Quantum calculation support program, quantum calculation support method, and information processing apparatus

Abstract

To improve the calculation efficiency of the thermal equilibrium expectation value of a physical quantity at a finite temperature.SOLUTION: An information processing apparatus 10 expands an imaginary time evolution equation into equations for each of a plurality of degrees, and generates a plurality of sets of degrees obtained by extracting the degrees twice from the plurality of degrees. The information processing apparatus 10, for each generated set, generates a quantum circuit 2 showing a procedure for the quantum calculation of the value of a physical quantity obtained by partial imaginary time evolution based on the equation of the first degree included in the set and the equation of the second degree included in the set. The information processing apparatus 10, for each generated set, makes an output state after the quantum calculation according to the quantum circuit 2 corresponding to the set into an input state in the subsequent quantum calculation, and allows a quantum computer to repeatedly execute the quantum calculation based on the quantum circuit 2 until the value of the physical quantity obtained from the result of the quantum calculation converges. Then, based on the value of the physical quantity after the convergence for each generated set, the information processing apparatus 10 calculates the thermal equilibrium expectation value of the physical quantity at a finite temperature.SELECTED DRAWING: Figure 1

Description

Technical Field

[0001] The present invention relates to a quantum computing support program, a quantum computing support method, and an information processing apparatus.

Background Art

[0002] Due to environmental noise and the like, errors are likely to occur in the states of qubits in a quantum computer. In the error correction technology of qubits (quantum error correction), information is made redundant and encoded. To realize quantum error correction at a practical level, about one million qubits are required. On the other hand, currently realized quantum computers are limited to small and medium-sized quantum computers (Noisy Intermediate Scale Quantum Computer, NISQ) with a maximum of about several hundred qubits that cannot perform error correction.

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

[0004] One of the fields where calculations on a quantum computer are effective is the calculation of physical quantities of a quantum system. Particularly significant in practical applications is the calculation of the thermal equilibrium expectation value at a finite temperature. To obtain the thermal equilibrium expectation value at a finite temperature, for example, calculations of expectations regarding a population of quantum states that represent the thermal equilibrium state are performed. One of the populations that represents the thermal equilibrium state at a finite temperature is the canonical statistical population (canonical ensemble). As a technique for efficiently generating the canonical statistical population, the METTS (Minimally Entangled Typical Thermal State) algorithm is known.

[0005] The METTS algorithm was originally devised as a computational method to be executed on classical computers, but an equivalent method can also be executed on a quantum computer. An algorithm equivalent to METTS that can be executed on a quantum computer is particularly called QMETTS (Quantum METTS).

[0006] In the METTS algorithm, the imaginary-time evolution algorithm is used to realize the Boltzmann weights (similarly in QMETTS). One of the methods of quantum imaginary-time evolution to realize imaginary-time evolution on a quantum computer is a method based on the Linear Combination of Unitaries (LCU). LCU is a quantum computing method that expands the imaginary-time evolution into a polynomial expansion, represents each-degree polynomial by a quantum circuit, and calculates physical quantities by the linear combination of them all.

[0007] When realizing LCU in early FTQC, the target physical quantity is calculated based on the calculation results obtained by applying each of a plurality of subcircuits extracted from the original quantum circuit to qubits. Since the execution target is a small-scale subcircuit, the calculation is possible even in early FTQC.

[0008] As a technology related to quantum computers, for example, a method for obtaining the excited state of a Hamiltonian has been proposed. Also, for problems where executable solutions cannot be obtained in FALQON (Feedback-based ALgorithm for Quantum OptimizatioN), a combinatorial optimization calculation method that enables obtaining executable solutions has been proposed. Furthermore, a quantum computer that improves quantum optimization by using peripheral data has also been proposed.

Prior Art Documents

Patent Documents

[0009]

Patent Document 1

Patent Document 2

Patent Document 3

Non-Patent Document

[0010]

Non-Patent Document 1

Non-Patent Document 2

Summary of the Invention

Problems to be Solved by the Invention

[0011] By using LCU, it becomes possible to calculate the thermal equilibrium expectation value of physical quantities at finite temperature by early FTQC. However, in order to calculate the thermal equilibrium expectation value of physical quantities at finite temperature, it is necessary to randomly sample a huge number of states from an exponentially large number of states with respect to the size of the target system of calculation, and the calculation efficiency is poor.

[0012] In one aspect, the present invention aims to improve the calculation efficiency of the thermal equilibrium expectation value of physical quantities at finite temperature.

