Quantum computation evaluation program, quantum computation evaluation method, and information processing device.

Abstract

To streamline the evaluation of errors in quantum gates. [Solution] The information processing device 10 acquires measurement data 14 showing the measured quantum state of a qubit measured after the quantum computer repeatedly executes a sequence of quantum gates 13 containing multiple quantum gates on the qubit. The information processing device 10 divides the space to which the matrix representing the quantum operation by the sequence of quantum gates 13 belongs into a subspace 15a containing a singularity 15c for which an eigenvalue satisfying certain conditions is calculated, and a subspace 15b that does not contain the singularity 15c. The information processing device 10 generates a function 16 within the subspace 15b that linearly approximates the effect of the error of the quantum gate 13a included in the multiple quantum gates on the measured value. The information processing device 10 uses the function 16 and the measurement data 14 to estimate the error when the quantum computer executes the quantum gate 13a.

Description

[Technical Field]

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

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

[0003] Implemented quantum computers typically have errors (errors) in their quantum computations, where the operation deviates from the ideal. Users may evaluate these quantum computation errors and, based on the evaluation results, perform calibration by adjusting the values ​​of the quantum computer's control parameters. For example, a quantum computer may have control parameters to adjust the duration, amplitude, and waveform of the microwave pulses emitted to the qubits. Adjusting the values ​​of these control parameters can improve the accuracy of quantum computations.

[0004] Here, since the measured values ​​of a qubit are generated through multiple types of quantum operations, it is not easy to directly evaluate the errors of individual quantum operations. Therefore, one technique for estimating the errors of quantum operations is quantum tomography. 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] Furthermore, there are techniques for performing tomography experiments using quantum processors according to quantum circuits, analyzing the experimental results to generate tomogram data, and visualizing the tomogram data. Additionally, there are techniques for performing quantum tomography of a target quantum system and analyzing quantum noise processes based on tensor transfer mapping that shows the dynamic evolution of the quantum noise process. [Prior art documents] [Patent Documents]

[0006] [Patent Document 1] U.S. Patent Application Publication No. 2019 / 0102917 [Patent Document 2] International Publication No. 2020 / 228470 [Overview of the Initiative] [Problems that the invention aims to solve]

[0007] Quantum tomography sometimes uses quantum circuits that include error amplification circuits, which repeat the same sequence of quantum gates, to evaluate errors in quantum gates. Error amplification circuits amplify the minute errors in quantum gates, making the error components in the measured values ​​more visible and improving the reliability of the evaluation results. On the other hand, because error amplification circuits have a nonlinear amplification effect on errors, data analysis may become more complex. As a result, the burden of data analysis may increase, and the numerical stability of the evaluation results may decrease.

[0008] For example, when approximating the amplification effect of an error amplification circuit with a linear function, singularities may appear during the calculation of the linear function, resulting in abnormal values ​​such as infinity, depending on the quantum gate. In this case, it is difficult to evaluate the error of the quantum gate using simple approximation methods. Therefore, in one aspect, the present invention aims to improve the efficiency of evaluating errors in quantum gates. [Means for solving the problem]

[0009] In one aspect, a quantum computation evaluation program is provided that causes a computer to perform the following steps: acquire measurement data showing the measured quantum state of a qubit after the quantum computer has repeatedly executed a sequence of quantum gates containing multiple quantum gates on a qubit; divide the space to which the matrix representing the quantum operation by the sequence of quantum gates belongs into a first subspace containing a singularity that satisfies certain conditions and a second subspace that does not contain a singularity, generate a function within the second subspace that linearly approximates the effect of the error of the first quantum gate contained in the multiple quantum gates on the measured value; and estimate the error when the quantum computer executes the first quantum gate using the function and the measurement data. [Effects of the Invention]

[0010] In one respect, it makes evaluating errors in quantum gates more efficient. [Brief explanation of the drawing]

[0011] [Figure 1] This is a diagram illustrating the information processing device of the first embodiment. [Figure 2] This figure shows an example of the hardware of the information processing system according to the second embodiment. [Figure 3] This figure shows an example of an evaluation quantum circuit that includes an error amplification circuit. [Figure 4] This figure shows an example of pseudocode for identifying subspaces affected by singularities. [Figure 5] This figure shows an example of a matrix representing a projection onto a subspace unaffected by singularities. [Figure 6] This is a block diagram showing examples of functions of an information processing device. [Figure 7] This figure shows an example of configuration data. [Figure 8] This figure shows examples of experimental and evaluation data. [Figure 9] This flowchart shows an example of a procedure for evaluating quantum computing. [Figure 10]This is a flowchart (continued) showing an example procedure for evaluating quantum computing. [Modes for carrying out the invention]

[0012] This embodiment will be described below with reference to the drawings. (a) First embodiment Figure 1 is a diagram illustrating an information processing device of the first embodiment. The information processing device 10 of the first embodiment evaluates the accuracy of quantum operations performed 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 computer or 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 RAM (Random Access Memory). Alternatively, the storage unit 11 may be a non-volatile storage such as an HDD (Hard Disk Drive) or flash memory.

[0014] The processing unit 12 is a processor, such as a CPU (Central Processing Unit), a GPU (Graphics Processing Unit), or a DSP (Digital Signal Processor). However, the processing unit 12 may also include electronic circuits such as an ASIC (Application Specific Integrated Circuit) or an FPGA (Field Programmable Gate Array). The processor executes a program stored in memory, such as RAM. The processor is sometimes called a processor circuit. A collection of processors is sometimes called a multiprocessor or simply a "processor." Different processes among the multiple processes described later may be executed by different processors.

[0015] The memory unit 11 stores the measurement data 14. The measurement data 14 represents the measured quantum state of a qubit, measured after the quantum computer has executed the quantum gate sequence 13 multiple times (N times) on the qubit. The quantum computer that executes the quantum gate sequence 13 is the quantum computer whose implementation accuracy is being evaluated. The information processing device 10 may be connected to the quantum computer, or it may read the measurement data from the quantum computer. The information processing device 10 may also receive the measurement data 14 from another information processing device connected to the quantum computer. Furthermore, the information processing device 10 may sequentially acquire the measurement data from the quantum computer while proceeding with the data analysis described below.

[0016] The quantum gate sequence 13 includes one or more quantum gates (typically, multiple quantum gates). An example of a quantum gate is a 180-degree rotation gate. The quantum gate sequence 13 includes the quantum gate 13a whose accuracy is to be evaluated. The implementation of quantum gate 13a on a quantum computer has an error that indicates a deviation from the ideal operation. The quantum gate sequence 13 may include two or more quantum gates to be evaluated, or it may include one or more quantum gates not to be evaluated. In the data analysis described below, the actual error of the quantum gate not to be evaluated may be known, or it may be assumed that the quantum gate not to be evaluated has no error, or it may be assumed that the quantum gate not to be evaluated has a certain error.

[0017] The quantum gate sequence 13 can be described, for example, by a quantum circuit. A quantum circuit that repeats the same quantum gate sequence multiple times is sometimes called an error amplification circuit. The quantum computer executes the quantum gates in the quantum gate sequence 13 sequentially for each qubit, from the first to the last. The quantum computer then takes over the quantum state represented by the qubit and executes the quantum gates in the quantum gate sequence 13 sequentially again, from the first to the last. The error of quantum gate 13a is amplified through the error amplification circuit. The final measurement of the qubit measured by the quantum computer includes the amplified error component.

[0018] In a quantum computer, even if the method of initializing qubits, the type of quantum gate, and the measurement method are the same, the measured values ​​are obtained probabilistically. Therefore, the measurement data 14 represents a probability distribution that enumerates the probability of occurrence of each of the multiple values ​​that a qubit can take. The information processing device 10 may have the quantum computer perform multiple trials of initializing qubits, executing the quantum gate sequence 13 a certain number of times, and measuring the value of the qubit. The information processing device 10 may calculate the probability of occurrence of each value by dividing the number of occurrences of each value by the number of trials.

[0019] The measurement data 14 may include multiple measurement results corresponding to different initialization methods or different measurement methods. Furthermore, the measurement data 14 may include multiple measurement results corresponding to different quantum gate sequences, and may include multiple measurement results corresponding to different iteration counts N.

[0020] The processing unit 12 analyzes the measurement data 14 to estimate the error that occurs when the quantum computer executes the quantum gate 13a. Here, the effect of the error of the quantum gate 13a on the measurement is strictly nonlinear. In response to this, the processing unit 12 generates a function 16 that linearly approximates the effect of the error of the quantum gate 13a on the measurement to facilitate data analysis. Function 16 is a linear approximation function and can be expressed, for example, using a matrix.

