Energy level difference estimation program, energy level difference estimation method, and quantum computing system

Abstract

Reduce the depth of quantum circuits. [Solution] The processing unit 12 causes the quantum computer 20 to execute the quantum circuit 30. The quantum circuit 30 includes a first control gate 32 of first polarity and a second control gate 33 of second polarity, each controlled by an auxiliary qubit. The first control gate 32 applies a first time evolution operator corresponding to the Hamiltonian H(λ) to a group of qubits corresponding to the quantum state |ψ>. The second control gate 33 applies a second time evolution operator corresponding to H(λ+δλ) to the same group of qubits. The processing unit 12 obtains a probability amplitude from the quantum computer 20 that corresponds to the event that the result of applying two types of time evolution operators to the quantum state |ψ> matches |ψ>. The processing unit 12 obtains a cumulative distribution function 40 of the difference in energy levels for H(λ) and H(λ+δλ) by performing a discrete Fourier transform using the probability amplitude, and estimates the difference based on the cumulative distribution function 40.

Description

[Technical Field]

[0001] This invention relates to an energy level difference estimation program, an energy level difference estimation method, and a quantum computing system. [Background technology]

[0002] Quantum computers are susceptible to errors in the state of their qubits due to environmental noise and other factors. Quantum error correction techniques address this by redundantly encoding the information. Achieving practical quantum error correction would require a massive number of qubits, around one million. However, currently, existing quantum computers are limited to small to medium-sized (Noisy Intermediate Scale Quantum Computers, NISQs) with a maximum of only a few hundred qubits, lacking the capability for error correction.

[0003] A quantum computer capable of error correction is called a fault-tolerant quantum computer (FTQC). Smaller FTQCs are expected to be realized relatively soon and are sometimes called early-FTQCs.

[0004] One of the typical tasks that quantum computers excel at is the simulation of quantum many-body systems. Quantum many-body system simulations are sometimes used, for example, to estimate intrinsic energies.

[0005] Here, a method for estimating the ground state energy by statistical phase estimation using early FTQC is proposed. There is also a proposal to extend this energy estimation method to the estimation of the expectation value of general physical quantities. Furthermore, there is a proposal to improve the efficiency of quantum amplitude estimation algorithms in systems including classical and quantum computers. In addition, a method called Bayesian Phase Difference Estimation (BPDE) using large-scale FTQC is proposed. [Preliminary Technology Documents] [License]

[0006] [License 1] U.S. Patent and Trademark Office Publication No. 2023 / 0081927 [License 2] U.S. Patent and Trademark Office Publication No. 2024 / 0112063 [License 3] International Publication No. 2020 / 252425 [License 4] Special Announcement No. 2023-39444 [Non-licensed literature]

[0007] [Non-licensed Document 1] L.Lin, Y.Tong, "Heisenberg-Limited Ground-State Energy Estimation for Early Fault-Tolerant Quantum Computers", [online], February 2, 2022, Physical Review Journals, PRX Quantum 3, 010318, [January 20, 2025], インターネット<URL:https: / / journals.aps.org / prxquantum / abstract / 10.1103 / PRXQuantum.3.010318> [Non-licensed Document 2] K.Sugisaki, No. 5, "Bayesian phase difference estimation: a general quantum algorithm for the direct calculation of energy gaps", Physical Chemistry Chemical Physics 23, 20152-20162, 2021 [Non-licensed Document 3] K.Sugisaki and 5 others, “Quantum Algorithm for Numerical Energy Gradient Calculations at the Full Configuration Interaction Level of Theory”, Journal of Physical Chemistry Letters, 13, 11105-11111, 2022 [Overview of the Initiative] [Problems that the invention aims to solve]

[0008] For a quantum many-body system, a Hamiltonian H=H(λ) can be considered that contains a certain parameter λ. Parameter λ can represent, for example, an electric field, a magnetic field, nuclear spin, or the position of a nuclear nucleus. The energy levels corresponding to the Hamiltonian H(λ) are denoted as E(λ). By calculating the derivative of the energy levels (=dE(λ) / dλ) in response to a small change in parameter λ, the response of matter to a small change in parameter λ can be obtained. Therefore, it is conceivable to use a quantum computer to calculate the derivative of the energy levels.

[0009] The derivative of an energy level can be approximated by a finite difference. Therefore, to approximate the derivative by a finite difference, one possible method is to individually estimate two energy levels E(λ) and E(λ+δλ) using statistical phase estimation with a quantum computer, and then calculate their difference (=E(λ+δλ)-E(λ)). Here, δλ represents the infinitesimal change in the parameter λ.

[0010] However, these minute differences are generally several orders of magnitude smaller than the energy levels. To estimate a difference several orders of magnitude smaller than the energy levels with guaranteed accuracy, extremely high precision is required in the estimation of the energy levels. The above method has the problem that the depth of the quantum circuits executed by the quantum computer becomes enormous in order to estimate the energy levels with high accuracy.

[0011] In one aspect, the present invention aims to reduce the depth of quantum circuits. [Means for solving the problem]

[0012] In one embodiment, an energy level difference estimation program is provided. The energy level difference estimation program causes a computer to perform the following operations: The computer causes a quantum computer or quantum circuit simulator to execute a quantum circuit including a first Hadamard gate acting on an auxiliary qubit, a first control gate of first polarity controlled by the auxiliary qubit that acts on a group of qubits corresponding to the quantum state of the system of interest with a first time evolution operator corresponding to a first Hamiltonian including predetermined parameters, a second control gate of second polarity opposite to the first polarity controlled by the auxiliary qubit that acts on the group of qubits with a second time evolution operator corresponding to a second Hamiltonian obtained by applying a change in parameters to the first Hamiltonian, a second Hadamard gate acting on an auxiliary qubit, and measurement of the auxiliary qubit. As a result of the execution of the quantum circuit by the quantum computer or quantum circuit simulator, the computer obtains a probability amplitude corresponding to the event that the result of acting on the quantum state with the first time evolution operator and the second time evolution operator matches the quantum state. The computer obtains the cumulative distribution function of the difference in energy levels for the first and second Hamiltonians by performing a Discrete Fourier Transform using the probability amplitude. The computer then estimates this difference based on the cumulative distribution function.

[0013] In one embodiment, a method for estimating energy level differences is provided. In one embodiment, a quantum computing system is provided that includes a quantum computer and an information processing device. [Effects of the Invention]

[0014] In one respect, it can reduce the depth of quantum circuits. [Brief explanation of the drawing]

[0015] [Figure 1] This is a diagram illustrating a quantum computing system according to the first embodiment. [Figure 2] This figure shows an example of a quantum computing system according to the second embodiment. [Figure 3] This figure shows an example of a quantum computing system's hardware. [Figure 4] This figure shows an example of a quantum circuit. [Figure 5] This figure shows an example of the functionality of a quantum computing system. [Figure 6] This flowchart shows an example of a quantum computing system's processing. [Figure 7] This flowchart shows a continuation of the processing example for the quantum computing system. [Figure 8] This diagram illustrates an example of binary search for a cumulative distribution function. [Figure 9] This figure shows an example of a binary search procedure. [Figure 10] This figure shows an example of the antiferromagnetic Heisenberg model. [Figure 11] The figure shows an example of a cumulative distribution function. [Figure 12] This figure shows a modified example of the quantum circuit according to the second embodiment. [Figure 13] This is a diagram showing a quantum circuit as an example. [Modes for carrying out the invention]

[0016] This embodiment will be described below with reference to the drawings. [First Embodiment] Figure 1 is a diagram illustrating a quantum computing system according to a first embodiment. The quantum computing system 1 comprises an information processing device 10 and a quantum computer 20. The information processing device 10 may also be called a classical computer or a von Neumann computer. The information processing device 10 has a storage unit 11 and a processing unit 12. The quantum computer 20 is, for example, an early FTQC.

[0017] The memory unit 11 may be a volatile semiconductor memory such as RAM (Random Access Memory), or a non-volatile storage such as an HDD (Hard Disk Drive) or flash memory. The processing unit 12 is a processor such as a CPU (Central Processing Unit), GPU (Graphics Processing Unit), or DSP (Digital Signal Processor). However, the processing unit 12 may also include application-specific electronic circuits such as an ASIC (Application Specific Integrated Circuit) or FPGA (Field Programmable Gate Array). The processor executes programs stored in memory such as RAM (which may also be the memory unit 11). A collection of multiple processors is sometimes called a "multiprocessor" or simply a "processor."

[0018] The quantum computer 20 performs quantum computations using qubits. A qubit is a unit of information that can exist in a superposition state between a |0> state and a |1> state. When a qubit is measured, its state changes probabilistically to either |0> or |1>. By measuring the state of a qubit multiple times, its state before measurement can be estimated based on the probability of occurrence of |0> and |1>.

[0019] The quantum computing system 1 may have a device with a quantum circuit simulator instead of a quantum computer 20. In that case, the quantum computing system 1 can perform the following processes using the quantum circuit simulator instead of the quantum computer 20. The quantum circuit simulator is software or hardware that simulates the execution of quantum circuits in the quantum computer 20. The device with the quantum circuit simulator is, for example, a classical computer.

[0020] Quantum computing system 1 performs calculations by changing the state of qubits and obtains the calculation result by statistically processing the measurement results of multiple calculations. The qubits can be changed to a desired state by applying a predetermined quantum gate.

[0021] The sequence of quantum gates applied to each qubit in order for the quantum computer 20 to perform quantum computations can be modeled using a quantum circuit. A quantum circuit corresponds to a program that outlines the steps of a quantum operation. The quantum computer 20 performs quantum gate operations on the qubits according to the quantum circuit and measures the final state of the qubits. The measurement results are statistically processed by the information processing device 10.

