device 100 applies these five iteration numbers N in common to the two quantum gate arrays.

[0255] Furthermore, the information processing device 100 prepares four initializations x0, y0, z0, and z1 as the initialization ρ. x0 is a quantum state corresponding to the eigenvector of σ1 (eigenvalue + 1). y0 is a quantum state corresponding to the eigenvector of σ2 (eigenvalue + 1). z0 is a quantum state corresponding to the eigenvector of σ3 (eigenvalue + 1). z1 is a quantum state corresponding to the eigenvector of σ3 (eigenvalue - 1). The information processing device 100 applies these four initializations ρ in common to the two quantum gate arrays described above.

[0256] Furthermore, the information processing device 100 prepares three measurements x, y, and z as measurements Π. x is a projection measurement corresponding to σ1. y is a projection measurement corresponding to σ2. z is a projection measurement corresponding to σ3. The information processing device 100 applies these three measurements Π in common to the two quantum gate arrays. Therefore, in this example, there are 2×5×4×3=120 combinations of quantum gate array a, iteration count N, initialization ρ, and measurements Π.

[0257] Since the measurement results of a quantum computer are obtained probabilistically, the information processing device 100 obtains multiple samples for each combination to generate the aforementioned experimental data. However, in this numerical simulation, the information processing device 100 generates generator errors for the quantum gate G1 using pseudorandom numbers instead of using the quantum computer 115. The information processing device 100 generates experimental data indicating a true probability distribution from these generator errors.

[0258] FIG. 7 is a diagram showing examples of correct and estimated values ​​of generator errors. Matrix 153 indicates the correct generator error of quantum gate G1. Matrix 153 is a representation matrix based on the above-mentioned Pauli matrix basis B. Information processing device 100 generates matrix 153 using pseudorandom numbers. Matrix 154 indicates the generator error of quantum gate G1 estimated from matrices 151 and 152 and experimental data. Matrix 154 is a representation matrix based on the above-mentioned Pauli matrix basis B. Comparing matrix 153 and matrix 154, many matrix elements are estimated with high accuracy. Note that some elements of the generator error are not sufficiently amplified using only the above-mentioned two quantum gate arrays, and therefore some matrix elements are affected.

[0259] Next, we will explain examples of quantum gates with singularities other than the 180-degree rotation gate. The CNOT·XI90 gate is a two-input quantum gate with order k=8. The ideal action of a quantum gate acting on a d-dimensional quantum system is expressed as in equation (113) using a Hermitian matrix H of size d×d. The matrix H is called the Hamiltonian. The matrix representation of the mapping shown in equation (113) is the unitary matrix U.

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[0261] In the following, we consider a two-qubit system corresponding to d=4. The Hamiltonian H of the controlled NOT gate is CX is given by equation (114). The Hamiltonian H of the XI90 gate XI90 is given by Equation (115). In Equations (114) and (115), the product operator is the tensor product. I, X, and Z are each one of the Pauli matrices defined in Equation (116). Here, we use the tensor product of normalized Pauli matrices as the representation basis B for the matrix representation of quantum gates and generators, as shown in Equation (117).

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[0266] Figure 8 shows an example of a quantum gate with a singularity. The CNOT·XI90 gate first performs a CNOT operation, followed by an XI90 operation. Matrix 155 is a unitary matrix U that represents the ideal behavior of the CNOT·XI90 gate.

[0267] Figure 9 shows an example of a generator of a quantum gate with a singularity. Matrix 156 is the generator L ideal The matrix 156 corresponds to the matrix logarithm of the matrix 155 in FIG.

[0268] The matrix 156 has 16 eigenvalues ​​as shown in equation (118). Among these 16 eigenvalues, the 13th and 14th pairs, the 15th and 16th pair, and the 15th and 12th pair contradict equation (91). Therefore, CNOT·XI90 corresponds to a singular point. Also, these 16 eigenvalues ​​a j are all exp(a j ) 8 = 1. On the other hand, if n is an integer less than or equal to 7, then exp(a j ) n Eigenvalue a that does not satisfy =1 j Therefore, the order k of the CNOT·XI90 gate is 8.