[0021] The processing unit 12 estimates the error of the quantum gate 13a using the measurement data 14 and the function 16. For example, the processing unit 12 extracts an amplification component from the measurement data 14 that is proportional to the number of iterations N. The processing unit 12 optimizes the error of the quantum gate 13a input to the function 16 so that the difference between the extracted amplification component and the output of the function 16 is minimized. For example, the processing unit 12 defines this optimization problem as a semi-definite quadratic programming problem and calculates the error of the quantum gate 13a using a numerical solver.

[0022] In generating function 16, the processing unit 12 represents the error-free, ideal quantum operations for the quantum gates included in the quantum gate sequence 13 using matrices. This matrix may also be called a generator (Lindbladian). A generator corresponds to the logarithm of the quantum gate matrix that represents the transformation of the quantum state. A matrix representing the entire quantum operation of the quantum gate sequence 13 is generated from the matrix representing the quantum gates included in the quantum gate sequence 13. For example, the processing unit 12 synthesizes the quantum gates included in the quantum gate sequence 13 in order from the beginning.

[0023] Furthermore, a matrix representing the quantum operation of the quantum gate sequence 13 is generated from the matrix representing the overall quantum operation of the quantum gate sequence 13. The processing unit 12 generates a function 16, which is a linear approximation function, by extracting the first-order component of the action that changes the error of the quantum gate 13a through, for example, synthesis or amplification. In extracting the first-order component, the processing unit 12 may perform spectral decomposition, which decomposes the matrix of each quantum gate into a sum of products of eigenvalues ​​and projection matrices. This spectral decomposition may also be called eigenvalue decomposition.

[0024] Here, the space 15 to which the matrix representing the quantum operation by the quantum gate sequence 13 belongs may contain a singularity 15c. A singularity 15c is a point in the matrix from which an eigenvalue satisfying certain conditions is calculated. A singularity 15c may also be called a critical point. For example, a singularity 15c corresponds to a generator with different eigenvalues ​​such that the values ​​of the exponential function with base e are the same. If the eigenvalues ​​are complex numbers, the values ​​of the exponential function calculated from different eigenvalues ​​may be the same. The processing unit 12 may determine that two eigenvalues ​​whose difference in exponential function values ​​is less than a threshold satisfy the above-mentioned conditions.

[0025] Depending on the type of quantum gate included in the quantum gate sequence 13, a matrix corresponding to singularity 15c may be calculated during the linear approximation process described above. An example of a quantum gate for which a matrix corresponding to singularity 15c can be calculated is the 180-degree rotation gate. At singularity 15c, an abnormal value, such as division by zero, may be calculated. This means that the linear approximation of the action of quantum gate sequence 13 does not hold at singularity 15c.

[0026] Therefore, the processing unit 12 divides space 15 into subspace 15a and subspace 15b. Subspace 15a is a subspace that contains the singularity 15c. Subspace 15b is a subspace that does not contain the singularity 15c. Subspace 15b is, for example, the orthogonal complement of subspace 15a and has no common region with subspace 15a.

[0027] For example, the processing unit 12 synthesizes the generators of the quantum gates included in the quantum gate sequence 13 in order from the beginning, and decomposes the synthesized generators at each stage into a sum of products of eigenvalues ​​and projection matrices. The processing unit 12 identifies the subspace 15a using projection matrices corresponding to eigenvalues ​​that satisfy certain conditions. For example, the processing unit 12 calculates a projection matrix that projects points in space 15 onto subspace 15a. The processing unit 12 calculates a matrix orthogonal to the projection matrix onto subspace 15a as the projection matrix that projects points in space 15 onto subspace 15b.

[0028] The processing unit 12 generates a function 16 within the subspace 15b that linearly approximates the effect of the error of the quantum gate 13a on the measured value. Function 16 suppresses the calculation of matrices belonging to the subspace 15a. For example, function 16 includes a projection matrix onto the subspace 15b, and uses this projection matrix to perform matrix calculations restricted to matrices belonging to the subspace 15b. The processing unit 12 also uses the projection matrix onto the subspace 15b to extract components corresponding to the subspace 15b from the measurement data 14. The processing unit 12 compares the output of function 16 with the components extracted from the measurement data 14 to estimate the error of the quantum gate 13a.

[0029] As described above, the information processing device 10 of the second embodiment acquires measurement data 14 that shows the measured quantum state of a qubit measured after the quantum computer has repeatedly executed a quantum gate sequence 13 containing multiple quantum gates on the qubit. The information processing device 10 divides the space to which the matrix representing the quantum operation by the quantum gate sequence 13 belongs into a subspace 15a that includes a singularity 15c for which an eigenvalue satisfying certain conditions is calculated, and a subspace 15b that does not include the singularity 15c.

[0030] The information processing device 10 generates a function 16 within the subspace 15b that linearly approximates the effect of the error of a quantum gate 13a included in a plurality of quantum gates on the measured value. Using the function 16 and the measurement data 14, the information processing device 10 estimates the error when the quantum computer executes the quantum gate 13a.

[0031] This allows the error of the quantum gate 13a implemented in the quantum computer to be estimated. Therefore, the user can perform calibration by adjusting the values ​​of the control parameters of the quantum computer based on the estimated error. Furthermore, because an error amplification circuit that repeats the quantum gate sequence 13 multiple times is used, the error is amplified, improving the accuracy of the error estimation.

[0032] Furthermore, function 16 provides a linear approximation of the action of the quantum gate sequence 13. This reduces the load on quantum tomography and improves the numerical stability of the estimated error. Additionally, function 16 performs a linear approximation within the subspace 15b, which does not include the singularity 15c associated with the quantum gate sequence 13. Therefore, even if the quantum gate sequence 13 includes a quantum gate for which a matrix corresponding to the singularity 15c is calculated during the linear approximation process, the approximation calculation becomes possible. Consequently, the evaluation of the error of quantum gate 13a becomes more efficient.

[0033] Furthermore, the component of the error in quantum gate 13a that is affected by singularity 15c is mainly the part that describes the dissipation of information in the qubit. The component that describes the noise during qubit control is less affected by singularity 15c. Therefore, even if the information processing device 10 estimates the error within the range of subspace 15b, it can generate useful estimation results. For example, the information processing device 10 can generate estimation results that are useful for calibrating a quantum computer.

[0034] (b) Second embodiment The information processing system of 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, adjusting the values ​​of the control parameters of the quantum computer 115 to improve accuracy. The information processing device 100 corresponds to the information processing device 10 of the first embodiment.

[0035] Figure 2 shows an example of the hardware of an information processing system according to the second embodiment. The information processing device 100 includes a CPU 101, RAM 102, HDD 103, GPU 104, input interface 105, media reader 106, communication interface 107, and interface 108. These units are connected to a bus. The CPU 101 corresponds to the processing unit 12 of the first embodiment. The RAM 102 or HDD 103 corresponds to the storage unit 11 of the first embodiment.

[0036] The CPU 101 is a processor that executes program instructions. The CPU 101 loads the program and data stored in the HDD 103 into the RAM 102 and executes the program. The information processing device 100 may have multiple processors.

[0037] RAM 102 is a volatile semiconductor memory that temporarily stores programs executed by CPU 101 and data used for calculations by CPU 101. The information processing device 100 may have a type of volatile memory other than RAM.

[0038] The HDD 103 is a non-volatile storage device that stores software programs such as the operating system (OS), middleware, and application software, as well as data. The information processing device 100 may have other types of non-volatile storage, such as flash memory or an SSD (Solid State Drive).

[0039] The GPU 104 works in conjunction with the CPU 101 to perform image processing and outputs the image to the display device 111 connected to the information processing device 100. The display device 111 is, for example, a CRT (Cathode Ray Tube) display, a liquid crystal display, an organic EL (Electro Luminescence) display, or a projector. Other types of output devices, such as a printer, may also be connected to the information processing device 100.

[0040] Furthermore, the GPU 104 may be used as a GPGPU (General Purpose Computing on Graphics Processing Unit). The GPU 104 can execute programs in response to instructions from the CPU 101. The information processing device 100 may have volatile semiconductor memory other than RAM 102 as GPU memory.

[0041] The input interface 105 receives input signals from an input device 112 connected to the information processing device 100. The input device 112 is, for example, a mouse, a touch panel, or a keyboard. Multiple input devices may be connected to the information processing device 100.

[0042] The media reader 106 is a reading device that reads programs and data recorded on the recording medium 113. The recording medium 113 is, for example, a magnetic disk, an optical disk, or semiconductor memory. Magnetic disks include flexible disks (FD) and HDDs. Optical disks include CDs (Compact Discs) and DVDs (Digital Versatile Discs). The media reader 106 copies the programs and data read from the recording medium 113 to other recording media such as RAM 102 or HDD 103. The read programs may be executed by the CPU 101.