[0022] Quantum computing system 1 estimates the difference in energy levels for small changes in a Hamiltonian with certain parameters. A Hamiltonian H(λ) with parameter λ can be expressed, for example, by equation (1).

[0023]

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[0024] H0 is a term that does not depend on λ. V is an element that represents the contribution to H depending on λ. However, H(λ) may be expressed by an equation other than equation (1). The energy levels of the eigenstates corresponding to the Hamiltonian H(λ) are expressed as E(λ).

[0025] Here, for example, by considering the derivative of energy levels, we can obtain the response of a substance to small changes in parameters. Estimating the derivative of energy levels is particularly common in molecular systems. Measuring the response of a substance to external fields such as electric and magnetic fields corresponds to experimentally natural situations. Furthermore, the derivative with respect to nuclear positions is also used in molecular dynamics methods.

[0026] For example, if λ represents an electric field, differentiating the energy level with respect to the electric field yields the electric dipole moment as a response of the material. If λ represents a magnetic field, differentiating the energy level with respect to the magnetic field yields the magnetic dipole moment as a response of the material. If λ represents nuclear spin, differentiating the energy level with respect to nuclear spin yields the hyperfine coupling constant as a response of the material. If λ represents the position of the atomic nucleus, differentiating the energy level with respect to the position of the atomic nucleus yields the force acting on the atomic nucleus as a response of the material.

[0027] The derivative of an energy level can be approximated by a finite difference. The simplest method is to use the lowest-order equation, which is given by equation (2). The accuracy of the approximation can be improved by using a higher-order general equation.

[0028]

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[0029] Therefore, the quantum computing system 1 directly estimates the difference in energy levels ΔE = E(λ + δλ) - E(λ) as follows. Note that the difference to be estimated can be generalized to a forward difference, a backward difference, a central difference, or a higher-order difference.

[0030] The processing unit 12 generates information for the quantum circuit 30. The quantum circuit 30 includes a first Hadamard gate 31, a first control gate 32, a second control gate 33, a quantum gate 34, a second Hadamard gate 35, and a measurement 36. Quantum gate operations are performed sequentially from left to right in the quantum circuit 30.

[0031] The auxiliary qubit is initialized to |0>. The first Hadamard gate 31 acts on the auxiliary qubit. The first control gate 32 is a controlled unitary gate controlled by an auxiliary qubit. The first control gate 32 is, for example, positively polarized. That is, the first control gate 32 acts on, for example, the positive polarity (|1>) of an auxiliary qubit. In quantum circuit 30, positive polarity is represented by a black circle. The first control gate 32 applies the first time evolution operator e to the group of qubits corresponding to the quantum state |ψ> of the system of interest. -ijτH(λ) Apply the first time evolution operator e -ijτH(λ) is the time evolution operator corresponding to the first Hamiltonian H(λ) containing a given parameter λ. The i in the exponent is the imaginary unit. τ is the time step of the time evolution. j is an integer multiplied by τ. j ∈ [-d, d]. d is the truncation order of the discrete Fourier transform, which will be described later. d is a positive integer. d is predetermined based on τ and the required error for the difference in energy levels.

[0032] The second control gate 33 is a controlled unitary gate controlled by an auxiliary qubit. The second control gate 33 is, for example, negative polarity. That is, the second control gate 33 acts on, for example, the negative polarity (|0>) of an auxiliary qubit. In quantum circuit 30, negative polarity is represented by a white circle. The second control gate 33 applies the second time evolution operator e to the qubit group of the system of interest. -ijτH(λ+δλ) Apply the second time evolution operator e. -ijτH(λ+δλ) This is the time evolution operator corresponding to the second Hamiltonian H(λ+δλ). The second Hamiltonian H(λ+δλ) is the Hamiltonian obtained by applying a change in parameter λ δλ to the first Hamiltonian H(λ). The auxiliary qubits can also be considered the control bits of the first control gate 32 and the second control gate 33, respectively.

[0033] Quantum gate 34 acts on an auxiliary qubit. Quantum gate 34 is denoted as "W". Quantum gate 34 is an I gate or S gate. † It is a gate. The I gate is the identity gate. Making quantum gate 34 an I gate is equivalent to quantum circuit 30 not containing quantum gate 34. S †is a phase shift gate that performs a shift operation of -π / 2 rotation in the Z-axis direction. When switching to either W = I or W = S † the real and imaginary parts of the probability amplitude measured in the quantum circuit 30 are switched. When W = I, the real part of the probability amplitude is measured. When W = S † the imaginary part of the probability amplitude is measured.

[0034] The second Hadamard gate 35 acts on the auxiliary qubit. The measurement 36 is a measurement of the auxiliary qubit. In the measurement 36, the measurement of the auxiliary qubit is performed for each of the cases of W = I and W = S as described above † The measurement may be referred to as a measurement operation.

[0035] The processing unit 12 instructs the quantum computer 20 to execute the quantum circuit 30 and causes the quantum computer 20 to execute the quantum circuit 30 (step S1). The quantum circuit 30 is executed for various values of j. The processing unit 12 obtains the probability amplitude <ψ|e ijτH(λ+δλ) e -ijτH(λ) |ψ> as the execution result of the quantum circuit 30 by the quantum computer 20 (step S2).

[0036]

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[0037] The probability amplitude of equation (3) is the probability amplitude corresponding to the event that the result of applying the first time evolution operator and the second time evolution operator to the quantum state |ψ> of the target system coincides with the original quantum state |ψ>.

[0038] For example, the processing unit 12 obtains the real part of the probability amplitude of equation (3) by causing the quantum computer 20 to execute the quantum circuit 30 with W = I for a certain j. Also, the processing unit 12 obtains the imaginary part of the probability amplitude of equation (3) by causing the quantum computer 20 to execute the quantum circuit 30 with W = S † for the said j.

[0039] The number of samples for the probability amplitude is predetermined based on the degree of overlap between the initial state |ψ> and the eigenstates of the Hamiltonian, and the allowable failure probability for estimating the difference in energy levels.

[0040] The processing unit 12 performs a discrete Fourier transform using the acquired probability amplitude (step S3). Specifically, the processing unit 12 uses Heaviside's step function F. The step function F is a step function with a period of 2π. The step function F is expressed by equation (4).

[0041]

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[0042] In equation (4), k is an integer. The step function F is equivalent to the ordinary step function on the interval x ∈ (-π, π). The processing unit 12 calculates the discrete Fourier expansion coefficients F^ of the step function for the probability amplitude. j Multiply by and perform the Discrete Fourier Transform. "F^" represents the letter F with a caret "^" above it. In the Discrete Fourier Transform, only orders less than or equal to the truncation order d are considered. The Discrete Fourier Transform is expressed by equation (5).

[0043]

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[0044] C ~ (x) gives a good approximation of the cumulative distribution function of the difference in energy levels under the assumption that δλ << 1. That is, the contribution of the m ≠ n component in equation (5) becomes 0, and equation (6) is obtained. Note that "C ~ " represents the letter C with a tilde "~" placed above it.

[0045]

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[0046] Thus, the processing unit 12 performs a discrete Fourier transform using the probability amplitude to obtain the cumulative distribution function C of the difference in energy levels. ~ (x) is obtained (step S4). The cumulative distribution function 40 is the cumulative distribution function C ~ This is an example. The x-axis of the cumulative distribution function is τ × ΔE. The y-axis of the cumulative distribution function is the cumulative distribution function C. ~ This is the value of . More specifically, the y-axis is the cumulative value of the contribution rate of each level included in the input state. Processing unit 12 is C ~ The difference in energy levels ΔE from the position where the value of (x) increases sharply. k Estimate ΔE (Step S5). k This is expressed by equation (7).

[0047]

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[0048] Specifically, the processing unit 12 is C ~ By dividing the x-coordinate of the position where the value of (x) increases sharply by τ, we obtain the difference in energy levels ΔE. k It is possible to specify C. ~ The sharp increase in (x) is due to C ~ This is equivalent to the derivative of (x) having a peak.

[0049] Here, by determining the initial state |ψ> such that equation (8) is satisfied, the difference ΔE of a particular energy level can be found. k_0 It is possible to selectively estimate this. "k_0" represents "k0".

[0050]

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[0051] In addition, in the quantum circuit 30, the first control gate 32 may be negative polarity and the second control gate 33 may be positive polarity. In that case, the processing unit 12 can obtain the same result as when the first control gate 32 is positive polarity and the second control gate 33 is negative polarity by inverting the sign of the difference in the final energy levels. Thus, when the first control gate 32 is the first polarity, the second control gate 33 is the second polarity, which is the opposite of the first polarity. The first polarity is either positive or negative polarity. The second polarity is either negative or positive polarity.

[0052] According to the information processing device 10 of the first embodiment, the quantum circuit 30 is executed by the quantum computer 20. However, as mentioned above, a quantum circuit simulator may be used instead of the quantum computer 20. The quantum circuit 30 includes, in order, a first Hadamard gate 31, a first control gate 32, a second control gate 33, a second Hadamard gate 35, and an auxiliary qubit measurement 36. As a result of the execution of the quantum circuit 30, the first time evolution operator e is applied to the quantum state |ψ> of the system of interest. -ijτH(λ) and the second time evolution operator e -ijτH(λ+δλ) A probability amplitude is obtained that corresponds to the event where the result of applying and coincides with the quantum state |ψ>. By performing the Discrete Fourier Transform using this probability amplitude, the cumulative distribution function C of the difference in energy levels for the first Hamiltonian H(λ) and the second Hamiltonian H(λ+δλ) is obtained. ~ (x) is obtained. Cumulative distribution function C ~ Based on (x), the difference in energy levels is estimated. This allows the information processing device 10 to reduce the depth of the quantum circuit. Specifically, this is as follows: Let D be the maximum depth required in the quantum circuit 30. D is proportional to the truncation order d. d and D are expressed by equation (9).