Means for Solving the Problems

[0013] In one proposal, a quantum computing support program for causing a computer to execute the following processing is provided. The computer expands the formula of imaginary-time evolution for calculating the thermal equilibrium expectation value of the physical quantity of the system to be calculated at a finite temperature into formulas for each of a plurality of orders. The computer generates a plurality of sets of orders obtained by extracting the orders twice from the plurality of orders. For each generated set, the computer generates a quantum circuit showing the procedure of quantum calculation of the value of the physical quantity obtained by partial imaginary-time evolution based on the formula of the first order included in the set and the formula of the second order included in the set. For each generated set, the computer repeatedly executes quantum calculation based on the quantum circuit on the quantum computer, using the output state after the quantum calculation according to the quantum circuit corresponding to the set as the input state in the subsequent quantum calculation, until the value of the physical quantity obtained from the result of the quantum calculation converges. Then, the computer calculates the thermal equilibrium expectation value of the physical quantity at the finite temperature based on the converged value of the physical quantity for each generated set.

Advantages of the Invention

[0014] According to one aspect, the calculation efficiency of the thermal equilibrium expectation value of the physical quantity at the finite temperature is improved.

Brief Description of the Drawings

[0015]

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Mode for Carrying Out the Invention

[0016] Hereinafter, this embodiment will be described with reference to the drawings. Note that multiple embodiments can be combined and implemented within a non - conflicting range. 〔First Embodiment〕 The first embodiment is a quantum - computing support method for efficiently calculating the thermal equilibrium expectation value of a physical quantity at a finite temperature.

[0017] FIG. 1 is a diagram showing an example of the quantum - computing support method according to the first embodiment. In FIG. 1, an information processing apparatus 10 for implementing the quantum - computing support method according to the first embodiment is shown. The information processing apparatus 10 can implement the quantum - computing support method according to the first embodiment by executing, for example, a quantum - computing support program.

[0018] The information processing apparatus 10 is connected to a quantum computer 1. The information processing apparatus 10 causes the quantum computer 1 to perform quantum computing. Further, the information processing apparatus 10 acquires the measurement result by quantum computing from the quantum computer 1.

[0019] The information processing apparatus 10 includes a storage unit 11 and a processing unit 12. The storage unit 11 is, for example, a memory or a storage device included in the information processing apparatus 10. The processing unit 12 is, for example, a processor or an arithmetic circuit included in the information processing apparatus 10. The information processing apparatus 10 is, for example, a classical computer.

[0020] The processing unit 12 calculates the thermal equilibrium expectation value of the physical quantity at a finite temperature according to the following procedure. The processing unit 12 expands the formula for imaginary time evolution for calculating the thermal equilibrium expectation value of the physical quantity of the system to be calculated (the system of interest) at a finite temperature into formulas for a plurality of orders respectively. The specific formula for expanding the imaginary time evolution will be described later (see Equation (2)). As a result, a plurality of n-th order (n is the order) formulas such as a 0-th order formula, a 1st order formula, and a 2nd order formula are obtained.

[0021] Next, the processing unit 12 generates a plurality of pairs of orders (a, b) obtained by extracting the order twice from the plurality of orders (a and b are integers indicating the extracted orders). Next, for each generated pair, the processing unit 12 generates a quantum circuit 2 showing the procedure of quantum calculation of the value of the physical quantity obtained by partial imaginary time evolution based on the formula of the first order included in the pair and the formula of the second order included in the pair. The details of the formula for calculating the value of the physical quantity will be described later (see Equation (1)). For example, the processing unit 12 generates a quantum circuit 2 having Hermiticity.

[0022] The quantum circuit 2 does not output the state after the imaginary time evolution, but outputs the state reflecting only the contribution of the extracted pair of orders. Therefore, the quantum circuit 2 for each pair of orders can also be considered as a partial circuit with respect to the quantum circuit showing the entire imaginary time evolution.

[0023] The quantum circuit 2 includes a first unitary gate showing the calculation of the unitary matrix corresponding to the formula of the first order and a second unitary gate showing the calculation of the unitary matrix corresponding to the formula of the second order. The first unitary gate and the second unitary gate act on the qubits of the system of interest according to the state of the auxiliary qubits used as the control qubits. By performing a predetermined conditioning on the first unitary gate and the second unitary gate, Hermiticity can be expressed in the quantum circuit 2.

[0024] Furthermore, for each generated set of orders, the processing unit 12 causes the quantum computer 1 to repeatedly execute quantum computation based on the quantum circuit 2 until the values of the physical quantities obtained from the results of the quantum computation converge. When causing the quantum computation to be executed, the processing unit 12 uses the output state after the quantum computation in the quantum circuit 2 corresponding to the set as the input state in the subsequent quantum computation. For example, the processing unit 12 uses the projective measurement result of the computational basis of the output state after the computation of the quantum circuit 2 based on the input state in the N-th (N is a natural number) quantum computation as the input state in the (N + 1)-th quantum computation.