[0043] The recording medium 113 may be a portable recording medium. The recording medium 113 may be used for distributing programs and data. The recording medium 113 and HDD 103 may also be referred to as computer-readable recording media.

[0044] The communication interface 107 communicates with other information processing devices via the network 114. The communication interface 107 may be a wired communication interface connected to a wired communication device such as a switch or router, or a wireless communication interface connected to a wireless communication device such as a base station or access point.

[0045] Interface 108 is connected to the quantum computer 115. Interface 108 sends commands to the quantum computer 115 in response to instructions from the CPU 101. Interface 108 also receives data from the quantum computer 115 and writes the received data to RAM 102 or HDD 103.

[0046] The quantum computer 115 has a qubit unit 116, a control unit 117, and a measurement unit 118. The qubit unit 116 includes a plurality of qubits that represent quantum states. The plurality of qubits are arranged, for example, in a lattice. Quantum operations on the qubits are implemented, for example, by irradiating the qubits with microwave pulses.

[0047] The control unit 117 receives a command from the information processing device 100 for calibration of the quantum operation. The command includes, for example, the name and value of a control parameter. Examples of control parameters include the time width, amplitude, and waveform of the microwave pulse. Changing the value of the control parameter changes the accuracy of the quantum operation. In response to the instruction from the information processing device 100, the control unit 117 performs a quantum operation equivalent to a quantum gate on the qubit unit 116. For example, the control unit 117 transmits a microwave pulse with a waveform corresponding to the value of the control parameter to the qubit unit 116.

[0048] The measurement unit 118 measures the value of a qubit in response to instructions from the information processing device 100 and stores the measured value in the memory of the quantum computer 115. The measurement unit 118 transmits the measured value to the information processing device 100 in response to a request from the information processing device 100.

[0049] Generally, quantum information processing includes quantum computing, quantum simulation, quantum communication, quantum cryptography, and quantum sensing. Examples of physical platforms for quantum information processing include superconducting quantum circuits, semiconductor quantum dots, diamond NV centers, NMR molecules, neutral atoms, trapped ions, and light. Typical quantum information processing protocols based on quantum circuits utilize three types of quantum operations: initialization, quantum gates, and measurement.

[0050] The quantum operations performed by the quantum computer 115 have errors that indicate a deviation 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 to reduce the errors based on the error information. The information processing device 100 may repeat the evaluation and calibration process.

[0051] The information processing device 100 evaluates quantum operations using quantum tomography. The information processing device 100 collects qubit measurements from the quantum computer 115 while varying the combinations of initialization, quantum gates, and measurements. The information processing device 100 analyzes the experimental data, which associates the tried combinations with the measurements, to estimate the errors in the quantum operations. Quantum tomography can estimate the errors of multiple quantum gates simultaneously.

[0052] The information processing device 100 generates a quantum circuit that includes an error amplification circuit that repeats the same quantum gate sequence multiple times, and has the quantum computer 115 execute this quantum circuit. The number of iterations can be, for example, 10, 100, or 1000 times. The errors of the quantum gates included in the quantum gate sequence are amplified through the error amplification circuit. This increases the error component included in the measured value, improving the evaluation accuracy. Examples of quantum tomography techniques that use error amplification circuits include GST (Gate-Set Tomography), IT (Idle Tomography), and HEAT (Hamiltonian Error Amplifying Tomography).

[0053] GST is mentioned, 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.

[0054] 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.

[0055] Error amplification circuits are also described in the following two non-patent documents: 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.

[0056] 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.

[0057] Figure 3 shows an example of an evaluation quantum circuit including an error amplification circuit. This quantum circuit includes an initialization circuit 141, a quantum gate sequence 142, and a measurement circuit 143. The initialization circuit 141 initializes one or more qubits to generate a desired quantum state. The quantum gate sequence 142 includes one or more quantum gates. The error amplification circuit repeats the quantum gate sequence 142 in series N times. The measurement circuit 143 measures the value of the qubits.

[0058] A non-repeating quantum gate may be included between the initialization circuit 141 and the quantum gate sequence 142. Similarly, a non-repeating quantum gate may be included between the quantum gate sequence 142 and the measurement circuit 143. In Figure 3, the X gate is a rotational gate that rotates around the x-axis, the Y gate is a rotational gate that rotates around the y-axis, and the Z gate is a rotational gate that rotates around the z-axis. The CR gate is a cross-resonance gate. The R gate is a rotational gate that rotates by a fixed amount.

[0059] Quantum tomography collects experimental data while varying the combination of the quantum gate sequence 142, the number of iterations N, the initialization circuit 141, and the measurement circuit 143. The quantum gate sequence 142 includes one or more quantum gates to be evaluated for error assessment. The quantum gate sequence 142 may also include one or more non-evaluated quantum gates for which error assessment is not performed.

[0060] The ideal behavior of the quantum gate under evaluation and the quantum gates not under evaluation is known. The error of the quantum gate under evaluation in quantum computer 115 is unknown. The error of the quantum gates not under evaluation in quantum computer 115 may be known or unknown. In the latter case, quantum tomography may consider the error of the quantum gates not under evaluation to be zero or assume a constant value.

[0061] Here, strictly defining the action of the error amplification circuit on quantum gate errors would require solving a highly nonlinear numerical optimization problem. This would increase the burden of experimental data analysis and potentially reduce the stability of the solution search. Therefore, the information processing device 100 linearly approximates the action of the error amplification circuit and defines the search for quantum gate errors as a semi-definite quadratic programming problem. A semi-definite quadratic programming problem is a numerical optimization problem that optimizes a quadratic function under linear constraints, where the eigenvalues ​​of the matrix are non-negative. The method of the second embodiment may also be called RLT (Robust Lindbladian Tomography).

[0062] The calculation of quantum tomography in the second embodiment will be explained below using mathematical formulas. The action of a quantum gate acting on a d-dimensional quantum system is described by the linear map shown in equation (1). This linear map is a map from a d×d-dimensional complex space to a d×d-dimensional complex space, and is a trace-preserving and completely positive map. Here, G is the representation matrix of the linear map of equation (1) under the orthonormal basis shown in equation (2). This orthonormal basis is a d-dimensional matrix with a norm of 1 and mutually orthogonal to each other. 2 Individual element B α This is the set of elements G in the α row and β column of matrix G. αβ As shown in equation (3), element B of the orthonormal basis corresponding to the α row. α And the element B of the orthonormal basis corresponding to the β column β This is calculated using the trace.

[0063]

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[0064]

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[0065]

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[0066] A linear map exists between the quantum gate defined by equation (1) and the exponential map exp that satisfies the relationship in equation (4). This linear map is sometimes called a generator (Lindbladian). If we denote the representation matrix of the linear map in equation (4) as L, then the relationship in 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. ideal If we denote the error representation matrix of the generator as δL, then G=e L This is expanded as shown in equation (5).

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[0069] When an error is amplified through an error amplification circuit, it is easier to analyze the error of the generator corresponding to the matrix logarithm than the error of the quantum gate itself. This is because when errors accumulate, rotations occur in the qubits, and the angular deviation may not be exactly proportional to the number of repetitions N. Therefore, in the second embodiment, the information processing apparatus 100 estimates the generator error from the 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. For simplicity of explanation, hereinafter, a linear mapping and its representation matrix may not be distinguished. For example, a quantum gate may be denoted as G, a generator as L, and a generator error as δL.

[0070] When preparing an error amplification circuit, a set I of quantum gates shown in Equation (6) g is given by the user. The quantum gates included in the set I g are identified by unique numbers. Among n g different quantum gates, the first n g,1 quantum gates are the quantum gates to be evaluated, and the remaining quantum gates are the quantum gates not to be evaluated.

[0071]

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[0072] The quantum gate sequence forming the error amplification circuit is a quantum circuit in which one or more (typically two or more) of the quantum gates included in the set I g are arranged in series. The quantum gate sequence a is defined as in Equation (7). In Equation (7), the length of the quantum gate sequence is m. The elements on the right side of Equation (7) are numbers identifying any of the quantum gates included in the set I g . The same type of quantum gate may appear two or more times in the quantum gate sequence a.

[0073]

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[0074] If we denote the combined quantum gate corresponding to the overall action of the quantum gate sequence a as G(a), then G(a) can be expanded as shown in equation (8). Here, the product of three or more matrices is calculated from right to left. If we denote the number of iterations of the error amplifier circuit as N, then the combined quantum gate corresponding to the overall action of the error amplifier circuit is G(a). N It is expressed as follows.

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[0076] In the second embodiment, the information processing device 100 is represented by the n shown in formula (9). g,1 The generator error δL of n is estimated. To collect measurements taken under various conditions, the user is given a set of quantum gate sequences a and a set of iteration counts N, as shown in equation (10). For the quantum gate sequences, n a Given a sequence of 1 different quantum gates, the number of iterations is n for each individual quantum gate sequence. N A different number of iterations is given.