[0053]

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[0054] ε ΔE_k ΔE kThis is the required error for . As shown in equation (9), the condition for D can be obtained from the condition for d. Here, as a comparative example, one can consider a method in which E(λ) and E(λ+δλ) are individually determined by statistical phase estimation (SPE), and then E(λ+δλ)-E(λ) is calculated based on these results. For SPE, the above-mentioned Non-Patent Document 1 is a useful reference. When determining the energy level by SPE, for example, in the comparative example quantum circuit in which the second control gate 33 is removed from the quantum circuit 30, the probability amplitude < φ0|e -ijτH |φ0> is measured. The quantum circuit of the comparative example will be described later. |φ0> is the quantum state of the system of interest. Probability amplitude <φ0|e -ijτH Based on |φ0> and the Fourier components of Heaviside's step function F, the discrete Fourier transform of equation (10) is performed.

[0055]

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[0056] C(x) in equation (10) is a good approximation of the cumulative distribution function, i.e., the convolution product of the density of states p(x) and F. The density of states p(x) is given by equation (11). The convolution product of p(x) and F is given by equation (12).

[0057]

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[0058]

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[0059] p k is the energy level E k Eigenstates |φ k > and the overlap with the initial state |φ0> |<φ k |φ0>| 2 This indicates. The cumulative distribution function C(x) corresponds to the position x = τE of the energy levels. k de p k The value increases sharply by only that much. Therefore, from the position where C(x) increases sharply, the energy level E k You can read the estimated value.

[0060] Here, in SPE, there is a constraint shown in equation (13).

[0061]

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[0062] Furthermore, in SPE, the truncation order d, which is proportional to the maximum depth D of the quantum circuit, must satisfy equation (14).

[0063]

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[0064] ε is the required error for the target to be estimated. Using the comparative example method, E k (λ) and E k (λ+δλ) is estimated individually, and ΔE k =E k When calculating (λ)-E(λ+δλ), the truncation order d must satisfy equation (15).

[0065]

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[0066] ε ΔE_k ΔE k This is the required error for the energy level E. k For this, the small difference ΔE k It is generally several orders of magnitude smaller. That is, equation (16) holds.

[0067]

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[0068] According to equations (15) and (16), the comparative example method yields the reciprocal of the required relative precision (ε ΔE_k / ΔE k A depth D(∝d) several orders of magnitude larger than )^(-1) is required. Therefore, the information processing device 10 directly estimates the energy level difference ΔE using the quantum circuit 30. When ΔE is directly estimated, the required depth D of the quantum circuit 30 is expressed by equation (9). Comparing equation (9) with equation (15), equation (9) does not contain the large factor expressed in equation (17). Thus, according to the information processing device 10, the depth of the quantum circuit is reduced by several orders of magnitude compared to the method of the comparative example.

[0069]

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[0070] Here, the condition for the depth D required by the comparative example method is expressed by equation (18) based on equation (15).

[0071]

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[0072] As shown in equation (18), the reason why the depth D becomes enormous in the comparative example method is that "in order to estimate a difference of several orders of magnitude with guaranteed accuracy, the energy levels must be estimated with very high relative accuracy."

[0073] Here, the larger the time step τ, the greater the required depth D∝(τε ΔE_k )^(-1) becomes smaller. When estimating the energy levels using the comparative example method, the constraint on τ based on equation (13) is expressed by equation (19).

[0074]

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[0075] On the other hand, the constraint on τ when directly estimating the difference in energy levels is expressed by equation (20).

[0076]

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[0077] Comparing equations (19) and (20), the constraints on τ are significantly relaxed in the direct estimation of the energy level difference compared to the comparative example method. That is, with the information processing device 10, τ can be made several orders of magnitude larger than in the comparative example method, to the extent expressed in equation (17). As a result, the information processing device 10 can reduce the depth D required for the quantum circuit 30.

[0078] [Second Embodiment] Figure 2 shows an example of a quantum computing system according to the second embodiment. The quantum computing system 300 is a hybrid computer system in which a classical computer 100 and a quantum computer 200 operate in conjunction. The classical computer 100 is also called a von Neumann type computer.

[0079] A terminal device 400 is connected to the classical computer 100 via a network 50. The terminal device 400 is a computer used by users who request quantum computations from the quantum computing system 300. The classical computer 100 may receive information indicating a quantum circuit from the terminal device 400. A quantum circuit indicates the sequence of operations on qubits by the arrangement of elements such as gates. A qubit is a bit that can represent a superposition state between a "0" state and a "1" state.

[0080] The classical computer 100 gives instructions to the quantum computer 200 to control the qubits according to the quantum circuit. The classical computer 100 also obtains the measurement results of each qubit from the quantum computer 200. The classical computer 100 is an example of the information processing device 10 of the first embodiment.

[0081] Quantum computer 200 has multiple qubits and devices for manipulating each of the multiple qubits. The multiple qubits of quantum computer 200 are realized using, for example, superconducting qubit devices. Alternatively, the qubits may be realized using other types of qubit devices, such as ion trap devices. A quantum computer is, for example, an early FTQC.

[0082] Figure 3 shows an example of the hardware of a quantum computing system. The classical computer 100 is controlled as a whole by a processor 101. The processor 101 is connected to memory 102 and several peripheral devices via a bus 109. The processor 101 may be a multiprocessor. The processor 101 is, for example, a CPU, an MPU (Micro Processing Unit), or a DSP. At least some of the functions realized by the processor 101 executing a program may be realized by electronic circuits such as ASICs or PLDs (Programmable Logic Devices). The processor that executes one of the multiple processes performed by the classical computer 100 may be different from the processor that executes a different process from the multiple processes. Also, at least some of the processes described below may be executed in parallel using multiple processors or processor cores. The processor 101 may also be called "processor circuitry". The processor 101 is an example of the processing unit 12 of the first embodiment.

[0083] Memory 102 is used as the main memory of the classical computer 100. Memory 102 temporarily stores at least a portion of the OS (Operating System) program and application programs that are to be executed by the processor 101. Memory 102 also stores various data used for processing by the processor 101. For memory 102, a volatile semiconductor memory device such as RAM is used.

[0084] Peripheral devices connected to bus 109 include storage device 103, GPU 104, input interface 105, optical drive device 106, device connection interface 107, and network interface 108a and communication interface 108b.

[0085] The storage device 103 electrically or magnetically writes and reads data from its built-in recording medium. The storage device 103 is used as an auxiliary storage device for the classical computer 100. The storage device 103 stores the OS program, application programs, and various data. For example, an HDD or SSD (Solid State Drive) can be used as the storage device 103. The memory 102 or storage device 103 is an example of the storage unit 11 in the first embodiment.

[0086] The GPU104 is a processing unit that performs image processing. The GPU104 is an example of a graphics controller. A monitor 51 is connected to the GPU104. The GPU104 displays images on the screen of the monitor 51 according to instructions from the processor 101. The monitor 51 can be a display device using organic EL (Electro Luminescence) or a liquid crystal display device.

[0087] The input interface 105 is connected to a keyboard 52 and a mouse 53. The input interface 105 transmits signals from the keyboard 52 and mouse 53 to the processor 101. Note that the mouse 53 is just one example of a pointing device; other pointing devices can also be used. Other pointing devices include touch panels, tablets, touchpads, and trackballs.

[0088] The optical drive device 106 uses laser light or the like to read data recorded on the optical disc 54 or write data to the optical disc 54. The optical disc 54 is a portable recording medium on which data is recorded in a way that makes it readable by the reflection of light. Examples of optical discs 54 include DVD (Digital Versatile Disc), DVD-RAM, CD-ROM (Compact Disc Read Only Memory), and CD-R (Recordable) / RW (ReWritable).

[0089] The device connection interface 107 is a communication interface for connecting peripheral devices to the classical computer 100. For example, a memory device 55 and a memory reader / writer 56 can be connected to the device connection interface 107. The memory device 55 is a recording medium equipped with a communication function with the device connection interface 107. The memory reader / writer 56 is a device that writes data to or reads data from the memory card 57. The memory card 57 is a card-type recording medium.

[0090] The network interface 108a is connected to the network 50. The network interface 108a transmits and receives data to and from other computers or communication devices via the network 50. The network interface 108a is a wired communication interface, for example, connected by cable to a wired communication device such as a switch or router. Alternatively, the network interface 108a may be a wireless communication interface, connected by radio waves to a wireless communication device such as a base station or access point.

[0091] The communication interface 108b is an interface for connecting to the quantum computer 200. The processor 101 transmits quantum circuits to the quantum computer 200 via the communication interface 108b, causing the quantum computer 200 to perform quantum computations. The processor 101 also obtains the results of the quantum computations via the communication interface 108b.

[0092] The classical computer 100 can realize the processing functions of the second embodiment with the hardware described above. The information processing device 10 shown in the first embodiment can also be realized with the same hardware as the classical computer 100 shown in Figure 3.