[0025] The processing unit 12 stores the values 3a, 3b,... of the physical quantities at the time of convergence for each generated set of orders, for example, in the storage unit 11. Then, the processing unit 12 calculates the thermal equilibrium expectation value of the physical quantity at a finite temperature based on the values 3a, 3b,... of the physical quantities after convergence for each generated set of orders. For example, the processing unit 12 calculates the weighted average of the values 3a, 3b,... of the physical quantities after convergence for each generated set of orders. The obtained weighted average becomes the thermal equilibrium expectation value of the physical quantity at a finite temperature.

[0026] In this way, by using the output state after the quantum computation in the quantum circuit 2 corresponding to the generated set of orders as the input state in the subsequent quantum computation, the input state in the case of repeatedly calculating the value of the physical quantity based on the quantum circuit 2 becomes a statistical ensemble suitable for the quantum circuit 2.

[0027] A statistical ensemble suitable for the quantum circuit 2 is a statistical ensemble that can cause the value of the physical quantity calculated based on the quantum circuit 2 to converge early. For example, it has been confirmed that by generating a quantum circuit 2 having Hermiticity, it becomes a statistical ensemble that can cause the value of the physical quantity calculated based on the quantum circuit 2 to converge early. Since an appropriate statistical ensemble is automatically generated, the value of the physical quantity obtained by the quantum computation converges early and the processing is made more efficient.

[0028] In addition, in the process of generating a plurality of degree pairs, the processing unit 12 generates, for example, a degree pair of combinations obtained by extracting degrees twice with duplication allowed from among a plurality of degrees. Since duplication of degrees is allowed, for example, a pair of 1st degree and 1st degree, and a pair of 2nd degree and 2nd degree can also be extracted. Also, the extraction of degrees is a combination that does not consider the order, and a pair of m-th degree and n-th degree (m and n are integers) and a pair of n-th degree and m-th degree are not extracted redundantly. Thereby, the number of degree pairs to be generated can be reduced, and the efficiency of the process can be improved.

[0029] In addition, when the errors of the quantum computer 1 cannot be sufficiently corrected, considering the extraction order, a pair of (m,n) (m-th degree first, n-th degree second) and a pair of (n,m) (n-th degree first, m-th degree second) may be extracted redundantly. When the pair of (m,n) is extracted, the quantum circuit 2 is a circuit that first acts a unitary gate corresponding to the m-th degree expression on the target quantum bit, and then acts a unitary gate corresponding to the n-th degree expression. Conversely, when the pair of (n,m) is extracted, the quantum circuit 2 is a circuit that first acts a unitary gate corresponding to the n-th degree expression on the target quantum bit, and then acts a unitary gate corresponding to the m-th degree expression. Thereby, the calculation accuracy when using the quantum computer 1 whose errors cannot be completely corrected can be improved.

[0030] 〔Second Embodiment〕 The second embodiment is a quantum computing system that efficiently calculates the thermal equilibrium expectation value of a physical quantity at a finite temperature.

[0031] FIG. 2 is a diagram showing an example of the configuration of the quantum computing system. The quantum computing system 300 is a hybrid computer system in which a classical computer 100 and a quantum computer 200 operate in cooperation. The classical computer 100 is also called a Neumann type computer.

[0032] A classical computer 100 has a terminal device 400 connected via a network 20. The terminal device 400 is a computer used by a user who requests quantum computing by a quantum computing system 300. The classical computer 100 receives a quantum circuit from the terminal device 400. The quantum circuit indicates the order of operations on quantum bits by the arrangement of elements such as gates. A quantum bit is a bit capable of expressing a superposition state of a state of "0" and a state of "1".

[0033] The classical computer 100 gives an instruction for controlling quantum bits to the quantum computer 200 according to the quantum circuit received from the terminal device 400. Also, the classical computer 100 acquires the measurement result of each quantum bit from the quantum computer 200.

[0034] The quantum computer 200 has a plurality of quantum bits and devices for operating each of the plurality of quantum bits. The plurality of quantum bits included in the quantum computer 200 are realized by, for example, a superconducting quantum device. Also, the quantum bits may be realized by a quantum device of another method such as an ion trap method.