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[0079] The different iteration counts N and N' given to the same quantum gate sequence satisfy the condition of equation (11). The effect of repeating an ideal quantum gate sequence without errors N times is the same as the effect of repeating the same quantum gate sequence N' times. This makes it easy to extract amplified error components from measurements with different iteration counts.

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[0081] Furthermore, for each quantum gate sequence, the user provides a set of initialization ρ and measurement Π pairs shown in equation (12). This set consists of n initialization ρ and measurement Π pairs. t,i It contains n elements. Therefore, the quantum computer 115 generates an initial quantum state by initialization ρ, repeatedly executes the quantum gate sequence a for N iterations, and obtains a measurement by measurement Π. The quantum computer 115 performs this experiment for all combinations of a, N, (ρ, Π).

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[0083] If we denote the vectorization of matrix X with respect to basis B as |X>>, the probability of obtaining a measurement x in each experiment is calculated as shown in equation (13). Matrix X is an element of a d×d complex space, and the vector |X>> is d 2 These are elements of a dimensional complex number space. Below, i may be used as the identifier of the quantum gate sequence a, j as the identifier of the number of iterations N, and k as the identifier of the initialization and measurement pair (ρ, Π). Due to the nature of quantum computation, the measured value is obtained probabilistically, so quantum computer 115 uses n for each of (i, j, k). i,j,k Obtain a sample of individual measurements.

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[0085] The frequency at which the measurement value x was obtained is f. i,j,k,x Expressed as above, the frequency distribution of formula (14) can be calculated for each of (i, j, k). Frequency f i,j,k,x n is the number of samples from which a measurement value x was obtained. i,j,k It is calculated by dividing by [a certain number].

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[0087] The information processing device 100 performs a data processing phase that analyzes the experimental data showing the frequency distribution, following the experimental phase in which it collects the frequency distribution as described above. The data processing phase includes step #1, which extracts the amplified component from the experimental data; step #2, which linearly approximates the action of the error amplifier circuit; and step #3, which solves a numerical optimization problem defined using the results of steps #1 and #2. However, steps #1 and #2 may be performed in reverse order. Furthermore, the information processing device 100 may omit step #2 by reusing the action of the error amplifier circuit calculated in previous quantum tomography.

[0088] In step #1, the information processing device 100 extracts a first-order component from the experimental data that is proportional to the number of iterations N. The experimental data includes a constant component independent of the number of iterations N, a first-order component proportional to the number of iterations N, and higher-order components of second order 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 amplification circuit corresponds to the first-order component. Below, we will explain how to extract the first-order component by extrapolation.

[0089] The information processing device 100 selects and fixes one combination of quantum gate sequence a, initialization and measurement pair (ρ, Π), and measurement value x. The information processing device 100 then uses the probability p as shown in equation (15). x Expand (i,j,k) into a series with respect to the number of iterations N. Probability p x (i,j,k) is the probability of obtaining the measured value x under the experimental conditions (i,j,k). The coefficient of the ν-th expansion is h ν Therefore, due to resolution accuracy, the information processing device 100 performs a series expansion up to the infinite order shown in formula (15) as n N We approximate it with a finite series sum of order -1.

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[0091] Formula (16) is given by probability p x(i,j,k) to n N The vector summarizing the number of iterations is shown. T represents the transpose. Using the Vandermonde matrix V with respect to the number of iterations N in equation (17), the finite series sum can be expressed as in equation (18). The Vandermonde matrix V ranges from 0 to n. N A row vector in which powers of N up to the -1 power, where n N n corresponds to the number of iterations. N This is a matrix containing row vectors. The coefficient vector h in equation (18) is the expansion coefficient h v From the 0th to the nth N This is a vector arranged up to the -1st order.

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[0095] n N If the number of iterations is different from each other, the van der Mond matrix V is a reversible matrix. The coefficient vector h is calculated using the inverse of the van der Mond matrix V as shown in equation (19). In experiments using quantum computer 115, probability p x Frequency f corresponds to the estimated value of (i,j,k). i,j,k,x The probability p is measured. Therefore, the information processing device 100 calculates the probability p x Replace (i,j,k) with frequency f i,j,k,x Use the formula (20), which is the frequency f. i,j,k,x to n N The vector summarizing the number of iterations is shown. Estimated value of coefficient vector h est This is calculated as shown in formula (21). In this way, the information processing device 100 uses the measured frequency f i,j,k,xFrom the 0th to the nth N Calculate the expansion coefficients up to the -1st order.

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[0099] However, the information processing device 100 may calculate the expansion coefficients by a method other than the method using the van der Mond matrix V described above. For example, the information processing device 100 may calculate the expansion coefficients by polynomial approximation using the least squares method.

[0100] The second element of the coefficient vector h is a linear component proportional to the number of iterations N. The information processing device 100 extracts the linear component while changing the measured value x. The information processing device 100 combines linear components with the same (i,k) but different measured values ​​x to obtain the coefficient vector h est The coefficient vector h is calculated. The information processing device 100 calculates 1(i,k). est Calculate 1(i,k) for all (i,k).

[0101] In step #2, the information processing device 100, n g The n errors of the individual generators that are being evaluated g,1 The errors of each generator are collectively parameterized into a parameter v. The parameter v is a vector. g,1 The conversion from the individual generator errors to the parameter v can be any predetermined affine conversion, and may be defined using matrices. As will be described later, from the optimal value of parameter v, n g,1To estimate the individual generator errors, this parameterization is defined in such a way that it is inversely transformable.

[0102] The information processing device 100 linearly approximates the action of the error amplification circuit for each (i,k) and generates a matrix C showing components proportional to the number of iterations N. i,k Generates matrix C. i,k The product of this and parameter v indicates the effect of the error of the generator being evaluated on the first-order component of the experimental data.

[0103] Furthermore, the information processing device 100 is n g The errors of the generators that are not being evaluated are vectorized in the same way as the parameter v. The same transformation as that used for the errors of the generators being evaluated is used for this vectorization. The information processing device 100 then applies matrix C to this vector. i,k Applying this, vector b i,k Generates vector b. i,k This shows the effect of the error of the unevaluated generator on the first component of the experimental data. Since the error of the unevaluated generator is provided by the user, vector b i,k The elements are calculated as specific numerical values. The linear approximation of the action of the error amplification circuit will be discussed later.

[0104] In step #3, the information processing device 100 processes the coefficient vector h generated in step #1. est 1(i,k) and the matrix C generated in step #2 i,k and vector b i,k Using this, we generate the objective function F(v) shown in equation (22). i,k This is a weight matrix set by the user for (i,k). The information processing device 100 minimizes the value of the objective function F(v) using the parameter v as shown in equation (23). est The information processing device 100 calculates the parameter v est This returns the error to the generator being evaluated.

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[0107] The numerical optimization problem to solve equation (23) is a least-squares fitting with physical constraints. Parameter v est The search is conducted to satisfy the physical constraints imposed by the generator error. Since the parameter v is an affine parameterization of the generator error, the physical constraints imposed by parameter v are expressed by linear positive semi-definite constraints. Furthermore, the objective function F(v) is a quadratic function of parameter v. For this reason, this numerical optimization problem is defined as a positive semi-definite quadratic programming problem and is solved using a positive semi-definite programming solver.

[0108] Next, we will explain the linear approximation in step #2. Consider n × n complex square matrices A, B, and P, as shown in equation (24). Matrix A is diagonalizable and corresponds to the ideal values ​​of its generators. Matrix A can be decomposed by eigenvalue decomposition or spectral decomposition as shown in equation (25). In equation (25), V is a matrix containing the eigenvectors as column vectors, and Λ is a diagonal matrix with the eigenvalues ​​arranged on the diagonal. a is a complex number that is an eigenvalue, and P is the projection matrix corresponding to the eigenvalue a. Matrix A can be decomposed into a sum of products of the eigenvalue a and the projection matrix P.

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[0111] The projection matrix P satisfies equation (26). δ jk This is the Kronecker delta. In the case of j=k, it is δ jk = 1, and if j and k are different, then δ jk= 0. Therefore, the product of different projection matrices corresponding to different eigenvalues ​​is the zero matrix. On the other hand, the square of the same projection matrix is ​​the projection matrix itself.

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[0113] Using the eigenvalue a of matrix A and the projection matrix P, six linear transformations are defined for matrix B, as shown in equations (27) to (32). dcl A This shows the decomposition to the left with respect to matrix A. A This shows the decomposition to the right with respect to matrix A. A This represents composition to the left with respect to matrix A.