[0093] The classical computer 100 implements the processing functions of the second embodiment by executing a program recorded on a computer-readable recording medium, for example. The program describing the processing to be executed by the classical computer 100 can be recorded on various recording media. For example, the program to be executed by the classical computer 100 can be stored in the storage device 103. The processor 101 loads at least a portion of the program in the storage device 103 into the memory 102 and executes the program. Alternatively, the program to be executed by the classical computer 100 can be recorded on a portable recording medium such as an optical disc 54, a memory device 55, or a memory card 57. The program stored on the portable recording medium becomes executable after being installed in the storage device 103, for example, under control from the processor 101. The processor 101 can also directly read and execute the program from the portable recording medium.

[0094] The quantum computer 200 comprises a control device 201 and a qubit device 202. The control device 201 performs gate operations on the qubits in the qubit device 202 according to a quantum circuit. The qubit device 202 has multiple qubits. The qubit device 202 is, for example, one or more QPUs (Quantum Processing Units).

[0095] The quantum computing system 300 efficiently estimates the difference in energy levels for a small change δλ in the parameter λ of a Hamiltonian H(λ) having parameter λ. Figure 4 shows an example of a quantum circuit. The quantum computing system 300 uses the quantum circuit 60 to calculate the probability amplitude <ψ|e in equation (3). ijτH(λ+δλ) e -ijτH(λ) We find |ψ>.

[0096] The quantum circuit 60 includes a Hadamard gate 61, control gates 62 and 63, a quantum gate 64, a Hadamard gate 65, and a measurement gate 66. Quantum gate operations are performed sequentially from left to right in the quantum circuit 60, that is, from the Hadamard gate 61 to the measurement gate 66.

[0097] The auxiliary qubit is initialized to |0>. The Hadamard gate 61 acts on the auxiliary qubit. The auxiliary qubit is also called an ancilla. Control gate 62 is a positive-polarity controlled unitary gate controlled by an auxiliary qubit. Control gate 62 assigns the time evolution operator e to the group of qubits corresponding to the quantum state |ψ> of the system of interest. -ijτH(λ) To apply the effect.

[0098] Control gate 63 is a negative-polarity controlled unitary gate controlled by an auxiliary qubit. Control gate 63 has a time evolution operator e applied to the qubit group of the system of interest. -ijτH(λ+δλ) To apply the effect.

[0099] Quantum gate 64 acts on auxiliary qubits. Quantum gate 64 is represented as "W". W=I or W=S † Therefore, W=I means that quantum gate 64 is an I gate. W=I is equivalent to not inserting anything at the location of quantum gate 64. In other words, W=I is equivalent to quantum circuit 60 not containing quantum gate 64. W=S † The quantum gate 64 is S † This means it will be used as a gate.

[0100] The Hadamard gate 65 acts on the auxiliary qubit. Measurement 66 is a projection measurement on the computational basis for the auxiliary qubit. When W=I, the real part of the probability amplitude in equation (3) is measured. W=S† In this case, the imaginary part of the probability amplitude in equation (3) is measured.

[0101] Note that the polarities of control gate 62 and control gate 63 may be reversed. That is, control gate 62 may be negative polarity and control gate 63 may be positive polarity. Also, control gates 62 and 63 may be represented by a single control gate that is equivalent to the action of both control gates 62 and 63.

[0102] Here, e for the qubit group -ijτH The application is e -iτH This corresponds to |j| applications. Therefore, the larger the value of |j| for control gates 62 and 63, the greater the depth of quantum circuit 60. The upper limit of |j| is the truncation order d of the discrete Fourier transform.

[0103] Figure 5 shows an example of the functions of a quantum computing system. The quantum computer 200 has a gate operation unit 210 and a measurement unit 220. The gate operation unit 210 performs gate operations on qubits in the qubit device 202. The gate operation unit 210 performs gate operations on qubits according to gate operation instructions from the classical computer 100. For example, the gate operation unit 210 irradiates the qubit with microwaves corresponding to the quantum gate to be acted upon. The measurement unit 220 is realized by a device that measures the state of qubits in the qubit device 202. The measurement unit 220, for example, irradiates the qubit to be measured with microwaves and performs a projection measurement of the computational basis.

[0104] The classical computer 100 includes a measurement result storage unit 110, a quantum circuit control unit 120, a truncation order determination unit 130, a Fourier expansion coefficient calculation unit 140, a discrete Fourier transform execution unit 150, a median calculation unit 160, and a cumulative distribution function analysis unit 170. The measurement result storage unit 110 uses the storage areas of memory 102 and storage device 103. The quantum circuit control unit 120, truncation order determination unit 130, Fourier expansion coefficient calculation unit 140, discrete Fourier transform execution unit 150, median calculation unit 160, and cumulative distribution function analysis unit 170 are realized by a program being executed by processor 101. This program is stored in memory 102.

[0105] The measurement result storage unit 110 stores the measurement results of the probability amplitude based on the quantum circuit 60. The k-th measurement result of the probability amplitude is Z k It is expressed as Z k =X k +iY k That is. Z k The 'i' included in X is the imaginary unit. k is Z k This is the real part of Y. k is Z k This is the imaginary part. The measurement result storage unit 110 stores the probability amplitude Z k Correspond to Z k The value of j used in the measurement is stored.

[0106] The quantum circuit control unit 120 generates information about the quantum circuit 60 for the Hamiltonian H(λ) and the infinitesimal change δλ. The input information, including the Hamiltonian H(λ) and the infinitesimal change δλ, is pre-input into the classical computer 100. The input information may also be input into the classical computer 100 by the terminal device 400. Furthermore, when generating information about the quantum circuit 60, the quantum circuit control unit 120 selects an integer j included in the time evolution operator according to the truncation order d of the discrete Fourier transform, which will be described later, and generates the quantum circuit 60 based on the selected j.

[0107] The quantum circuit control unit 120 controls the qubit device 202 based on the quantum circuit 60. Specifically, the quantum circuit control unit 120 transmits information about the quantum circuit 60 to the quantum computer 200 and instructs the quantum computer 200 to execute the quantum circuit 60. As a result, the quantum circuit 60 is executed by the quantum computer 200. In response to the execution of the quantum circuit 60 by the quantum computer 200, the measurement result of the probability amplitude measured by the measurement unit 220 is stored in the measurement result storage unit 110.

[0108] The truncation order determination unit 130 determines the truncation order d of the discrete Fourier transform. The truncation order d is determined based on equation (21).

[0109]

number

[0110] τ is the time interval of time evolution. ε ΔE η is the required error for the difference in energy levels. η is the overlap between the eigenstates of the Hamiltonian H and the initial state |ψ>. ~ These are index values ​​defined such that both equations (22) and (23) are satisfied for τ, ε. ΔE And η is included in the input information.

[0111]

number

[0112]

number

[0113] The input information also includes the tolerance value ν for the failure probability of estimating the difference in energy levels. ν and η are parameters N that determine the number of samples for the probability amplitude. b ,N s It is used in the decision of N. b This is determined based on equation (24). N s This is determined based on equation (25).

[0114]

number

[0115]

number

[0116] Here, N b N is the number of blocks into which each sample of the probability amplitude is divided. For each block, one trial result of the Discrete Fourier Transform is obtained. The final result of the Discrete Fourier Transform is obtained from the median of multiple trial results corresponding to multiple blocks. s This is the number of samples contained in one block.

[0117] The truncation order determination unit 130 outputs the determined truncation order d to the quantum circuit control unit 120 and the Fourier expansion coefficient calculation unit 140. The Fourier expansion coefficient calculation unit 140 calculates the discrete Fourier expansion coefficients F^ of the Heaviside step function F represented by equation (4). i Calculate F^ i The range of integers i in this case is -d ≤ i ≤ d.

[0118] The Discrete Fourier Transform execution unit 150 performs a Discrete Fourier Transform using the measurement results of the probability amplitude stored in the measurement result storage unit 110 and the Discrete Fourier expansion coefficients of the step function F. Specifically, the Discrete Fourier Transform execution unit 150 performs a Discrete Fourier Transform on the Z in the measurement results. k ni e iarg(F^_k) Multiply by and perform the discrete Fourier transform of equation (26). F^ k is Z k These are the discrete Fourier expansion coefficients of the order corresponding to j used in the measurement.

[0119]

number

[0120] The i included in the exponent is the imaginary unit. r is the block identification number. The discrete Fourier transform execution unit 150 calculates G for each block based on Equation (26). r G r is a function of x. G r is one trial result of the discrete Fourier transform.

[0121] G r The expected value <G j} of the random variable {F^ r > reproduces the result of the discrete Fourier transform. <G r > is expressed by Equation (27).

[0122]

Equation

[0123] The median calculation unit 160 estimates the cumulative distribution function C b (x) using the median of N r generated G ~ samples. C ~ (x) is expressed by Equation (6). The median calculation unit 160 determines the median of the values of N b G r = G r (x) for each value x a of the variable x, and sets it as the value of C b for the value of C r (x a ) corresponding to the value of the said x a , thereby obtaining C ~ (x a ) and obtaining C ~ (x).

[0124] The cumulative distribution function analysis unit 170 reads the parameter point x ~ at which the value of the cumulative distribution function C * (x) rapidly increases. For example, the cumulative distribution function analysis unit 170 determines x * that satisfies the conditions of Equation (28) and Equation (29) by binary search.

[0125]

number

[0126]

number

[0127] The cumulative distribution function analysis unit 170 is x * The difference in energy levels ΔE is estimated by dividing by τ. The cumulative distribution function analysis unit 170 outputs the difference in energy levels ΔE. The cumulative distribution function analysis unit 170 may display an image showing ΔE on the monitor 51. The cumulative distribution function analysis unit 170 may transmit information showing ΔE to the terminal device 400.