[0035] FIG. 3 is a diagram showing an example of the hardware of a classical computer. The classical computer 100 is controlled as a whole by a processor 101. A memory 102 and a plurality of peripheral devices are connected to the processor 101 via a bus 109. The processor 101 may be a multiprocessor. The processor 101 is, for example, a CPU (Central Processing Unit), an MPU (Micro Processing Unit), or a DSP (Digital Signal Processor). At least a part of the functions realized by the processor 101 executing a program may be realized by an electronic circuit such as an ASIC (Application Specific Integrated Circuit) or a PLD (Programmable Logic Device).

[0036] The memory 102 is used as the main memory device of the classical computer 100. At least a part of the OS (Operating System) program and application programs to be executed by the processor 101 are temporarily stored in the memory 102. Also, various data used for the processing by the processor 101 are stored in the memory 102. As the memory 102, for example, a volatile semiconductor memory device such as RAM (Random Access Memory) is used.

[0037] Peripheral devices connected to the bus 109 include a storage device 103, a GPU (Graphics Processing Unit) 104, an input interface 105, an optical drive device 106, a device connection interface 107, and a network interface 108.

[0038] The storage device 103 writes and reads data electrically or magnetically to and from the built-in recording medium. The storage device 103 is used as the auxiliary storage device of the classical computer 100. The OS program, application programs, and various data are stored in the storage device 103. Note that as the storage device 103, for example, an HDD (Hard Disk Drive) or an SSD (Solid State Drive) can be used.

[0039] The GPU 104 is an arithmetic device that performs image processing. The GPU 104 is an example of a graphic controller. A monitor 21 is connected to the GPU 104. The GPU 104 displays an image on the screen of the monitor 21 according to an instruction from the processor 101. As the monitor 21, there are a display device using organic EL (Electro Luminescence), a liquid crystal display device, and the like.

[0040] The input interface 105 is connected to a keyboard 22 and a mouse 23. The input interface 105 transmits signals sent from the keyboard 22 or the mouse 23 to the processor 101. Note that the mouse 23 is an example of a pointing device, and other pointing devices can also be used. Other pointing devices include touch panels, tablets, touch pads, trackballs, and the like.

[0041] The optical drive device 106 reads data recorded on the optical disc 24 or writes data to the optical disc 24 using a laser beam or the like. The optical disc 24 is a portable recording medium on which data is recorded so as to be readable by light reflection. Examples of the optical disc 24 include DVDs (Digital Versatile Discs), DVD-RAMs, CD-ROMs (Compact Disc Read Only Memories), CD-Rs (Recordable) / RWs (ReWritable).

[0042] The device connection interface 107 is a communication interface for connecting peripheral devices to the classic computer 100. For example, a memory device 25 and a memory reader / writer 26 can be connected to the device connection interface 107. The memory device 25 is a recording medium equipped with a communication function with the device connection interface 107. The memory reader / writer 26 is a device that writes data to the memory card 27 or reads data from the memory card 27. The memory card 27 is a card-type recording medium.

[0043] Network interface 108 is connected to network 20. Network interface 108 transmits and receives data to and from other computers or communication devices via network 20. Network interface 108 is a wired communication interface that is connected by cable to a wired communication device such as a switch or router. Network interface 108 may also be a wireless communication interface that is communicatively connected by radio waves to a wireless communication device such as a base station or access point.

[0044] Classical computer 100 can realize the processing functions of the second embodiment by the above hardware. Note that the information processing apparatus 10 shown in the first embodiment can also be realized by hardware similar to classical computer 100 shown in FIG. 3.

[0045] Classical computer 100 realizes the processing functions of the second embodiment by executing a program recorded on a computer-readable recording medium, for example. Programs describing the processing content to be executed by classical computer 100 can be recorded on various recording media. For example, a program to be executed by classical computer 100 can be stored in storage device 103. Processor 101 loads at least a part of the program in storage device 103 into memory 102 and executes the program. Programs to be executed by classical computer 100 can also be recorded on portable recording media such as optical disk 24, memory device 25, and memory card 27. The program stored in the portable recording medium becomes executable after being installed in storage device 103 under the control of processor 101, for example. Also, processor 101 can directly read and execute the program from the portable recording medium.

[0046] Next, a method for calculating the thermal equilibrium expectation value of a physical quantity of a quantum system at a finite temperature will be described in detail. One of the important application areas of the quantum computer 200 is the calculation of physical quantities of a quantum system. Particularly meaningful in practical applications is the calculation of the thermal equilibrium expectation value at a finite temperature. Here, the finite temperature refers to a temperature other than absolute zero.