[0114] cmr A This indicates composition to the right with respect to matrix A. A This represents the sum of spectral projections with respect to matrix A. A This represents the sum of spectral projection complements with respect to matrix A.

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[0121] The coefficients used in equations (27) to (30) are defined as shown in equation (33). Projection matrix P j ,P k If they are the same, l jk = 1. Projection matrix P j ,P k If they are different, jk P is a projection matrix. j ,P k The corresponding eigenvalue a j ,a k These six linear transformations are calculated from the following. These six linear transformations can be represented as matrices using an orthonormal basis on an n × n complex space. The representation matrices for these six linear transformations are n 2 ×n 2 It is a matrix and is expressed as shown in formula (34).

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[0124] Using dcl in equation (27), the matrix exponents of the sum of matrices A and B are calculated as shown in equation (35). The second term on the right-hand side is a higher-order component of matrix B (order 2 or higher). Therefore, the matrix exponents of the sum of matrices A and B are linearly approximated as shown in the first term on the right-hand side. Also, using dcr in equation (28), the matrix exponents of the sum of matrices A and B are calculated as shown in equation (36). Therefore, the matrix exponents of the sum of matrices A and B are also linearly approximated as shown in the first term on the right-hand side.

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[0127] Furthermore, using cml in equation (29), the product of the matrix exponents of matrix A and matrix exponents of matrix B is calculated as shown in equation (37). Thus, the product of the matrix exponents of matrix A and matrix exponents of matrix B is linearly approximated as shown in the first term on the right-hand side. Also, using cmr in equation (30), the product of the matrix exponents of matrix B and matrix exponents of matrix A is calculated as shown in equation (38). Thus, the product of the matrix exponents of matrix B and matrix exponents of matrix A is linearly approximated as shown in the first term on the right-hand side.

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[0130] Using the linear transformations in equations (27) to (30), we get the quantum gate G i ,G i’ The action of the composite quantum gate formed by combining L can be approximated by equation (39). ideal This is a synthetic generator that corresponds to the matrix logarithm of the ideal value of a synthetic quantum gate, and is defined as shown in equation (40).

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[0133] Therefore, for the quantum gate G i the generator error δL i changes as shown in Equation (41) through the composite quantum gate. In Equation (41), the linear maps dcl L , cml L are used to linearly approximate the change in δL i . Also, for the quantum gate G i’ the generator error δL i’ changes as shown in Equation (42) through the composite quantum gate. In Equation (42), the linear maps dcr L , cmr L are used to linearly approximate the change in δL i’ . Equation (43) is the representation matrix of the change in δL i shown in Equation (41). Equation (44) is the representation matrix of the change in δL i’ shown in Equation (42).

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[0138] When the quantum gate sequence includes three or more quantum gates, the information processing apparatus 100 synthesizes the quantum gates one by one from the head to the tail. The information processing apparatus 100 first synthesizes the first quantum gate and the second quantum gate. Next, the information processing apparatus 100 synthesizes the synthesized quantum gate and the third quantum gate. By repeating the synthesis, the information processing apparatus 100 generates a synthesized quantum gate corresponding to the entire quantum gate sequence. As a result, the change in the generator error through the quantum gate sequence is linearly approximated.

[0139] Next, the information processing apparatus 100 calculates the repeated action of the quantum gate sequence. Repeating a certain quantum gate G N times is defined as in Equation (45). The synthesized quantum gate obtained by synthesizing the quantum gate sequence generally takes the form of e A+B Assuming that there exists a c such that mod(nA)=cA, the N - fold repetition of this synthesized quantum gate is defined as in Equation (46) using ssp and sspc of Equations (31) - (32). c is a constant independent of the number of repetitions N. The first term on the right - hand side is the linear component with respect to the matrix B. The second term on the right - hand side is the higher - order component of second order or higher with respect to the matrix B.

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[0142] Therefore, the generator error δL i for the quantum gate G changes as in Equation (47) through N repetitions. In Equation (47), using the linear mappings ssp and sspc, the change in δL i is linearly approximated. The first term on the right - hand side indicates the component that is not amplified by N repetitions. The second term on the right - hand side indicates the linear component proportional to the number of repetitions N. Equation (48) shows δL iThe representation matrix for the action that does not amplify is shown. Equation (49) shows that δL over N iterations. i The representation matrix for the amplification effect is shown.

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[0146] Finally, for an error amplification circuit that repeats a sequence of quantum gates N times, the information processing device 100 calculates the effect on generator errors by integrating the effect of quantum gate composition and the effect of repeating the sequence of quantum gates. The information processing device 100 generates the representation matrix for the composition effect and the representation matrix for the repeating effect as described above, and multiplies the representation matrix for the composition effect by the representation matrix for the repeating effect from the left.

[0147] In this way, the information processing device 100 linearly approximates the amplification effect of the error amplification circuit, which amplifies the generator error. This amplification effect is the effect in the generator space. For numerical optimization using the objective function F(v), the information processing device 100 converts the effect in the generator space into the effect in the space of the probability distribution of the measured values.

[0148] The information processing device 100 generates a transformation matrix U by rearranging the elements of the representation matrices for initialization ρ and measurement Π used in the experiment, and multiplies the representation matrix of the error amplification circuit by the transformation matrix U from the left. The representation matrices for initialization ρ and measurement Π used here include known or assumed errors related to ρ and Π, and are sometimes called model values. This results in the matrix C included in the objective function F(v). i,k The following is calculated. In addition, matrix C is used for errors of generators that are not evaluated.i,k By applying, vector b i,k is calculated.

[0149] Here, matrix C i,k and vector b i,k are supplemented with explanations. The probability p x (i,j,k) shown in formula (13) is linearly approximated as in formula (50). The amplification component proportional to the number of iterations N is rewritten as in formulas (51) and (52). The F in the notation in formulas (50)-(52) represents a mapping that describes the first-order approximation of the combined action of quantum gate synthesis and repetition. The F in the last line of formula (52) represents the representation matrix of the mapping.

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[0159] Next, we will describe the improvement of the linear approximation in Step #2. For the linear approximation of the action of the quantum gate sequence, 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 given by the coefficient l shown in equation (33). jk It includes the reciprocal of .

[0160] Here, the ideal values ​​of the generators are different eigenvalues ​​a j ,a k It has, and e's a j A of the exponents and e k There exists a quantum gate such that the exponentiation and the value are the same. Eigenvalue a j ,a k If is a complex number, then as in formula (57), e's a j A of the exponents and e k The multiplication factor and the exponentiation factor can sometimes be the same value. An example of such a quantum gate is the 180-degree rotation gate (π-pulse gate).

[0161] In this case, ljk = 0, its reciprocal is infinity, and the values ​​of the linear maps cml and cmr become indeterminate. This means that at a point representing the above generator contained in the generator space, the differential map of the exponential map has a zero eigenvalue. Such points in the generator space are sometimes called singularities or critical points. At singularities, it can also be said that the linear approximation of the action of the quantum gate sequence breaks down. If the quantum gate sequence contains a quantum gate corresponding to a singularity, or if a composite quantum gate corresponding to a singularity is generated during the synthesis process, the action of the error amplifier circuit cannot be calculated using the aforementioned linear approximation method.

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[0163] Here, the action of the linear transformation cml at a singularity can be formally decomposed as shown in equation (58). The first term on the right-hand side is l jk These are components calculated from combinations of eigenvalues ​​that do not equal 0. The second term on the right-hand side is l jk These are components calculated from combinations of eigenvalues ​​that equal 0. Similarly, the action of the linear transformation cmr is l kj Components that do not equal 0 and l kj The error is decomposed into components that equal 0. The information processing device 100 then removes the part affected by divergence at the singularity, such as the second term on the right-hand side, and generates a linear approximation function of the part not affected by divergence, such as the first term on the right-hand side. The information processing device 100 uses this linear approximation function to estimate the component of the generator error that is not affected by divergence.

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[0165] According to the inventor's logical analysis in this application, the component of the generator error affected by divergence at the singularity is part of the portion describing the dissipation of information in the qubit. The component resulting from the control of quantum operations is not affected by divergence. Therefore, estimates of the component not affected by divergence are important information for the calibration of quantum computers.

[0166] Equation (59) shows the set of ideal values ​​for the generators corresponding to the quantum gate. Corresponding to the ideal value of each generator, the Hamiltonian H i This is defined. Quantum gates G, generators L, and Hamiltonians H are mutually convertible by specific transformation formulas.

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[0168] The information processing device 100 performs the following process for each quantum gate sequence a of length m. First, the information processing device 100 calculates the ideal value of the Hamiltonian for each synthesized quantum gate generated during the process of synthesizing the quantum gates included in the quantum gate sequence a. (1) Let H be the ideal value of the Hamiltonian for the first quantum gate (i1). H is the ideal value of the Hamiltonian corresponding to the composite quantum gate obtained by combining the first quantum gate and the second quantum gate (i2). (1,2) It is calculated as shown in formula (60).