[0128] Figure 6 is a flowchart showing an example of processing in a quantum computing system. (S10) The quantum circuit control unit 120 controls H,|ψ>,η,ν,ε ΔE Accepts inputs τ, H,|ψ>,η,ν,ε ΔE τ is included in the input information that is input to the classical computer 100. Here, τ is, for example, ΔE k The approximate result of the estimation and equation (20) are predetermined. τ may be calculated by the quantum circuit control unit 120 or the truncation order determination unit 130 based on equation (20).

[0129] (S11) The termination order determination unit 130 determines τ and ε ΔE The truncation order d of equation (21) is calculated based on this. (S12) The Fourier expansion coefficient calculation unit 140 loops through step S13 for i = -d to d.

[0130] (S13) The Fourier expansion coefficient calculation unit 140 calculates the discrete Fourier expansion coefficients F^ of the step function F. i Calculate the result and store it in the array list_F. (S14) The Fourier expansion coefficient calculation unit 140 terminates the loop for i = -d to d.

[0131] (S15) The quantum circuit control unit 120 processes N from η,ν b and N s Calculate N b This is calculated based on equation (24). N s This is calculated based on equation (25). (S16) The quantum circuit control unit 120 controls r=1~N b N s For this, steps S17 to S19 are looped.

[0132] (S17) The quantum circuit control unit 120 probabilistically selects an integer j with an absolute value less than or equal to d. Specifically, the quantum circuit control unit 120 selects an integer j ∈ [-d, d] |F^ j The selection is made with a probability proportional to |.

[0133] (S18) The quantum circuit control unit 120 prepares a quantum circuit 60 including two types of time evolution operators and has the quantum computer 200 execute it. The two types of time evolution operators are the aforementioned time evolution operator e -ijτH(λ) and the time evolution operator e -ijτH(λ+δλ) That is the case.

[0134] (S19) The quantum circuit control unit 120 controls the probability amplitude Z in response to the execution of the quantum circuit 60. r The measurement results are stored in the array list_R held in the measurement result storage unit 110. The quantum circuit control unit 120 controls the probability amplitude Z r Correspond to Z r The value of j used in the measurement is stored in the measurement result storage unit 110.

[0135] (S20) The quantum circuit control unit 120 controls r=1~N b N s The loop regarding that is terminated. Then, the process proceeds to step S21. Figure 7 is a flowchart showing a continuation of an example of the processing of a quantum computing system.

[0136] (S21) The discrete Fourier transform execution unit 150 performs r=1~N b The process is repeated in step S22. (S22) The discrete Fourier transform execution unit 150 calculates G in Equation (26) and stores it in the array list_G. r

[0137] (S23) The discrete Fourier transform execution unit 150 ends the loop for r = 1 to N. b (S24) The median calculation unit 160 determines the cumulative distribution function C(x) based on the median of the data in list_G. ~

[0138] (S25) The cumulative distribution function analysis unit 170 reads the parameter point x at which the value of the cumulative distribution function C(x) rapidly increases. The cumulative distribution function analysis unit 170 divides x by τ to estimate the energy level difference ΔE. The cumulative distribution function analysis unit 170 outputs ΔE. Then, the processing of the quantum computing system 300 ends. ~ * *

[0139] Here, the cumulative distribution function analysis unit 170 can obtain x as follows, for example, based on the cumulative distribution function C(x). ~ * FIG. 8 is a diagram for explaining an example of a binary search for a cumulative distribution function.

[0140] The cumulative distribution function analysis unit 170 identifies the x - coordinate at which the value of the cumulative distribution function C(x) rapidly increases for the cumulative distribution function C(x). The graph 500 is an example of the cumulative distribution function C(x). For the vicinity of the location where the value of C(x) rapidly increases, it is defined that C(x) = C at x < x and C(x) = C at x > x. This definition means that the cumulative distribution function rapidly increases from C to C from x to x. ~ ~ ~ ~ L ~ small R ~ large L R small large ​

[0141] Figure 9 is a diagram showing an example of a binary search procedure. Algorithm 501 shows the search procedure for x with respect to graph 500. * In algorithm 501, δ = τε ΔE is true. Algorithm 501 shows the following procedure.

[0142] x R - x L > 2δ, the following process is repeatedly executed. x M is substituted with (x L + x R ). C ~ (x M + (2 / 3)δ) is greater than (C small + C large ), the value of x R is updated to x M + (2 / 3)δ. C ~ (x M + (2 / 3)δ) is less than or equal to (C small + C large ), the value of x L is updated to x M - (2 / 3)δ.

[0143] And, when x R - x L ≤ 2δ, x * = (x L + x R ) / 2 is returned. Based on algorithm 501, the cumulative distribution function analysis unit 170 can obtain the parameter point x ~ at which the value of the cumulative distribution function C * (x) rapidly increases.

[0144] Here, the processing capabilities of quantum computing system 300 were verified through numerical simulations. The antiferromagnetic Heisenberg model was used as the specific system in the numerical simulations. The antiferromagnetic Heisenberg model is a fundamental theoretical model that describes the quantum mechanical behavior of magnetic materials. The numerical simulations investigated the case where a small external magnetic field was added to the antiferromagnetic Heisenberg model.

[0145] Figure 10 shows an example of the antiferromagnetic Heisenberg model. In the antiferromagnetic Heisenberg model 600, sites 601, 602, and 603 are located on the lattice points. A spin is defined for each of sites 601, 602, and 603. Interactions act between adjacent spins. Using the antiferromagnetic Heisenberg model 600, the spin orientations of sites 601, 602, and 603 when an external magnetic field is applied can be calculated by numerical simulation. For example, the Hamiltonian H of the antiferromagnetic Heisenberg model 600 is given by equation (30).

[0146]

number

[0147] In equation (30), X i ,Y i ,Z i These are the Pauli operators at site i, respectively. The Pauli operators are operators that describe the spin component. That is, X i is the Pauli operator that describes the X-direction component of the spin at the i-th site. i is the Pauli operator that describes the Y-direction component of the spin at the i-th site. i is the Pauli operator describing the Z-direction component of the spin at the i-th site. J is a parameter representing the Heisenberg interaction. J is a real number. h is a parameter representing a small external magnetic field. h is a real number.

[0148] In numerical simulations, the difference in energy levels before and after applying a small external magnetic field is estimated based on the quantum circuit 60 relating to the Hamiltonian H in equation (30). The parameter that is shifted by a small amount is h. The difference in energy levels for the cases h=0 and h≠0 is calculated. The calculation conditions are as follows:

[0149] The assumed antiferromagnetic Heisenberg model 600 is a 5-site, one-dimensional model. J = 1.0. The value of h when h ≠ 0 is h = 0.1. The initial state |ψ> is the ground state when the external magnetic field is zero (h = 0) with exp(i(π / 20)Y0) applied. Applying exp(i(π / 20)Y0) corresponds to rotating the zeroth spin by a small amount. This operation was performed to intentionally add the contribution of the excited state. The initial state |ψ> can be prepared numerically using an iterative method.

[0150] Furthermore, the following assumptions were made in the numerical simulation: the number of executions of the quantum circuit is infinite and measurement fluctuations can be ignored; and the number of samplings of various orders is infinite and sampling errors can be ignored.

[0151] Furthermore, ε ΔE = 0.01. ε ΔE τ is the required error for the difference in energy levels. τ = 1.0. τ is the time step of the time evolution. Furthermore, the expansion coefficients when approximately performing a discrete Fourier expansion of a step function F with period 2π include a parameter β. The parameter β governs the accuracy of the approximation. The larger β is, the better the accuracy of the expansion coefficients. In this example, β = 700. The expansion coefficients when approximately performing a discrete Fourier expansion of a step function F with period 2π are expressed by equation (31).

[0152]

number

[0153] I j (β) is the j-th order modified Bessel function of the first kind. The middle equation F^ of equation (31) 2j+1=… is the value when j < d. Also, when j is an even number other than 0, F j = 0.

[0154] As a result of numerical simulation based on the above calculation conditions, the following cumulative distribution function was obtained. FIG. 11 is a diagram showing an example of the cumulative distribution function. Graph 701 is the cumulative distribution function C ~ (x) = C ~ (τΔE) shown based on the calculation conditions described in FIG. 10. Graph 702 shows the derivative of C ~ (x). However, graph 702 is normalized so that 0.1 times the derivative of C ~ (x) is output as a value of 1 or less. The horizontal axis (x-axis) of graphs 701 and 702 is the difference in energy levels ΔE×τ. Note that "ACDF" is an abbreviation for "Approximate Cumulative Distribution Function".

[0155] Also, x = x0 is the value obtained by multiplying τ by the exact difference in the ground state levels (E0(h = 0) - E0(h = 0.1)) calculated by exactly diagonalizing the Hamiltonian. x = x1 is the value obtained by multiplying τ by the exact difference in the first excited state levels (E1(h = 0) - E1(h = 0.1)) calculated by exactly diagonalizing the Hamiltonian. Also, the "ground state level" indicates the energy level of the ground state. The "first excited state level" indicates the energy level of the first excited state.

[0156] The cumulative distribution function C ~ (x) shows a sharp increase in value at two positions. The positions where the value sharply increases in the cumulative distribution function C ~ (x) correspond to the peak positions of the derivative of C ~ (x) shown in graph 702. Graph 702 has two peaks.

[0157] The cumulative distribution function C ~The position on the right side where the value of (x) rapidly increases coincides with the position corresponding to the difference in energy levels precisely calculated for the ground state (i.e., x = x0). That is, the peak position on the right side of graph 702 coincides with x = x0. Also, the cumulative distribution function C ~ The position on the left side where the value of (x) rapidly increases coincides with the position corresponding to the difference in energy levels precisely calculated for the first excited state (i.e., x = x1). That is, the peak position on the left side of graph 702 coincides with x = x1.