[0047] When obtaining the thermal equilibrium expectation value at a finite temperature, for example, the expectation value of a physical quantity regarding a population of quantum states that represents a thermal equilibrium state is calculated. As a population of quantum states that represents a thermal equilibrium state at a finite temperature, there is a canonical statistical ensemble. The canonical statistical ensemble is a population of quantum states in which eigenstates |E> having an energy eigenvalue E appear according to a probability distribution called the Boltzmann weight.

[0048] When expressing the canonical statistical ensemble ρ by an equation, it becomes "ρ = Σ E (e -βE / Z)|E><E|". The "e -βE / Z" in this equation is the Boltzmann weight. The thermal equilibrium expectation value of a physical quantity regarding the canonical statistical ensemble <o>is, " <o>=Σ E (e -βE / Z)<E|O|E>”. Here, Z is the partition function “Z = Σ E e -βE ”. β is the inverse temperature (the reciprocal of the temperature).

[0049] The canonical statistical ensemble can be efficiently generated by the METTS algorithm. When the classical computer 100 generates the canonical statistical ensemble using METTS, the processing is performed in the following procedure. 1. The classical computer 100 selects an input state from among the computational bases. 2. The classical computer 100 executes the imaginary-time evolution algorithm to realize the Boltzmann weight. 3. The classical computer 100 calculates the expected value of the physical quantity to be obtained. 4. The classical computer 100 calculates the probability distribution corresponding to the projective measurement regarding the computational basis of the output state in order to generate the probability distribution following the Boltzmann weight. 5. The classical computer 100 repeats steps 2 to 4, using the state probabilistically selected according to the probability distribution obtained in step 4 as the next input state.

[0050] The population of quantum states used as the input state in the repetition of steps 2 to 4 constitutes the canonical statistical ensemble. That is, in the case of the METTS algorithm, the canonical statistical ensemble is automatically generated in the process of calculating the expected value of the physical quantity.

[0051] Such a METTS algorithm is a computational method assuming execution only by the classical computer 100, but in the case of the quantum computing system 300, a method called QMETTS, which is equivalent to METTS, can be executed.

[0052] Also in QMETTS, imaginary-time evolution is to be performed to realize the Boltzmann weight. There are various methods for quantum imaginary-time evolution that realizes imaginary-time evolution in the quantum computer 200.

[0053] As effective quantum imaginary-time evolution methods in NISQ, there are variational imaginary-time evolution, restricted imaginary-time evolution, stochastic imaginary-time evolution, etc. Variational imaginary-time evolution is a method of variably optimizing the parameters of a quantum circuit so as to reproduce the imaginary-time evolution. Restricted imaginary-time evolution is a method of determining the coefficients of simple quantum gates from equations so as to reproduce the imaginary-time evolution. Stochastic imaginary-time evolution is a method of post-selecting only events in which the added auxiliary qubits satisfy specific conditions.

[0054] As effective quantum imaginary-time evolution methods in FTQC, there are methods based on the quantum singular value transformation algorithm, methods based on LCU, etc. The method based on the quantum singular value transformation algorithm is a method of approximately implementing the imaginary-time evolution by polynomially transforming the eigenvalues of the Hamiltonian. LCU is a method of polynomially expanding the imaginary-time evolution, representing each-degree polynomial by a quantum circuit, and obtaining the linear combination of them all. LCU can be realized even in relatively small-scale early FTQC.

[0055] In the quantum computing system 300 according to the second embodiment, the thermal equilibrium expectation value of the physical quantity of the quantum system at a finite temperature is calculated by QMETTS involving the imaginary-time evolution by LCU. The quantum circuit for the imaginary-time evolution can be executed by dividing it into a plurality of subcircuits with a small number of qubits used, and it has good compatibility with early FTQC.

[0056] FIG. 4 is a diagram showing a first example (first implementation method) of a quantum circuit for realizing LCU. In the quantum circuit 30, a plurality of qubits representing the state |ψ> of the system of interest and gate operations on the auxiliary qubits (each qubit with an initial state of |0>) are shown.

[0057] Unitary gates 33a, 33b, ···, 33k corresponding to the polynomials for each degree when the imaginary-time evolution is polynomially expanded are arranged on the qubits of the system of interest. Gate operations of a predetermined unitary gate 31 are performed on the auxiliary qubits.

[0058] Each of the unitary gates 33a, 33b, ···, 33k in the target system uses an auxiliary qubit as a control qubit, and is a gate that acts when the state of the control qubit satisfies a predetermined condition. The auxiliary qubits that serve as the control qubits for each of the unitary gates 33a, 33b, ···, 33k are indicated by white circles or black circles. A white circle has a negative polarity and indicates that a gate operation on the target qubit (qubit in the target system) is applied when the state is "0". A black circle has a positive polarity and indicates that a gate operation on the target qubit (qubit in the target system) is applied when the state is "1".