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[0170] Next, the information processing device 100 calculates the Hamiltonian H (1,2) The Hamiltonian of the third quantum gate (i3) is combined with the Hamiltonian H (1,2,3) The information processing device 100 generates the Hamiltonian. The information processing device 100 repeatedly synthesizes Hamiltonians up to the mth quantum gate. This generates the Hamiltonian sequence shown in equation (61).

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[0172] Next, the information processing device 100 converts the ideal values ​​of the Hamiltonian in the synthesis process into ideal values ​​of the generators. The action of the generator matrix X corresponding to the Hamiltonian H is defined as shown in equation (62). For a pair of square matrices (X,Y), [X,Y] is defined as a mapping sometimes called a commutator, and [X,Y] = XY - YX. Therefore, the components of the generator representation matrix L under the basis B shown in equation (2) are calculated as shown in equation (63).

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[0175] Hamiltonian H at the stage where the tth quantum gate has been synthesized. (1,…,t) The corresponding generator is L (1,…,t) When written this way, the Hamiltonian sequence in equation (61) is transformed into the generator sequence in equation (64). Note that the generators are calculated via the Hamiltonian to ensure numerical stability in the case of singularities.

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[0177] Next, the information processing device 100 determines the singularity that affects the quantum gate sequence from the set of generators in equation (59) and the sequence of generators in equation (64). The information processing device 100 determines the generator L included in the set of equation (59). i Each is spectrally decomposed, and the coefficient l of equation (33) is calculated for each set of eigenvalues. jkThe information processing device 100 calculates the generator L as shown in formula (65). i Regarding l jk The set S of (j,k) such that = 0 i Identify S i If the set is not empty, then the generator L i This would mean that it is at a singularity. Equation (66) is included in the set of equation (59) n g S corresponding to individual generators i This shows the set.

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[0180] Furthermore, the information processing device 100 includes the generator L in the generator sequence of formula (64). (1,…,t) Each is spectrally decomposed, and the coefficient l of equation (33) is calculated for each set of eigenvalues. jk The information processing device 100 calculates the generator L as shown in formula (67). (1,…,t) Regarding l jk The set T of (j,k) such that = 0 (1,…,t) Identify the T that corresponds to the m generators included in the generator sequence of equation (64). (1,…,t) This shows the set.

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[0183] Next, the information processing device 100 identifies a subspace from the generator space that is affected by the singularity discovered above. Figure 4 shows an example of pseudocode for identifying a subspace affected by a singularity. Pseudocode 144 shows an algorithm for identifying a subspace affected by a singularity. First, the information processing device 100 initializes the sets Y and Z to empty sets. The elements of sets Y and Z are matrices. The information processing device 100 repeats the following process from t=1 to t=m-1.

[0184] The information processing device 100 determines the t-th set T from equation (68). (1,…,t) Read out the set T (1,…,t) Determine if set T is an empty set. (1,…,t) If the set is empty, the information processing device 100 adds the identity matrix I to the set Y. (1,…,t) The fact that is an empty set is determined by the generator L (1,…,t) This indicates that it is not a singularity.

[0185] On the other hand, set T (1,…,t) If the set T is not empty, the information processing device 100 will determine the set T (1,…,t) The data processor 100 reads out pairs (j,k) one by one and performs the following processing for each (j,k). The data processor 100 generates L (1,…,t) The spectral decomposition of the generator L (1,…,t) The set of projection matrices {P i Calculate the projection matrix P. i This is sometimes called spectral projection.

[0186] The information processing device 100 processes the set T (1,…,t) The projection matrix P corresponding to the pair (j,k) in the matrix. j ,P k The information processing device 100 identifies the projection matrix P. i For each of them, P j Multiply from left to right and P k Multiply from the right, and the resulting product is the projection matrix P i The matrix P is calculated by adding the values. The information processing device 100 calculates the matrix Q in the form Q = IP.

[0187] Furthermore, the information processing device 100 generates L (1,…,t)The representation matrix of the linear transformation dcr using is calculated as matrix X. The information processing device 100 calculates matrices QX and PX and adds them to the set Y. QX represents the components that are not affected by singularities and corresponds to the first term on the right-hand side of equation (58). PX represents the components that are affected by singularities and corresponds to the second term on the right-hand side.

[0188] set T (1,…,t) Once the processing of the elements is complete, the information processing device 100 generates the generator L of the (t+1)th quantum gate of the quantum gate sequence a. i(t+1) The corresponding set S i(t+1) Read this from formula (66), and set S i(t+1) Determine if the set S is empty. i(t+1) If the set S is empty, the information processing device 100 adds the identity matrix I to the set Z. i(t+1) The fact that is an empty set is determined by the generator L i(t+1) This indicates that it is not a singularity.

[0189] On the other hand, set S i(t+1) If the set S is not empty, the information processing device 100 will determine the set S i(t+1) The data processor 100 reads out pairs (j,k) one by one and performs the following processing for each (j,k). The data processor 100 generates L (1,…,t) The spectral decomposition of the generator L (1,…,t) The set of projection matrices {P i Calculate the set T. (1,…,t) When processing {P i If {P} has already been calculated, the information processing device 100 will determine its {P} i You may reuse}.

[0190] The information processing device 100 is set S i(t+1) The projection matrix P corresponding to the pair (j,k) in the matrix. j ,P k The information processing device 100 identifies the projection matrix P. i For each of them, P j Multiply from left to right and P k Multiply from the right, and the resulting product is the projection matrix P iThe matrix P is calculated by adding the values. The information processing device 100 calculates the matrix Q in the form Q = IP.

[0191] Furthermore, the information processing device 100 generates L i(t+1) The representation matrix of the linear transformation dcl using is calculated as matrix X. The information processing device 100 calculates matrices QX and PX and adds them to the set Z. QX represents the components that are not affected by singularities and corresponds to the first term on the right-hand side of equation (58). PX represents the components that are affected by singularities and corresponds to the second term on the right-hand side.

[0192] The sets Y and Z above represent subspaces affected by singularities at each stage of quantum gate synthesis. Next, the information processing device 100 uses sets Y and Z to identify subspaces affected by singularities at the stage when the synthesis of the quantum gate sequence is complete.

[0193] First, the information processing device 100 generates set Y' by rearranging the matrices contained in set Y, as shown in equation (69). Set Y' is a sequence of m' matrices. Furthermore, the information processing device 100 generates set Z' by rearranging the matrices contained in set Z, as shown in equation (69). Set Z' is a sequence of m' matrices.

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[0195] The information processing device 100 generates all possible combinations between the elements of set Y' and the elements of set Z'. The information processing device 100 uses the elements of set Y' as a matrix in place of cmr·dcr shown in equation (44). The information processing device 100 also uses the elements of set Z' as a matrix in place of cml·dcl in equation (43).

[0196] The information processing device 100 generates matrix C by substituting equations (43) and (44) and calculating the amplification effect through the synthesis and iteration of quantum gates as described above. This matrix C represents the effect in the space of generators and is before the transformation to the space of the probability distribution of measured values. The information processing device 100 removes matrix C0, which is not affected by singularities, from the set of matrices C calculated for all possible combinations. Matrix C0 is calculated from the combinations of matrices QX described above.

[0197] In other words, the sets Y',Z' are formed by elements obtained by applying matrix Q in pseudocode 144 and elements obtained by applying matrix P in pseudocode 144. Since P is a projection matrix onto a subspace related to a singularity, the elements obtained by applying Q correspond to the application to a subspace unrelated to a singularity. Therefore, matrix C0 corresponds to the matrix calculated from the elements obtained by applying matrix Q to a combination of elements from set Y' and elements from set Z'.

[0198] The information processing device 100 uses the remaining matrix C after removing matrix C0 to obtain the matrix X shown in equation (70). in The information processing device 100 calculates the matrix X. in Matrix R showing the orthogonal projection onto the base in Calculate the matrix R. in This is matrix X in We show the orthogonal projection of the matrix R onto the eigenspace corresponding to the non-zero eigenvalues. in For example, matrix X in This is calculated by numerically diagonalizing the matrix R. in This projects the generator error onto the subspace affected by the singularity in the generator space.

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[0200] Next, the information processing device 100 identifies a subspace that is not affected by singularities once the synthesis of the quantum gate sequence is complete. Here, the information processing device 100 projects the subspace affected by singularities onto the orthogonal complement W in the generator space. in This is calculated as shown in equation (71). This orthogonal complement is a subspace unaffected by singularities.