[0158] Therefore, the cumulative distribution function analysis unit 170 ~ reads the value x0 of the x - coordinate corresponding to the position on the right side where the value of C * (x) rapidly increases, and by dividing x0 * by τ, ΔE0 = E0(h = 0) - E0(h = 0.1) can be obtained. Also, the cumulative distribution function analysis unit 170 ~ reads the value x1 of the x - coordinate corresponding to the position on the left side where the value of C * (x) rapidly increases, and by dividing x1 * by τ, ΔE1 = E1(h = 0) - E1(h = 0.1) can be obtained.

[0159] Note that in the binary search for the cumulative distribution function C ~ (x), the difference in energy levels for the above - mentioned ground state and the difference in energy levels for the first excited state can be distinguished as follows. The cumulative distribution function analysis unit 170 ~ identifies the interval in which C ~ (x) rapidly increases and has the largest increase width. The cumulative distribution function analysis unit 170 determines that the identified location corresponds to the difference in energy levels of the eigenstate with the largest overlap with the input state. That is, if the input state is close to the ground state, the location with the largest increase width corresponds to the difference in energy levels of the ground state. In the example of graph 701, the initial state is close to the ground state. Therefore, the cumulative distribution function analysis unit 170 can associate the right - hand position, which has the larger increase width among the two positions where C ~ (x) rapidly increases, with the difference in energy levels of the ground state.

[0160] Here, the specific values ​​of the energy levels and their differences for the above ground state are as follows: The exact value of the energy level of the ground state when h=0 is E0(h=0)=-7.711. The exact value of the energy level of the ground state when h=0.1 is E0(h=0.1)=-7.811.

[0161] The exact difference between E0(h=0) and E0(h=0.1) is ΔE0 = 0.100. The relative ratio of the difference to the absolute value of the ground state energy level is ΔE0 / |E0| ≈ 1 / 80.

[0162] Required error ε for the difference in energy levels ΔE When fixed, the required truncation order d in the method of the second embodiment is reduced to ΔE0 / |E0|≈ 1 / 80 times compared to when E0(h=0) and E0(h=0.1) are individually determined by the aforementioned SPE. Therefore, the maximum depth D(∝d) required for the quantum circuit 60 is also reduced to ΔE0 / |E0|≈ 1 / 80 times.

[0163] [Differentiation] Figure 12 shows a modified example of the quantum circuit of the second embodiment. The quantum computing system 300 may use quantum circuit 70 instead of quantum circuit 60. Quantum circuit 70 is useful when the initial state |ψ> does not have a large overlap with a single eigenstate, that is, when the initial state |ψ> is a superposition of various states.

[0164] The quantum circuit 70 includes a Hadamard gate 71, control gates 72, 73, and 74, a quantum gate 75, a Hadamard gate 76, and a measurement gate 77. Quantum gate operations are performed sequentially from left to right in the quantum circuit 70, that is, from the Hadamard gate 71 to the measurement gate 77.

[0165] The auxiliary qubit is initialized to |0>. The Hadamard gate 71 acts on the auxiliary qubit. Control gate 72 is a positive-polarity controlled unitary gate controlled by an auxiliary qubit. Control gate 72 assigns the time evolution operator e to the group of qubits corresponding to the quantum state |ψ> of the system of interest. -ikτH(λ) The following is applied. τ is the time step of the time evolution for estimating the difference in energy levels. A value for τ is predetermined for estimating the difference in energy levels. k is an integer multiplied by τ. The range of k is -d ≤ k ≤ d. As mentioned above, d is the truncation order of the discrete Fourier transform for estimating the difference in energy levels.

[0166] Control gate 73 is a negative-polarity controlled unitary gate controlled by an auxiliary qubit. Control gate 73 has a time evolution operator e applied to the qubit group of the system of interest. -ikτH(λ+δλ) To apply the effect.

[0167] Control gate 74 is a positive-polarity controlled unitary gate controlled by an auxiliary qubit. Control gate 74 has a time evolution operator e applied to the qubit group of the system of interest. -ijτ’H(λ) The following is applied. τ' is the time step size for the time evolution of the energy levels. A value for τ' is predetermined for the energy level estimation. j is an integer multiplied by τ'. The range of j is -d'≦j≦d'. d' is the truncation order of the discrete Fourier transform for energy level estimation. d,d' are determined by the truncation order determination unit 130. The method for determining d,d' will be described later. Note that the positive and negative polarities of control gates 72, 73, and 74 may be reversed.

[0168] Quantum gate 75 acts on the auxiliary qubit. Quantum gate 75 is represented as "W". W=I or W=S † Therefore, W=I means that quantum gate 75 is an I gate. W=I is equivalent to not inserting anything at the location of quantum gate 75. In other words, W=I is equivalent to quantum circuit 70 not containing quantum gate 75. W=S † This is quantum gate 75 S † This means it will be used as a gate.

[0169] The Hadamard gate 76 acts on the auxiliary qubit. Measurement 77 is a projection measurement on the computational basis for the auxiliary qubit. When W=I, the probability amplitude <ψ|e ikτH(λ+δλ) e -ijτ’H(λ) e -ikτH(λ) The real part of |ψ> is measured. W=S † In this case, probability amplitude < ψ|e ikτH(λ+δλ) e -ijτ’H(λ) e -ikτH(λ) The imaginary part of |ψ> is measured. Probability amplitude <ψ|e ikτH(λ+δλ) e -ijτ’H(λ) e -ikτH(λ) |ψ> is expressed by equation (32).

[0170]

number

[0171] The probability amplitude in equation (32) is the probability amplitude corresponding to the event that the result of applying the first time evolution operator, the second time evolution operator, and the third time evolution operator to the quantum state |ψ> of the system of interest matches the original quantum state |ψ>.

[0172] The quantum circuit control unit 120 causes the quantum computer 200 to execute the quantum circuit 70. The measurement result storage unit 110 stores the measurement results of the probability amplitude of equation (32) according to the execution of the quantum circuit 70.

[0173] The discrete Fourier transform execution unit 150 applies F^ to the measurement result of the probability amplitude. j ,F^ k Multiply by and perform the Discrete Fourier Transform. This Discrete Fourier Transform is expressed by equation (33).

[0174]

number

[0175] F^ j ,F^ k is the discrete Fourier expansion coefficient of the step function F. j ,F^k This is calculated by the Fourier expansion coefficient calculation unit 140. The result of equation (33) approximates the joint cumulative distribution function of the energy levels and the difference between the energy levels, under the assumption that δλ << 1. Joint cumulative distribution function C ~ (x,y) is expressed by equation (34).

[0176]

number

[0177] δ² = δ²(x,y) is the delta function of two variables. F² = F²(x,y) is the step function of two variables. F²(x,y) is the product of a single-variable step function F(x) and a single-variable step function F(y). That is, F²(x,y) is expressed by equation (35).

[0178]

number

[0179] The discrete Fourier transform execution unit 150 performs N in the same manner as described above. b Individual G r Calculate (x,y). G r( x,y) is G in equation (26) r =G r This is an extension of (x) to two variables. The median calculation unit 160 is N b Individual G r By calculating the median of (x,y), the joint cumulative distribution function C ~ Obtain (x,y). C ~ (x,y) is the two-dimensional cumulative distribution function.

[0180] The cumulative distribution function analysis unit 170 analyzes the simultaneous cumulative distribution function C ~ The energy level and its difference can be simultaneously estimated from the position where the value of (x,y) increases sharply. For example, (x * ,y * ) in C ~If the value of (x,y) is rapidly increasing, the cumulative distribution function analysis unit 170 calculates the difference ΔE = x * / τ is the energy level E=y * It can be identified that this corresponds to an eigenstate of / τ'. In this way, the quantum computing system 300, by using the quantum circuit 70, can calculate ΔE even if the initial state |ψ> is a superposition of various states. k The correspondence between this and the eigenstate k can be appropriately identified.

[0181] The termination order determination unit 130 determines the termination order d based on equation (36).

[0182]

number

[0183] ε ΔE_k This is the difference in energy levels ΔE k This is the required error. The termination order determination unit 130 determines the termination order d' based on equation (37).

[0184]

number

[0185] ε E_k is the energy level E k This is the required error. The time step size τ is predetermined based on equation (38).

[0186]

number

[0187] The time step size τ' is predetermined based on equation (39).

[0188]

number

[0189] τ,τ',ε ΔE_k ,ε E_k Each of these values ​​is included in the input information. This input information is input to the classical computer 100. The truncation order determination unit 130 determines τ, τ', ε included in the input information. ΔE_k ,ε E_k Based on the respective values, d and d' are calculated. In this case, d' in equation (37) can be made smaller than the truncation order expressed in equation (15). Therefore, even in the modified example, the quantum computing system 300 can reduce the depth of the quantum circuit 70.

[0190] [Comparative Example] As a comparative example to the second embodiment, a method can be considered in which E(λ) and E(λ+δλ) are individually determined using SPE.

[0191] Figure 13 shows a quantum circuit of a comparative example. The comparative example quantum circuit 80 has an Adamard gate 81, a control gate 82, a quantum gate 83, an Adamard gate 84, and a measurement gate 85. Quantum gate operations are performed sequentially from left to right in the quantum circuit 80.