[0059] For each of the unitary gates 33a, 33b, ···, 33k, when the states of all the control qubits are in the state for which the gate operation acts, a gate operation on the qubit in the target system is performed according to the corresponding unitary gate.

[0060] After the gate operations of the unitary gates 33a, 33b, ···, 33k, a gate operation of the unitary gate 32 is performed on the auxiliary qubit. The quantum circuit 30 realizes a calculation in which the quantum circuits of the respective components after polynomial expansion are sequentially applied in the form of a controlled unitary and linearly combined. In the quantum circuit 30, in addition to the qubits in the target system that are the operation targets of the unitary gates 33a, 33b, ···, 33k, a large number of auxiliary qubits are used. Therefore, the number of qubits to be used increases.

[0061] FIG. 5 is a diagram showing a second example (second implementation method) of a quantum circuit for realizing an LCU. The sub-circuit 40 is a simplified quantum circuit obtained by extracting gate operations corresponding to two of the degrees when the formula of imaginary time evolution is polynomially expanded. In the sub-circuit 40, one auxiliary qubit is used. First, a gate operation of the Hadamard gate 41 is performed on the auxiliary qubit.

[0062] When the state of the auxiliary qubit superposed by the Hadamard gate 41 is |0>, the gate operation of the unitary gate 43a corresponding to one of the two selected degrees of the qubit of interest is performed. Next, when the state of the auxiliary qubit is |1>, the gate operation of the unitary gate 43b corresponding to the other of the two selected degrees of the qubit of interest is performed. After the two unitary gates 43a and 43b, the gate operation of the Hadamard gate 42 is performed on the auxiliary qubit.

[0063] By using the subcircuit 40, it becomes possible to extract two-component quantum circuits, operate on the simplified subcircuits, calculate physical quantities, and then calculate the linear combination of measurement results using the classical computer 100 later.

[0064] As shown in FIGS. 4 and 5, there are two implementation methods for the LCU. These implementation methods have their own advantages and disadvantages in terms of performance and the load on implementation resources. Therefore, an appropriate implementation method is used considering the advantages and disadvantages. In particular, in implementing quantum imaginary time evolution, there are the following differences between the first implementation method (coherent superposition of all degrees) shown in FIG. 4 and the second implementation method (extraction of contributions of two degrees) shown in FIG. 5.

[0065] [Suitable Quantum Computer] In the first implementation method, execution on a large-scale FTQC is required. On the other hand, in the second implementation method, it is possible to implement even with a relatively small-scale FTQC (early FTQC).

[0066] [Depth of Circuit] In the first implementation method, the depth of the circuit is the sum of the depths of each unitary gate of all degrees, and the overall depth of the circuit is deep. On the other hand, in the second implementation method, the depth of the circuit is the sum of the depths of the two extracted-degree unitary gates, and the depth of the circuit is shallower than that of the first implementation method.

[0067] [Implementation Resources] In the first implementation method, a large number of non-Clifford gates are used, and a large amount of hardware resources are consumed (resource-intensive). On the other hand, in the second implementation method, the number of non-Clifford gates is small, and less hardware resources are required compared to the first implementation method (resource-relatively light).

[0068] [Number of quantum circuits] In the first implementation method, one quantum circuit is used. In the second implementation method, the number of quantum circuits is equal to the number of sets of extracted degrees. When multiple quantum circuits are used as in the second implementation method, it is also possible to execute quantum calculations based on the quantum circuits in parallel processing.

[0069] [Method for realizing linear combination] In the first implementation method, a linear combination is realized by coherent superposition. On the other hand, in the second implementation method, a linear combination is realized by summing the results of each sub-circuit with a classical computer 100.

[0070] [Output state] In the first implementation method, the state after imaginary time evolution is output. In the second implementation method, the state with a part of imaginary time evolution applied is output.

[0071] [Compatibility with QMETTS] In the first implementation method, a thermal equilibrium state is generated by projective measurement, so the compatibility with QMETTS is good. In the second implementation method, a thermal equilibrium state is not generated even by projective measurement, and the compatibility with QMETTS is not good.

[0072] The differences between the first implementation method and the second implementation method are as above. Here, it takes decades to realize large-scale FTQC. Therefore, it is realistic to first realize it with the second implementation method that can also be implemented with relatively small-scale FTQC.