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[0270] The CNOT·XI90 gate with k=8 has a 45-degree rotation component because it returns the quantum bit to its original quantum state in eight operations. The information processing device 100 can perform the linear approximation of the second embodiment for a quantum gate array including the CNOT·XI90 gate. Next, the functions and processing procedures of the information processing device 100 will be described.

[0271] 10 is a block diagram showing an example of functions of an information processing device. The information processing device 100 has a setting data storage unit 121, an experiment data storage unit 122, an evaluation data storage unit 123, an experiment unit 124, and an analysis unit 125. The setting data storage unit 121, the experiment data storage unit 122, and the evaluation data storage unit 123 are implemented using, for example, RAM 102 or HDD 103. The experiment unit 124 and the analysis unit 125 are implemented using, for example, CPU 101 or GPU 104 and a program.

[0272] The setting data storage unit 121 stores setting data. The setting data indicates the content of an experiment using the quantum computer 115. The setting data indicates each quantum gate G and its generator L. The setting data also indicates the generator error δL of a quantum gate not being evaluated. The setting data also indicates the quantum gate sequence a to be tried and the number of iterations N. The setting data also indicates the initialization ρ and measurement Π to be tried.

[0273] The experimental data storage unit 122 stores experimental data. The experimental data is a compilation of measurement values ​​read from the quantum computer 115. The experimental data indicates a frequency distribution f for each combination of the quantum gate array a, the number of iterations N, the initialization ρ, and the measurement Π. The evaluation data storage unit 123 stores evaluation data. The evaluation data indicates the generator error δL' of the quantum gate to be evaluated, estimated from the setting data and the experimental data. However, the evaluation data may indicate other indices converted from the generator error δL'.

[0274] The experiment unit 124 instructs the quantum computer 115 to perform quantum computation based on the setting data stored in the setting data storage unit 121. The experiment unit 124 reads measurement values ​​from the quantum computer 115. The experiment unit 124 aggregates the read measurement values ​​to generate experimental data, and stores the experimental data in the experimental data storage unit 122.

[0275] The analysis unit 125 analyzes the setting data stored in the setting data storage unit 121 and the experimental data stored in the experimental data storage unit 122 to estimate the generator error of the quantum gate to be evaluated. At this time, the analysis unit 125 generates an objective function and searches for a generator error that minimizes the value of the objective function using a mathematical programming solver. The analysis unit 125 generates evaluation data and stores it in the evaluation data storage unit 123. However, the analysis unit 125 may display the evaluation data on the display device 111 or transmit it to another information processing device.

[0276] 11 is a diagram showing an example of the setting data. The setting data storage unit 121 stores a quantum gate table 131, a quantum gate array table 132, an iteration count table 133, and an initialization measurement table .

[0277] The quantum gate table 131 associates the identifier of a quantum gate with the representation matrix of the quantum gate and the representation matrix of the generator. g n quantum gates are registered. g n quantum gates g,1 The quantum gates are evaluated, and the remaining quantum gates are not evaluated. For the quantum gates that are not evaluated, a representation matrix of the generator error is also registered.

[0278] The quantum gate array table 132 associates an identifier i with a quantum gate array. A quantum gate array of length m is described by arranging m identifiers included in the quantum gate table 131. The iteration count table 133 associates an identifier (i, j) with the iteration count. For each quantum gate array included in the quantum gate array table 132, one or more iteration counts (typically, two or more iteration counts) are registered. The iteration count is an integer multiple of the period of the quantum gate. The initialization / measurement table 134 associates an identifier (i, k), a representation matrix indicating the initialization of a quantum state, and a representation matrix indicating the measurement of the quantum state. For each quantum gate array included in the quantum gate array table 132, one or more pairs of initialization and measurement are registered.