[0192] The auxiliary qubit is initialized to |0>. The Hadamard gate 81 acts on the auxiliary qubit. The control gate 82 is a positive-polarity controlled unitary gate controlled by the auxiliary qubit. The control gate 82 applies the time evolution operator e to the group of qubits corresponding to the quantum state |φ0> of the system of interest. -ijτH It applies the following. Quantum gate 83 acts on the auxiliary qubit. Quantum gate 83 is represented as "W". W=I or W=S † The Hadamard gate 84 acts on the auxiliary qubit. Measurement 85 is a projection measurement on the computational basis for the auxiliary qubit.

[0193] By using quantum circuit 80, the probability amplitude < φ0|e -ijτH |φ0> is measured. If W=I, the probability amplitude <φ0|e -ijτH The real part of |φ0> is measured. W=S † In this case, probability amplitude < φ0|e-ijτH The imaginary part of |φ0> is measured.

[0194] Probability amplitude < φ0|e -ijτH Based on |φ0>, the cumulative distribution function C in equation (12) is obtained. In the comparative example, E(λ) and E(λ+δλ) are calculated individually using the quantum circuit 80. For example, E(λ) is calculated based on the probability amplitude obtained by setting H=H(λ) in the quantum circuit 80. Also, E(λ+δλ) is calculated based on the probability amplitude obtained by setting H=H(λ+δλ) in the quantum circuit 80. By taking the difference between the individually calculated E(λ) and E(λ+δλ), the difference in energy levels is calculated.

[0195] However, in the comparative example method, as expressed in equation (15), the reciprocal of the relative accuracy required for the difference in energy levels (ε) ΔE_k / ΔE k A depth D(∝d) several orders of magnitude larger than )^(-1) is required.

[0196] Therefore, the quantum computing system 300 directly estimates the energy level difference ΔE using the quantum circuit 60. When ΔE is directly estimated, the required depth D of the quantum circuit 60 is expressed by equation (9). Comparing equation (9) with equation (15), equation (9) does not contain the large factor expressed in equation (17). Thus, according to the quantum computing system 300, the depth of the quantum circuit is reduced by several orders of magnitude compared to the method of the comparative example.

[0197] Here, the execution time for each quantum gate is, for example, a few nanoseconds to several hundred nanoseconds. The greater the depth of the quantum circuit, the greater the computation time. Also, since a single quantum computation must be completed within the duration of the qubit, if the depth of the quantum circuit is excessive, the quantum computation may not be completed. Furthermore, since an error occurs with a certain probability for each quantum gate, the greater the depth of the quantum circuit, the greater the probability of an error in the output of the quantum circuit. To address these problems, the quantum computing system 300 can increase the probability of completing a quantum computation within the duration of the qubit by reducing the depth of the quantum circuit. In addition, the quantum computing system 300 can reduce the probability of an error in the output of the quantum circuit by reducing the depth of the quantum circuit.

[0198] As explained above, the quantum computing system 300 can obtain a guaranteed-accuracy estimate of the difference in energy levels for small parameter shifts in a quantum many-body system simulated on the quantum computer 200.

[0199] Furthermore, when the quantum computing system 300 estimates the difference in energy levels for a small shift in the parameters of a quantum system using a quantum computer, it can implement quantum circuits with a depth approximately equal to the reciprocal of the relative accuracy for that difference. In particular, by directly estimating the difference in energy levels rather than the energy levels themselves, the quantum computing system 300 can eliminate the dependence on the amount of small shift from the required relative accuracy and reduce the order of the Fourier components required for statistical processing. As a result, the quantum computing system 300 can reduce the depth of the quantum circuits.

[0200] In the example of estimating the difference in energy levels in the ferrodiamagnetic Heisenberg model, it was confirmed that the maximum depth required for the quantum circuit could be reduced to about 1 / 80th compared to the comparative example method. Quantum computing system 300 can reduce the depth of the quantum circuit compared to the comparative example method without degrading the estimation accuracy when estimating the difference in energy levels for small parameter shifts.

[0201] Incidentally, when using a large-scale FTQC instead of an early medium-scale FTQC, it is conceivable to use BPDE to estimate the difference in energy levels. In BPDE, the probability amplitude related to the time evolution operator is estimated using quantum circuits. In addition, in BPDE, the ancilla rotation angle is updated based on Bayesian inference.

[0202] However, using BPDE presents several problems, including the inability to separate energy levels analytically, the need for highly precise control over ancilla rotations, and the increased depth of the quantum circuit compared to phase estimation.

[0203] On the other hand, according to the quantum computing system 300 of the second embodiment, analytical energy level separation is possible. Furthermore, the quantum computing system 300 does not require ancilla rotation control. In addition, the quantum computing system 300 reduces the truncation order of the discrete Fourier transform and reduces the depth of the quantum circuit.

[0204] Thus, the quantum computing system 300 of the second embodiment can estimate the difference in energy levels more efficiently than the comparative example method, i.e., the method that estimates the difference in energy levels using the results of SPE. Furthermore, the quantum computing system 300 can estimate the difference in energy levels more efficiently than BPDE.

[0205] The quantum computing system 300 may have a device that has a quantum circuit simulator for simulating the execution of quantum circuits in the quantum computer 200, instead of the quantum computer 200. In that case, the quantum computing system 300 can perform the processing of the second embodiment using the quantum circuit simulator instead of the quantum computer 200.

[0206] The quantum computing system 300 performs, for example, the following process: The processor 101 causes the quantum computer 200 or quantum circuit simulator to execute the quantum circuit 60. The quantum circuit 60 includes, in order, an Adamard gate 61, a control gate 62, a control gate 63, an Adamard gate 65, and a measurement 66. Here, the Adamard gate 61 is the first Adamard gate. The control gate 62 is the first control gate. The control gate 63 is the second control gate. The Adamard gate 65 is the second Adamard gate.

[0207] The Hadamard gate 61 acts on the auxiliary qubit. Control gates 62 and 63 are controlled by the auxiliary qubit. Control gate 62 is first polarity. Control gate 63 is second polarity, opposite to the first polarity. Control gate 62 acts on the group of qubits corresponding to the quantum state |ψ> of the system of interest with the first time evolution operator corresponding to the first Hamiltonian, which includes predetermined parameters. Control gate 63 acts on the group of qubits with the second time evolution operator corresponding to the second Hamiltonian, which is obtained by applying a change in parameters to the first Hamiltonian. The Hadamard gate 65 acts on the auxiliary qubit. Measurement 66 is a measurement of the auxiliary qubit. The processor 101 obtains a probability amplitude corresponding to the event that the result of applying the first time evolution operator and the second time evolution operator to the quantum state |ψ> matches the original quantum state |ψ>, as a result of the execution of the quantum circuit 60 by the quantum computer 200 or the quantum circuit simulator. The processor 101 obtains the cumulative distribution function of the difference in energy levels for the first Hamiltonian and the second Hamiltonian by performing a discrete Fourier transform using the probability amplitude. The processor 101 estimates the difference based on the cumulative distribution function.

[0208] This allows the quantum computing system 300 to reduce the depth of the quantum circuit. Specifically, by directly estimating the difference in energy levels, as in the quantum computing system 300, the constraints on τ are significantly relaxed compared to the comparative example method, as shown in equation (20). That is, according to the quantum computing system 300, τ can be made several orders of magnitude larger than in the comparative example method, to the extent shown in equation (17). As a result, the quantum computing system 300 can reduce the truncation order d of the discrete Fourier transform, as shown in equations (9) and (21). There is a relationship D∝d between the truncation order d and the depth D of the quantum circuit. Therefore, by reducing the truncation order d, the depth D of the quantum circuit is reduced.

[0209] Furthermore, the difference estimated by the quantum computing system 300 can be generalized to forward difference, backward difference, central difference, and higher-order difference. The processor 101 inserts an identity gate or a phase-shift gate at a predetermined location in the quantum circuit 60. The predetermined location is, for example, the position of the quantum gate 64. The processor 101 obtains the real part of the probability amplitude from the quantum computer 200 or quantum circuit simulator by having the quantum circuit 60 with the identity gate inserted run by the quantum computer 200 or quantum circuit simulator. The processor 101 obtains the imaginary part of the probability amplitude from the quantum computer 200 or quantum circuit simulator by having the quantum circuit 60 with the phase-shift gate inserted run by the quantum computer 200 or quantum circuit simulator. As a result, the quantum computing system 300 can efficiently obtain the probability amplitude.

[0210] Furthermore, the processor 101 has a required error ε for the difference in energy levels. ΔE The processor accepts inputs for the time step size τ of the time evolution in the first time evolution operator and the second time evolution operator. The processor 101 then processes the requested error ε ΔE Based on the time step τ, we calculate the truncation order d of the discrete Fourier transform.

[0211] This allows the quantum computing system 300 to reduce the depth of the quantum circuit. As mentioned above, the quantum computing system 300 allows for a larger τ compared to the comparative example method. As a result, the truncation order d can be reduced, and the depth D of the quantum circuit is reduced.

[0212] In estimating the difference in energy levels, processor 101 uses the cumulative distribution function C ~ The value of the variable x in (x), and the cumulative distribution function C ~ The value of the variable x at the point where the value of (x) increases. * The value of the identified variable x is determined based on a predetermined algorithm. The processor 101 then processes the value of the identified variable x. * The difference is calculated by dividing by the time step width τ. This allows the quantum computing system 300 to efficiently estimate the difference in energy levels. The processor 101 calculates the value of the variable x at the point where the value of the cumulative distribution function increases. * This may be identified by a binary search, such as algorithm 501. Alternatively, the processor 101 identifies the peak position of the derivative of the cumulative distribution function and obtains the value of the variable x at that peak position, based on an algorithm that determines the value of the variable x at the point where the value of the cumulative distribution function increases. * You may specify it.