[0073] In the second implementation method, since the output state of the quantum circuit is not the state itself after imaginary-time evolution, a thermal equilibrium state is not generated even if projective measurement is performed in the manner of QMETTS. In order to calculate the thermal equilibrium expectation value of a physical quantity in the second implementation method, a huge number of states must be randomly sampled from an exponentially large number of states with respect to the size of the system, and calculations must be performed for each sample, resulting in poor computational efficiency.

[0074] Here, it is noted that it is not essential to generate a canonical statistical ensemble that describes the thermal equilibrium state in order to calculate the thermal equilibrium expectation value of a physical quantity at finite temperature. Therefore, in the quantum computing system 300, the above-described second implementation method is improved, and for each of the subcircuits from which the contributions of two-degree sets are extracted, a statistical ensemble suitable for that subcircuit is efficiently generated, and a value indicating the contribution of that subcircuit is obtained.

[0075] FIG. 6 is a diagram showing an example of an efficient calculation method for the thermal equilibrium expectation value of a physical quantity at finite temperature. In the quantum computing system 300, the classical computer 100 polynomially expands the imaginary-time evolution to obtain polynomials for each degree. The polynomials for each degree are represented by unitary matrices U1, U2, U3, U4, U5, U6, ···.

[0076] The classical computer 100 extracts a plurality of two-degree sets from among the plurality of degrees of the expansion destination by polynomial expansion. Then, the classical computer 100 generates subcircuits 51, 52, 53, ··· for each two-degree set.

[0077] The subcircuit 51 shows gate operations on a plurality of qubits representing the system of interest and one auxiliary qubit. Let the input state of the plurality of qubits representing the system of interest be |ψ>, and the input state of the auxiliary qubit be |0>.

[0078] In the partial circuit 51, first, a Hadamard gate 51a is arranged on the auxiliary qubit. Next, a unitary gate 51b controlled by the auxiliary qubit as a negative-polarity control qubit is arranged on the qubit of the target system. Further, a unitary gate 51c controlled by the auxiliary qubit as a positive-polarity control qubit is arranged on the qubit of the target system. The two unitary gates 51b and 51c correspond to each of the two extracted degrees, and are quantum circuits for causing the quantum computer 200 to calculate polynomials of the corresponding degrees.

[0079] Next to the unitary gate 51c, a Hadamard gate 51d is arranged on the auxiliary qubit. And in the partial circuit 51, measurements 51e and 51f of the states of the auxiliary qubit and the qubit of the target system are shown respectively.

[0080] Regarding the partial circuit 52 as well, a Hadamard gate 52a on the auxiliary qubit, two unitary gates 52b and 52c on the qubit representing the state of the target system, and a Hadamard gate 52d on the auxiliary qubit are arranged. And in the partial circuit 52, measurements 52e and 52f of the states of the auxiliary qubit and the qubit of the target system are shown respectively.

[0081] Regarding the partial circuit 53 as well, a Hadamard gate 53a on the auxiliary qubit, two unitary gates 53b and 53c on the qubit representing the state of the target system, and a Hadamard gate 53d on the auxiliary qubit are arranged. And in the partial circuit 53, measurements 53e and 53f of the states of the auxiliary qubit and the qubit of the target system are shown respectively.

[0082] The classical computer 100 causes the quantum computer 200 to repeatedly execute quantum calculations corresponding to the partial circuits 51, 52, 53, ··· corresponding to each set of degrees until the physical quantity obtained from the measurement results converges. If the physical quantity does not converge, the classical computer 100 causes the quantum computer 200 to perform a projective measurement in the computational basis of the output state of the qubit of the target system, and uses the measurement result as the input state in the quantum calculation of the next iteration step.

[0083] By using the result of the projective measurement of the output state as the input state in the next step in this way, a statistical population suitable for quantum computation corresponding to the extracted degree combination is generated as the input state. For example, the input state of the partial circuit 51 is a statistical population suitable for the calculation of the unitary matrices U1 and U2 corresponding to each of the extracted degrees. The input state of the partial circuit 52 is a statistical population suitable for the calculation of the unitary matrices U3 and U4 corresponding to each of the extracted degrees. The input state of the partial circuit 53 is a statistical population suitable for the calculation of the unitary matrices U5 and U6 corresponding to each of the extracted degrees.

[0084] The classical computer 100 calculates the thermal equilibrium expectation value at a finite temperature based on the physical quantities at the convergence of each of the partial circuits 51, 52, 53, ···. In this way, the calculation of the thermal equilibrium expectation value at a finite temperature is performed by efficiently and automatically generating an input state that becomes an appropriate statistical population.