[0279] 12 is a diagram showing an example of experimental data and evaluation data. The experimental data storage unit 122 stores an experimental result table 135. The experimental result table 135 associates identifiers (i, j, k) with frequency distributions. Quantum gate array a i and the number of iterations N i,j and the set of initialization and measurement (ρ i,k ,Π i,k ) a frequency distribution is registered for each combination of the two. A frequency distribution is a list of the frequencies of multiple measurements.

[0280] The analysis unit 125 generates an objective function table 136. The objective function table 136 associates an identifier (i, k), a matrix C, and a vector b. i and the set of initialization and measurement (ρ i,k ,Π i,k ) and one combination of them will produce one matrix C and one vector b. i ,(ρ i,k ,Π i,k ) and vector b are used to define an objective function that indicates the quality of the generator error estimation.

[0281] The evaluation data storage unit 123 stores an evaluation result table 137. The evaluation result table 137 associates the identifier of a quantum gate with an estimated value of a generator error. The identifier is the same as that of the quantum gate table 131. The evaluation result table 137 stores n g Among the quantum gates, n g,1 Contains the generator errors of quantum gates.

[0282] Fig. 13 is a first flowchart showing an example of the procedure for quantum operation evaluation. The flowchart in Fig. 13 shows the overall flow of quantum operation evaluation performed by the information processing device 100. In step S10, the experiment unit 124 receives setting data from the user. The setting data includes the quantum gate array a to be tried and the number of iterations N.

[0283] In step S11, the experiment unit 124 performs an experiment using the quantum computer 115 based on the setting data of step S10. The experiment unit 124 generates a quantum circuit including an error amplifier circuit based on the setting data, and instructs the quantum computer 115 to execute the quantum circuit. The experiment unit 124 reads measurement values ​​from the quantum computer 115. The experiment unit 124 generates experimental data by statistically processing the measurement values.

[0284] In step S12, analysis unit 125 performs preprocessing on the experimental data generated in step S11. Through the preprocessing, analysis unit 125 extracts a first-order component proportional to the number of iterations N from the experimental data. In step S13, analysis unit 125 reads the setting data from step S10. The setting data includes information on quantum gates to be evaluated for implementation accuracy and information on quantum gates not to be evaluated for implementation accuracy.

[0285] In step S14, the analysis unit 125 uses the loaded setting data to generate a linear approximation function that linearly approximates the effect of the error amplifier circuit on errors in the quantum gate. In step S15, the analysis unit 125 performs data fitting using the linear approximation function of step S14 and the preprocessed data of step S12. The analysis unit 125 fits the preprocessed data to the linear approximation function to estimate the error of the quantum gate to be evaluated.

[0286] In step S16, the analysis unit 125 generates and outputs evaluation data indicating the estimated error. Note that steps S11 and S12 and steps S13 and S14 may be executed in reverse order or in parallel. Also, steps S11 and S12 and steps S13 and S14 may be executed by different information processing devices. Also, if a linear approximation function has already been generated in the past, the analysis unit 125 may omit steps S13 and S14.

[0287] Fig. 14 is a second flowchart showing an example of the procedure for evaluating quantum operations. The flowchart in Fig. 14 shows a flow of quantum operation evaluation in more detail than that in Fig. 13. In step S20, the experiment unit 124 reads out setting data. Based on the setting data, the experiment unit 124 identifies a combination of the quantum gate array a, the number of iterations N, the initialization ρ, and the measurement Π.

[0288] In step S21, the experiment unit 124 repeatedly instructs the quantum computer 115 to perform quantum calculations and read out measurement values ​​for each combination identified in step S20. In step S22, the experiment unit 124 aggregates the measurement values ​​read out in step S21 for each combination identified in step S20 to calculate a frequency distribution f.