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[0202] The transformation matrix U represents the transformation from the generator space to the probability distribution space of the measured values. The information processing device 100 uses matrix C and the transformation matrix U to obtain the matrix X shown in equation (72). out The information processing device 100 calculates the matrix X. out Matrix R showing the orthogonal projection onto the base out Calculate the matrix R. out This is matrix X out We show the orthogonal projection of the matrix R onto the eigenspace corresponding to the non-zero eigenvalues. out For example, matrix X out This is calculated by numerically diagonalizing the matrix R. out This represents the projection of the probability distribution of measured values ​​onto a subspace affected by a singularity. The information processing device 100 projects the subspace affected by a singularity onto the orthogonal complement W in the probability distribution of measured values. out This is calculated using formula (73).

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[0205] Ultimately, the action from a subspace of the generator space unaffected by singularities to a subspace of the measurement probability distribution space unaffected by singularities is linearly approximated by the matrix shown in equation (74). The information processing device 100 uses matrix W for each combination of quantum gate sequence a, initialization ρ, and measurement Π. in ,W out The information processing device 100 calculates U and C0. The device replaces the objective function F(v) in equation (22) with the objective function F(v) shown in equation (75), and searches for the value of parameter v that minimizes the value of the replaced objective function F(v).

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[0208] Next, we will explain a numerical example of the matrix used in the above calculation. Here, we consider a sequence of quantum gates that executes an X180 gate followed by a Z90 gate. The X180 and Z90 gates each act on a 1-qubit system with d=2. The X180 gate is a quantum gate that rotates 180 degrees around the X axis and is an example of a quantum gate with a singularity. The Z90 gate is a quantum gate that rotates 90 degrees around the Z axis. The X180 gate is the quantum gate being evaluated, while the Z90 gate is a quantum gate that is not being evaluated.

[0209] First, we define 2x2 matrices called Pauli matrices, σ0, σ1, σ2, and σ3, as shown in equation (76). Using these matrices σ0, σ1, σ2, and σ3, we select the Pauli matrix basis shown in equation (77) as a basis for the matrix representation of quantum gates and generators.

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[0212] Furthermore, as a basis for the vector representations of quantum gates and generators, S1 to S shown in equations (78) to (91) are used. 16 We select a basis S defined by the set of . In equations (78) to (91), the overline on B indicates the complex conjugate, and the product operator indicates the tensor product.

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[0227] The ideal value H of the Hamiltonian for the X180 gate under evaluation is given by equation (92). The ideal action of the X180 gate on a quantum state ρ is given by equation (93) using the ideal value H.

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[0230] The representation matrix G corresponding to equation (93) under the basis of equation (77) is a 4x4 complex matrix and is given as shown in equation (94). The ideal value representation matrix L of the generator corresponding to the ideal value representation matrix G of the quantum gate is calculated as shown in equation (95). The representation matrix L of equation (95) has eigenvalues ​​α1, α2, and α3 shown in equation (96). Equation (97) shows the projection matrix P1 corresponding to eigenvalue α1. Equation (98) shows the projection matrix P2 corresponding to eigenvalue α2. Equation (99) shows the projection matrix P3 corresponding to eigenvalue α3.

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[0237] The information processing device 100 uses the coefficient l shown in formula (33) between the eigenvalues ​​α1, α2, and α3. jk The coefficient l is calculated comprehensively. Formula (100) is the calculated coefficient l jkThe matrix l is shown, where the values ​​are arranged. The first row and first column of matrix l correspond to the eigenvalue α1. The second row and second column of matrix l correspond to the eigenvalue α2. The third row and third column of matrix l correspond to the eigenvalue α3. As shown in equation (101), in the pairs of eigenvalues ​​(α2, α3) and (α3, α2), l jk = 0. Therefore, this X180 gate has a singularity.

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[0240] Figure 5 shows an example of a matrix representing a projection onto a subspace unaffected by singularities. Matrix 145 represents the aforementioned projection W. in This is the representation matrix. Matrix 145 shows the projection onto the orthogonal complement of a subspace affected by a singularity in the generator space. The rank of matrix 145 is 14. The two-dimensional subspace affected by a singularity is the subspace spanned by two basis components, S5-S6 and S9. The projection shown by matrix 145 is the projection onto the subspace excluding the two-dimensional subspace affected by the singularity. Next, the functions and processing procedures of the information processing device 100 will be described.

[0241] Figure 6 is a block diagram illustrating an example of the functions of an information processing device. The information processing device 100 includes a setting data storage unit 121, an experimental data storage unit 122, an evaluation data storage unit 123, an experimental unit 124, and an analysis unit 125. The setting data storage unit 121, the experimental data storage unit 122, and the evaluation data storage unit 123 are implemented using, for example, RAM 102 or HDD 103. The experimental unit 124 and the analysis unit 125 are implemented using, for example, a CPU 101 or GPU 104 and a program.

[0242] The configuration data storage unit 121 stores configuration data. The configuration data indicates the details of the experiment using the quantum computer 115. The configuration data shows the Hamiltonian H and generator L of each quantum gate G. The configuration data also shows the generator error δL of quantum gates that are not being evaluated. The configuration data also shows the quantum gate sequence a and the number of iterations N to be tried. The configuration data also shows the initialization ρ and measurement Π to be tried.

[0243] The experimental data storage unit 122 stores experimental data. The experimental data is an aggregate of measurements read from the quantum computer 115. The experimental data shows the frequency distribution f for each combination of quantum gate sequence a, iteration count N, initialization ρ, and measurement Π. The evaluation data storage unit 123 stores evaluation data. The evaluation data shows the generator error of the quantum gate under evaluation, estimated from the configuration data and experimental data. However, the evaluation data may also show the quantum gate error δG, the quantum gate G containing the error, the Hamiltonian error δH, the Hamiltonian H containing the error, or the generator L containing the error.

[0244] The experimental unit 124 instructs the quantum computer 115 to perform a quantum computation based on the setting data stored in the setting data storage unit 121. The experimental unit 124 reads out the measured values ​​from the quantum computer 115. The experimental unit 124 aggregates the read-out measured values ​​to generate experimental data and stores the experimental data in the experimental data storage unit 122.

[0245] 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 the 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.

[0246] Figure 7 shows an example of configuration data. The configuration data storage unit 121 stores the quantum gate table 131, the quantum gate sequence table 132, the iteration count table 133, and the initialization measurement table 134.

[0247] The quantum gate table 131 associates the identifier of a quantum gate with the representation matrix of the quantum gate, the representation matrix of the Hamiltonian, and the representation matrix of the generator. The quantum gate table 131 contains different n g n quantum gates are registered. g n of the quantum gates g,1 These quantum gates are evaluated, while the remaining quantum gates are not. For the quantum gates not evaluated, a representation matrix of the generator error is also registered.

[0248] The quantum gate sequence table 132 associates identifier i with quantum gate sequences. A quantum gate sequence of length m is described by arranging m identifiers from the quantum gate table 131. The iteration count table 133 associates identifier (i,j) with iteration counts. For each quantum gate sequence in the quantum gate sequence table 132, one or more iteration counts (typically two or more) are registered. The initialization and measurement table 134 associates identifier (i,k) with a representation matrix indicating the initialization of a quantum state and a representation matrix indicating the measurement of a quantum state. For each quantum gate sequence in the quantum gate sequence table 132, one or more initialization and measurement pairs are registered.

[0249] Figure 8 shows examples of experimental and evaluation data. The experimental data storage unit 122 stores the experimental results table 135. The experimental results table 135 associates identifiers (i, j, k) with frequency distributions. Quantum gate sequence a i And, the number of iterations N i,j And the set of initialization and measurement (ρ i,k ,Π i,k For each combination of ), one frequency distribution is registered. The frequency distribution is a list of the frequencies of multiple measured values.

[0250] The analysis unit 125 generates an eigenvalue table 136 and an objective function table 137. The eigenvalue table 136 associates the identifiers of quantum gates with the set of eigenvalues. The identifiers are the same as those in the quantum gate table 131. The set of eigenvalues ​​is calculated by eigenvalue decomposition or spectral decomposition of the generator representation matrix.

[0251] Objective function table 137 consists of identifiers (i,k) and matrices C0,W in ,W out Associate vector b with quantum gate sequence a. i And the set of initialization and measurement (ρ i,k ,Π i,k From one combination of ), matrices C0,W in ,W out And one set of vector b is calculated. i ,(ρ i,k ,Π i,k Matrix C0,W for all combinations of ) in ,W out And using vector b, an objective function is defined that indicates the quality of the generator error estimation.

[0252] The evaluation data storage unit 123 stores the evaluation result table 138. The evaluation result table 138 associates the identifier of a quantum gate with the estimated value of the generator error. The identifier is the same as that of the quantum gate table 131. The evaluation result table 138 is n g Of the quantum gates, the n that are being evaluated are g,1 This includes the generator error of the quantum gate.