[0213] The exponents of the first time evolution operator and the exponents of the second time evolution operator include the product of the time step τ and the first integer value. The processor 101 selects from a plurality of integer values ​​(-d~d) whose absolute value is less than or equal to the truncation order d, the plurality of discrete Fourier expansion coefficients F^ used in the discrete Fourier transform. j Based on (j∈[-d,d]), the first integer value is probabilistically selected. Multiple discrete Fourier expansion coefficients F^ j (j∈[-d,d]) corresponds to multiple integer values ​​(-d~d). Specifically, processor 101 calculates the discrete Fourier expansion coefficient F^ of the order corresponding to the integer value j. j The absolute value of |F^ j With a probability proportional to |, an integer value j is selected as the first integer value.

[0214] This allows the quantum computing system 300 to efficiently acquire probability amplitudes. In other words, the quantum computing system 300 can perform the Σ of equation (27) j=-d d ...In order to find a sum of this form, instead of using all of the 2d quantum circuits, which is generally a large number, N b N s A few samples may suffice. Therefore, the number of samples obtained for the probability amplitude, i.e., the number of iterations of steps S16 to S20, is reduced. However, as a trade-off for probabilistic sampling, the processor 101 allows estimation failures with a probability less than or equal to the tolerance value ν.

[0215] The processor 101 repeatedly performs a first process of selecting a first integer value j, and a second process of having the quantum computer 200 or a quantum circuit simulator execute a quantum circuit 60 based on the first integer value j. This allows the quantum computing system 300 to efficiently sample probability amplitudes. Specifically, the quantum computing system 300 can reduce the number of repetitions between the first and second processes, i.e., the number of samples. As a result, the quantum computing system 300 can speed up the estimation of the difference in energy levels.

[0216] Furthermore, the quantum circuit 70 may further include a first-polarity control gate 74 controlled by an auxiliary qubit. The control gate 74 may also be called a third control gate. The control gate 74 is a third time evolution operator corresponding to the first Hamiltonian, and acts on the qubit group with a third time evolution operator having a time step width τ' different from that of the first and second time evolution operators. As a result of executing the quantum circuit 70 by the quantum computer 200 or a quantum circuit simulator, the processor 101 obtains a probability amplitude corresponding to the event that the result of acting the first time evolution operator, the second time evolution operator, and the third time evolution operator on the quantum state |ψ> matches the quantum state |ψ>. The processor 101 performs a discrete Fourier transform using this probability amplitude to obtain the joint cumulative distribution function of the difference in energy levels for the first Hamiltonian and the second Hamiltonian, and the energy level corresponding to the first Hamiltonian. The processor 101 estimates the difference and the energy levels corresponding to the difference and the first Hamiltonian, based on the simultaneous cumulative distribution function.

[0217] As a result, the quantum computing system 300 can calculate ΔE even if the initial state |ψ> is a superposition of various states. k and energy level E k The correspondence with the corresponding eigenstate k can be appropriately identified.

[0218] Although the embodiments have been illustrated above, the configuration of each part shown in the embodiments has the same function. It can be replaced with other things. Also, any other components or processes may be added. Furthermore, by combining any two or more configurations (features) from the embodiments described above... It's okay to have it. [Explanation of Symbols]

[0219] 1. Quantum Computing System 10 Information Processing Devices 11 Storage section 12 Processing Units 20 Quantum Computers 30 Quantum circuit 31. First Adamar Gate 32 First control gate 33 Second control gate 34 Quantum Gates 35. Second Adamar Gate 36 measurements 40. Cumulative Distribution Function S1, S2, S3, S4, S5 Steps

Claims

1. A quantum circuit is executed by a quantum computer or quantum circuit simulator, comprising: a first Hadamard gate acting on an auxiliary qubit; a first control gate of first polarity controlled by the auxiliary qubit, which acts on a group of qubits corresponding to the quantum state of the system of interest with a first time evolution operator corresponding to a first Hamiltonian including predetermined parameters; a second control gate of second polarity opposite to the first polarity, controlled by the auxiliary qubit, which acts on the group of qubits with a second time evolution operator corresponding to a second Hamiltonian obtained by applying the change in parameters to the first Hamiltonian; a second Hadamard gate acting on the auxiliary qubit; and measurement of the auxiliary qubit. As a result of executing the quantum circuit by the quantum computer or the quantum circuit simulator, a probability amplitude corresponding to the event that the result of applying the first time evolution operator and the second time evolution operator to the quantum state matches the quantum state is obtained. By performing a discrete Fourier transform using the aforementioned probability amplitude, we obtain the cumulative distribution function of the difference in energy levels for the first Hamiltonian and the second Hamiltonian. The difference is estimated based on the cumulative distribution function. An energy level difference estimation program that uses a computer to perform the processing.

2. The system accepts inputs of the required error for the difference and the time step size for the time evolution in the first time evolution operator and the second time evolution operator. Based on the required error and the time step size, the truncation order of the discrete Fourier transform is calculated. An energy level difference estimation program according to claim 1, which causes the computer to perform the processing.

3. In the estimation of the difference, the values ​​of the variable in the cumulative distribution function that increase in the cumulative distribution function are identified based on a predetermined algorithm, and the difference is calculated by dividing the identified values ​​of the variable by the time step size. The energy level difference estimation program according to claim 2, which causes the computer to perform the processing.

4. The exponents of the first time evolution operator and the exponents of the second time evolution operator include the product of the time step size and the first integer value. From among a plurality of integer values ​​whose absolute value is less than or equal to the truncation order, the first integer value is probabilistically selected based on a plurality of discrete Fourier expansion coefficients corresponding to the plurality of integer values ​​used in the discrete Fourier transform. The energy level difference estimation program according to claim 2, which causes the computer to perform the processing.

5. The process of selecting the first integer value and the process of causing the quantum computer or quantum circuit simulator to execute the quantum circuit based on the first integer value are repeated. The energy level difference estimation program according to claim 4, which causes the computer to perform the processing.

6. The quantum circuit further includes a third control gate of the first polarity, which is controlled by the auxiliary qubit. The third control gate is a third time evolution operator corresponding to the first Hamiltonian, which has a time step size for time evolution different from that of the first time evolution operator and the second time evolution operator, and applies this third time evolution operator to the qubit group. As a result of executing the quantum circuit by the quantum computer or the quantum circuit simulator, the probability amplitude corresponding to the event that the result of applying the first time evolution operator, the second time evolution operator, and the third time evolution operator to the quantum state matches the quantum state is obtained. By performing the Discrete Fourier Transform using the aforementioned probability amplitude, the simultaneous cumulative distribution function of the difference and the energy level corresponding to the first Hamiltonian is obtained. Based on the aforementioned joint cumulative distribution function, the difference and the energy levels corresponding to the difference and the first Hamiltonian are estimated. An energy level difference estimation program according to claim 1, which causes the computer to perform the processing.

7. An identity gate or a phase shift gate is inserted at a predetermined location in the quantum circuit. By having the quantum circuit into which the identity gate is inserted be executed by the quantum computer or the quantum circuit simulator, the real part of the probability amplitude is obtained from the quantum computer or the quantum circuit simulator. By having the quantum circuit into which the phase-shift gate is inserted run on the quantum computer or the quantum circuit simulator, the imaginary part of the probability amplitude is obtained from the quantum computer or the quantum circuit simulator. An energy level difference estimation program according to claim 1, which causes the computer to perform the processing.

8. Computers A quantum circuit is executed by a quantum computer or quantum circuit simulator, comprising: a first Hadamard gate acting on an auxiliary qubit; a first control gate of first polarity controlled by the auxiliary qubit, which acts on a group of qubits corresponding to the quantum state of the system of interest with a first time evolution operator corresponding to a first Hamiltonian including predetermined parameters; a second control gate of second polarity opposite to the first polarity, controlled by the auxiliary qubit, which acts on the group of qubits with a second time evolution operator corresponding to a second Hamiltonian obtained by applying the change in parameters to the first Hamiltonian; a second Hadamard gate acting on the auxiliary qubit; and measurement of the auxiliary qubit. As a result of executing the quantum circuit by the quantum computer or the quantum circuit simulator, a probability amplitude corresponding to the event that the result of applying the first time evolution operator and the second time evolution operator to the quantum state matches the quantum state is obtained. By performing a discrete Fourier transform using the aforementioned probability amplitude, we obtain the cumulative distribution function of the difference in energy levels for the first Hamiltonian and the second Hamiltonian. The difference is estimated based on the cumulative distribution function. Energy level difference estimation method.

9. A device having a quantum computer or quantum circuit simulator that executes a quantum circuit including: a first Hadamard gate acting on an auxiliary qubit; a first control gate of first polarity controlled by the auxiliary qubit, which acts on a group of qubits corresponding to the quantum state of the system of interest with a first time evolution operator corresponding to a first Hamiltonian including predetermined parameters; a second control gate of second polarity opposite to the first polarity controlled by the auxiliary qubit, which acts on the group of qubits with a second time evolution operator corresponding to a second Hamiltonian obtained by applying the change in parameters to the first Hamiltonian; a second Hadamard gate acting on the auxiliary qubit; and measurement of the auxiliary qubit. An information processing device that, as a result of executing the quantum circuit by the quantum computer or the quantum circuit simulator, obtains a probability amplitude corresponding to the event in which the result of applying the first time evolution operator and the second time evolution operator to the quantum state matches the quantum state, performs a discrete Fourier transform using the probability amplitude to obtain a cumulative distribution function of the difference in energy levels for the first Hamiltonian and the second Hamiltonian, and estimates the difference based on the cumulative distribution function, A quantum computing system having [a certain feature].

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