[0289] In step S23, the analysis unit 125 generates a Vandermonde matrix V for the number of iterations N. For each combination of the quantum gate array a, the initialization ρ, the measurement Π, and the measurement value x, the analysis unit 125 estimates a coefficient vector h of the series expansion from the Vandermonde matrix V and the frequency of the measurement value x included in the frequency distribution f calculated in step S22.

[0290] In step S24, the analysis unit 125 extracts a first-order component from the coefficient vector h of step S23 and estimates a coefficient vector h1 for each combination of the quantum gate array a, the initialization ρ, and the measurement Π. In step S25, the analysis unit 125 reads out the settings of the combination of the quantum gate array a, the number of iterations N, the initialization ρ, and the measurement Π from the setting data.

[0291] In step S26, the analysis unit 125 defines the generator error δL' of each quantum gate. The generator error δL' of the quantum gate to be evaluated corresponds to a variable indicating an unknown quantity. This generator error δL' is defined as shown in Equation (51). In step S27, the analysis unit 125 synthesizes one quantum gate included in the quantum gate array a, preferentially from the beginning. At this time, the analysis unit 125 performs the approximate calculation shown in Equations (53) and (54), taking advantage of the fact that the ideal value of the quantum gate is a unitary matrix and the generator error δL' is small.

[0292] In step S28, the analysis unit 125 determines whether all quantum gates included in the quantum gate array a have been combined. If all quantum gates have been combined, the quantum gate array a is represented by a single combined quantum gate. If all quantum gates have been combined, the process proceeds to step S29; otherwise, the process returns to step S27.

[0293] In step S29, the analysis unit 125 calculates the period k of the ideal value of the combined quantum gate. The period k is sometimes called the order. The method for calculating the period k will be described later. In step S30, the analysis unit 125 approximately calculates a matrix indicating the repetitive action from the ideal value of the combined quantum gate and the period k, as shown in equations (56) and (57). In this case, the analysis unit 125 utilizes the fact that the ideal value of the combined quantum gate is a unitary matrix, that the ideal value of the combined quantum gate has the cyclic property of equation (55), and that the generator error δL' is small.

[0294] 15 is a second flowchart (continued) showing an example of the procedure for evaluating quantum operations. In step S31, the analysis unit 125 converts the matrix calculated in step S30 into a matrix C indicating the action in the space of probability distributions of measurement values, using the initialization ρ and the measurement Π. In step S32, the analysis unit 125 calculates a vector b using a known generator error δL' for the quantum gate not being evaluated.

[0295] In step S33, analysis unit 125 uses matrix C from step S31, vector b from step S32, and coefficient vector h1 from step S24 to generate objective function F. In step S34, analysis unit 125 uses a mathematical programming solver to calculate the value of parameter v that minimizes the value of objective function F generated in step S33.

[0296] In step S35, analysis unit 125 estimates the generator error δL′ of the quantum gate to be evaluated from the value of parameter v calculated in step S34. In step S36, analysis unit 125 generates evaluation data indicating the generator error δL′ estimated in step S35, and outputs the evaluation data.

[0297] Note that step S20 corresponds to step 10 in Figure 13. Steps S21 and S22 correspond to step S11 in Figure 13. Steps S23 and S24 correspond to step S12 in Figure 13. Step S25 corresponds to step S13 in Figure 13. Steps S26 to S33 correspond to step S14 in Figure 13. Steps S34 and S35 correspond to step S15 in Figure 13. Step S36 corresponds to step S16 in Figure 13.

[0298] 16 is a diagram showing an example of a pseudo program for determining the period k of a quantum gate. Pseudo program 138 shows an example of an algorithm for determining the period k of a quantum gate. The algorithm shown by pseudo program 138 is executed in step S29 described above.

[0299] The analysis unit 125 receives a unitary matrix U, a maximum value of the period k, and an error threshold. The analysis unit 125 initializes k=-1 and a=1. The analysis unit 125 calculates the matrix norm of the difference between the ath power of U and the unit matrix I. The matrix norm is a real scalar value. For example, the Frobenius norm is used as the matrix norm.