[0253] Figure 9 is a flowchart illustrating an example of the procedure for evaluating quantum computation. In step S10, the experimental unit 124 reads out the configuration data. Based on the configuration data, the experimental unit 124 identifies combinations of quantum gate sequence a, iteration count N, initialization ρ, and measurement Π. In step S11, for each combination identified in step S10, the experimental unit 124 repeatedly instructs the quantum computer 115 to perform a quantum computation and reads out the measured value.

[0254] In step S12, the experimental unit 124 aggregates the multiple measurements read out in step S11 for each combination identified in step S10 and calculates the frequency distribution f. In step S13, the analysis unit 125 generates a van der Mond matrix V with respect to the number of iterations N. For each combination of quantum gate sequence a, initialization ρ, measurement Π, and measurement x, the analysis unit 125 estimates the coefficient vector h of the series expansion from the van der Mond matrix V and the frequency of measurement x included in the frequency distribution f calculated in step S12.

[0255] In step S14, the analysis unit 125 extracts a first-order component from the coefficient vector h of step S13 and estimates the coefficient vector h1 for each combination of quantum gate sequence a, initialization ρ, and measurement Π. In step S15, the analysis unit 125 calculates the composite Hamiltonian at each stage in which the quantum gates included in quantum gate sequence a are synthesized in order from the beginning.

[0256] In step S16, the analysis unit 125 converts the composite Hamiltonian of each step calculated in step S15 into a composite generator. In step S17, the analysis unit 125 performs spectral decomposition on the generator L of each quantum gate and obtains the eigenvalue a j and projection matrix P j It is decomposed into a sum of products. In step S18, the analysis unit 125 analyzes different eigenvalues ​​a calculated from the same quantum gate. j Between, the exponential function exp(a j The values ​​of ) are compared. The analysis unit 125 calculates exp(a j Eigenvalue a for which ) are equal j Identify the pair S. However, the analysis unit 125 uses exp(a j If the difference between ) is less than the threshold, exp(a j ) may be considered equal.

[0257] In step S19, the analysis unit 125 performs spectral decomposition on the composite generator of each composite quantum gate and determines the eigenvalue a j and projection matrix P jIt is decomposed into a sum of products. In step S20, the analysis unit 125 analyzes the different eigenvalues ​​a calculated from the same composite quantum gate in step S19. j Between, the exponential function exp(a j The values ​​of ) are compared. The analysis unit 125 calculates exp(a j Eigenvalue a for which ) are equal j The pair T is identified. In step S21, the analysis unit 125 generates sets of matrices Y and Z for each quantum gate sequence a according to the algorithm shown in the pseudocode 144 mentioned above.

[0258] Figure 10 is a flowchart (continued) showing an example of the procedure for evaluating quantum operations. In step S22, the analysis unit 125 comprehensively generates combinations of elements of set Y and elements of set Z for each quantum gate sequence a. The analysis unit 125 considers the elements of set Y as matrices corresponding to cmr·dcr and the elements of set Z as matrices corresponding to cml·dcl.

[0259] In step S23, the analysis unit 125 calculates a matrix C representing a first-order approximation of the action of the error amplification circuit on the generator error δL for each combination of quantum gate sequence a, initialization ρ, and measurement Π, using the matrix from step S22. In step S24, the analysis unit 125 separates matrix C0, which is not affected by singularities, from the set of matrix C calculated in step S23. In step S25, the analysis unit 125 separates matrix X in the generator space from the set of matrix C other than matrix C0. in ,R in ,W in Calculate.

[0260] In step S26, the analysis unit 125 obtains the above matrix X from the transformation matrix U which depends on the initialization ρ and measurement Π, and the set of matrices C other than matrix C0. in ,R in ,W in The matrix X in the space of the probability distribution of the measured values ​​corresponds to this. out ,R out ,W outThe following is calculated. In step S27, the analysis unit 125 calculates vector b using the generator error δL of the quantum gate not under evaluation. In step S28, the analysis unit 125 calculates the matrix C0 from step S24 and the matrix W from step S25. in , matrix W in step S26 out Then, using the vector b from step S27 and the coefficient vector h1 from step S14, the objective function F is generated.

[0261] In step S29, the analysis unit 125 calculates the value of parameter v that minimizes the value of the objective function F generated in step S28. In step S30, the analysis unit 125 estimates the generator error δL of the quantum gate to be evaluated from the value of parameter v calculated in step S29. In step S31, the analysis unit 125 generates evaluation data showing the generator error δL estimated in step S30 and outputs the evaluation data.

[0262] As described above, the information processing device 100 of the second embodiment uses quantum tomography to estimate the errors of the quantum gates of the quantum computer 115. Therefore, the information processing device 100 can efficiently estimate the errors of multiple quantum gates at once. Furthermore, the information processing device 100 causes the quantum computer 115 to execute an error amplification circuit that repeats the same sequence of quantum gates. This amplifies the minute errors of the quantum gates, improving the accuracy of estimating errors from experimental data. In addition, the user can adjust the values ​​of the control parameters of the quantum computer 115 based on the estimated errors, thereby improving the accuracy of the quantum operations of the quantum computer 115.

[0263] Furthermore, the information processing device 100 converts quantum gates into generators and linearly approximates the action of the error amplification circuit on generator errors. This reduces the burden of data analysis and improves the numerical stability of the estimated errors. The information processing device 100 also divides the generator space into a subspace containing a singularity where a value diverging to infinity is calculated, and a subspace that does not contain a singularity. Then, within the subspace that does not contain a singularity, the information processing device 100 linearly approximates the action of the error amplification circuit. As a result, the information processing device 100 can also estimate the errors of quantum gates with singularities through the above data analysis.

[0264] Furthermore, the component of the quantum gate error that is attributable to the control of quantum operations is mainly described within a subspace that does not contain singularities. For this reason, the error estimated by the method of the second embodiment has sufficient accuracy to be useful for calibrating the quantum computer 115. [Explanation of Symbols]

[0265] 10 Information Processing Devices 11 Storage section 12 Processing Units 13 Quantum Gate Sequence 13a Quantum Gates 14. Measurement data 15 Space 15a,15b subspace 15c Singularity 16 Functions

Claims

1. A process for obtaining measurement data that shows the measured quantum state of a qubit measured after a quantum computer repeatedly executes a sequence of quantum gates including multiple quantum gates on the qubit, The space to which the matrix representing the quantum operation by the quantum gate sequence belongs is divided into a first subspace containing a singularity that satisfies certain conditions and a second subspace that does not contain the singularity, and a function that linearly approximates the effect of the error of the first quantum gate included in the plurality of quantum gates on the measured value is generated within the range of the second subspace, A process for estimating the error when the quantum computer executes the first quantum gate, using the function and the measurement data, A quantum computing evaluation program that allows a computer to perform the following operations.

2. The aforementioned singularities correspond to points in the space that have different eigenvalues ​​such that the exponential function has the same value. The quantum computation evaluation program according to claim 1.

3. The generation process includes a process for detecting the singularity by sequentially combining multiple matrices representing each of the multiple quantum gates one step at a time from the beginning, and calculating the eigenvalues ​​of the combined matrices at each step. The quantum computation evaluation program according to claim 1.

4. The function includes a projection matrix that projects the error onto the second subspace, The estimation process includes extracting components corresponding to the second subspace from the measurement data and comparing the output of the function with the components. The quantum computation evaluation program according to claim 1.

5. A process for obtaining measurement data that shows the measured quantum state of a qubit measured after a quantum computer repeatedly executes a sequence of quantum gates including multiple quantum gates on the qubit, The space to which the matrix representing the quantum operation by the quantum gate sequence belongs is divided into a first subspace containing a singularity that satisfies certain conditions and a second subspace that does not contain the singularity, and a function that linearly approximates the effect of the error of the first quantum gate included in the plurality of quantum gates on the measured value is generated within the range of the second subspace, A process for estimating the error when the quantum computer executes the first quantum gate, using the function and the measurement data, A method for evaluating quantum computation performed by a computer.

6. A storage unit that stores measurement data indicating the measured quantum state of a qubit measured after a quantum computer repeatedly executes a sequence of quantum gates including multiple quantum gates on the qubit, A processing unit that performs the following steps: divides the space to which the matrix representing the quantum operation by the sequence of quantum gates belongs into a first subspace containing a singularity that satisfies certain conditions and a second subspace that does not contain the singularity; generates a function within the range of the second subspace that linearly approximates the effect of the error of the first quantum gate included in the plurality of quantum gates on the measured value; and estimates the error when the quantum computer executes the first quantum gate using the function and the measurement data. An information processing device having

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