[0300] The analysis unit 125 determines whether the matrix norm is less than a threshold. If the matrix norm is less than the threshold, the analysis unit 125 determines a as the period k of the unitary matrix U. On the other hand, if the matrix norm is equal to or greater than the threshold, the analysis unit 125 increments a by 1 and recalculates the matrix norm. If the period k cannot be determined even when a reaches its maximum value, the analysis unit 125 outputs a warning.

[0301] It should be noted that the quantum gate G containing an error can also be specified using the generator error δL″ instead of the generator error δL′, as shown in equation (78). When the generator error δL′ is used, the matrix exponent of the generator error δL′ is multiplied from the left side of the unitary matrix U indicating the ideal value of the quantum gate. This means that the generator error is decomposed to the left side. On the other hand, when the generator error δL″ is used, the matrix exponent of the generator error δL″ is multiplied from the right side of the unitary matrix U. This means that the generator error is decomposed to the right side.

[0302] The generator error δL″ may not coincide with the generator error δL or the generator error δL′. However, the information processing device 100 can evaluate the generator error δL″ using the same method as for the generator error δL′. The information processing device 100 may output the generator error δL″ instead of the generator error δL′ or together with the generator error δL′. Furthermore, the information processing device 100 may convert the generator error δL″ into the generator error δL. Furthermore, the information processing device 100 may calibrate the quantum computer 115 using the estimated generator error δL″.

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[0304] Between the generator error δL″ and the generator error δL, the relationship shown in Equation (79) is established using the unitary matrix U. Furthermore, between the generator error δL′ and the generator error δL, the relationship shown in Equation (80) is established within the range of first-order approximation using the linear map dcl. Furthermore, between the generator error δL″ and the generator error δL, the relationship shown in Equation (81) is established within the range of first-order approximation using the linear map dcr.

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[0308] Therefore, by replacing the above-mentioned δL' with dcl(δL), a linear approximation to δL is established, and the evaluation method for evaluating δL' can be used as an evaluation method for evaluating δL. Similarly, by replacing δL'' with dcr(δL), a linear approximation to δL is established, and the evaluation method for evaluating δL'' can be used as an evaluation method for evaluating δL.

[0309] As described above, the information processing device 100 of the second embodiment uses quantum tomography to estimate errors in quantum gates of the quantum computer 115. Therefore, the information processing device 100 can efficiently estimate errors in multiple quantum gates all at once. Furthermore, the information processing device 100 causes the quantum computer 115 to implement an error amplification circuit that repeats the same quantum gate array. This amplifies minute errors in the quantum gates, improving the accuracy of error estimation from experimental data.

[0310] Furthermore, based on the estimated error, the user can adjust the values ​​of the control parameters of the quantum computer 115, thereby improving the accuracy of the quantum operations of the quantum computer 115. Furthermore, the information processing device 100 converts quantum gates into generators and linearly approximates the effect of the error amplifier circuit on the generator error. This reduces the load of data analysis and improves the numerical stability of the estimated error.

[0311] Furthermore, the information processing device 100 defines a generator error so that a quantum gate can be expressed as the product of a unitary matrix representing the ideal value of the quantum gate and the matrix exponent of the generator error. The information processing device 100 then synthesizes quantum gates without using the linear mappings cml and cmr, taking advantage of the properties of unitary matrices and the fact that the generator error is small. This allows the information processing device 100 to complete the synthesis even if a matrix corresponding to a singularity appears during synthesis, thereby increasing the variety of quantum gates that can be synthesized.

[0312] Furthermore, the information processing device 100 linearly approximates the repetitive action without using the linear mappings cml and cmr by taking advantage of the fact that the ideal value of the composite quantum gate is a unitary matrix and has cyclicity. This allows the information processing device 100 to complete the approximate calculation while avoiding the influence of singular points, and increases the number of quantum gates to which linear approximation can be applied.

[0313] In particular, the information processing device 100 can estimate errors in various rotation gates with cyclicity, such as a 180-degree rotation gate, a 90-degree rotation gate, and a 45-degree rotation gate. Furthermore, the information processing device 100 does not need to distinguish between calculation methods for repeated actions depending on whether the combined quantum gate corresponds to a singularity, thereby simplifying approximation calculations. Furthermore, the information processing device 100 does not need to perform high-load spectral decomposition in either the synthesis of quantum gates or the iteration of the combined quantum gate, thereby reducing the amount of calculation. [Explanation of symbols]

[0314] 10. Information processing equipment 11 Storage section 12 Processing section 21 Quantum circuit 22 Quantum Gate Array 23 Quantum Gates 24 Measurement data 25 Linear Approximation Functions 26 Error 27 Queue 28 repetitions 29 cycles

Claims

1. A process of acquiring measurement data indicating the execution result of a quantum computer for a quantum circuit that repeats a quantum gate array including a first quantum gate N times (N is an integer equal to or greater than 1); a process of determining, from a first matrix indicating ideal values ​​of the quantum gate array, a period of k (k is an integer equal to or greater than 1) at which the first matrix is ​​raised to the kth power to become a unit matrix; a process of generating a linear approximation function that linearly approximates the influence of the error on the measurement data by approximating the repetition of the quantum gate array with a matrix exponent of a first conversion result obtained by converting an error that the quantum computer has for the first quantum gate using the first matrix, the N, and the period; a process of estimating the error using the linear approximation function and the measurement data; A quantum computing evaluation program that runs the above on a computer.

2. the first transformation result is generated by summing products of the mth power of the first matrix (m is a non-negative integer less than k), the error, and the mth power of the adjoint matrix of the first matrix for different values ​​of m, and multiplying the sum by the reciprocal of N and k; The quantum operation evaluation program according to claim 1.

3. N is a multiple of k. The quantum operation evaluation program according to claim 1.

4. the generating step includes a step of defining the first quantum gate as a product of a second matrix representing an ideal value of the first quantum gate and a matrix exponent of the error. The quantum operation evaluation program according to claim 1.

5. the generating process includes a process of approximating the quantum gate array by a product of a second matrix indicating an ideal value of the first quantum gate, a third matrix indicating an ideal value of a second quantum gate included in the quantum gate array, and a matrix exponent of a second transformation result obtained by transforming the error using the third matrix. The quantum operation evaluation program according to claim 1.

6. A process of acquiring measurement data indicating the execution result of a quantum computer for a quantum circuit that repeats a quantum gate array including a first quantum gate N times (N is an integer equal to or greater than 1); a process of determining, from a first matrix indicating ideal values ​​of the quantum gate array, a period of k (k is an integer equal to or greater than 1) at which the first matrix is ​​raised to the kth power to become a unit matrix; a process of generating a linear approximation function that linearly approximates the influence of the error on the measurement data by approximating the repetition of the quantum gate array with a matrix exponent of a first conversion result obtained by converting an error that the quantum computer has for the first quantum gate using the first matrix, the N, and the period; a process of estimating the error using the linear approximation function and the measurement data; A quantum operation evaluation method performed by a computer.

7. a storage unit that stores measurement data indicating the execution result of a quantum computer for a quantum circuit that repeats a quantum gate array including a first quantum gate N times (N is an integer equal to or greater than 1); a processing unit that executes the following processes: determining, from a first matrix indicating an ideal value of the quantum gate array, a period of k (k is an integer equal to or greater than 1) that results in a unit matrix when the first matrix is ​​raised to the kth power; approximating the repetition of the quantum gate array by a matrix exponent of a first conversion result obtained by converting an error that the quantum computer has for the first quantum gate using the first matrix, N, and the period, thereby generating a linear approximation function that linearly approximates the influence of the error on the measurement data; and estimating the error using the linear approximation function and the measurement data. An information processing device having the above.

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