[0085] Hereinafter, the principle of the calculation method shown in FIG. 6 will be described. Although the overall action of each of the partial circuits 51, 52, 53, ··· shown in FIG. 6 is complex, a basic symmetry called Hermiticity appears under certain conditions. To have Hermiticity means that the matrix representing the operation is equal to the Hermitian conjugate of that matrix. By utilizing this Hermiticity, the measurement result for the output state is linked to the generation of the statistical population.

[0086] FIG. 7 is a diagram showing an example of the conditions under which Hermiticity appears in the partial circuit. Each of the unitary gates 61 and 62 included in the partial circuit 60 further constitutes a smaller-scale LCU. That is, the partial circuit 60 has a double structure of LCU.

[0087] The LCU of a single unitary gate 61 shows gate operations on a plurality of auxiliary qubits and the qubit of interest, similar to the quantum circuit 30 shown in FIG. 4. For the qubit of interest, a plurality of unitary gates 61a, 61b, ···, 61k are arranged. The quantum calculation of the Hamiltonian of the system of interest is represented by these unitary gates 61a, 61b, ···, 61k. These unitary gates 61a, 61b, ···, 61k correspond to each term of the Hamiltonian. Each of the unitary gates 61a, 61b, ···, 61k is a quantum circuit for causing the quantum computer 200 to execute the calculation of the corresponding term.

[0088] The sub-circuit 60 exhibits Hermiticity by appropriately conditioning the states of the auxiliary qubits that make up the LCU inside the unitary gates 61 and 62. The condition for the state of the auxiliary qubits for Hermiticity to appear is to select only the event that the input and output are both in the |0> state for all of the auxiliary qubits that make up the LCU inside the unitary gates 61 and 62.

[0089] The expected value of the physical quantity O for the statistical ensemble introduced by the calculation method shown in FIG. 6 is given by the following equation (1).

[0090]

Equation

[0091] Equation (1) represents the contribution of the sub-circuit including the unitary gates U a , U b corresponding to two degrees a and b (a, b are non-negative integers) respectively. |Φ ab ik > is the state (the state of the system of interest) obtained by post-selecting the event that k ∈ {0, 1} is obtained by projective measurement of the auxiliary qubits in the computational basis after applying the sub-circuit with the input state of the system of interest as |i>, and is normalized. W ab ik is |Φ ab ik > is the probability obtained as the result of the measurement.

[0092] The average of the statistical population obtained by the calculation of the sub - circuit is |Φ ab ik > is the probability "W ab ik / Σ j W ab jk " and is the expected value with respect to the statistical population obtained in such a way. Next, the polynomial expansion of the imaginary - time evolution will be described in detail.

[0093] For the expansion of the imaginary - time evolution, for example, the Chebyshev polynomial T n (x) can be used. n is an integer indicating the degree. x is the argument of the polynomial, and in this case, the Hamiltonian H is substituted. The Chebyshev polynomial is the best - approximation polynomial and approximates a function with the best accuracy for arguments within a finite interval. The specific form of the expansion is given by the following formula (2).

[0094]

Equation

[0095] Here, the expansion coefficient c n (β) is the modified Bessel function of the first kind. The finite - temperature expected value of the physical quantity <o> β It is expanded as shown in formula (3).

[0096]

Number

[0097] m is an integer indicating the order. In this way, the polynomial expansion of the imaginary-time evolution becomes possible. Here, among the two selected orders (m, n), the unitary gate corresponding to the m-th order polynomial is U a is denoted as, and the unitary gate corresponding to the n-th order polynomial is U b is denoted as. In this case, from the measurement results of the sub-circuit corresponding to the two selected orders (m, n) pair, as shown in formula (1) <o> ab k is obtained. <o> ab k Expression (3) has the following relationship with it.

[0098]

Number

[0099] The extraction of two-degree pairs from polynomial expansion is performed, for example, as follows. In the polynomial expansion of the finite-temperature expectation value of a physical quantity, two-degree pairs (m, n) are extracted in order from the low-order pairs with large contributions.

[0100] When the truncation degree of the expansion is small, the total number of generable degree pairs is limited. In this case, all generable degree pairs may be exhaustively extracted. When the truncation degree of the expansion is large, the total number of generable degree pairs becomes extremely large. In this case, only the low-order pairs with large contributions need to be extracted. In particular, if the degree pairs are extracted probabilistically according to a probability distribution proportional to the absolute value of the expansion coefficient "c m (β / 2)c n (β / 2)", the thermal equilibrium expectation value can be correctly obtained.

[0101] The subcircuits representing each of the extracted degree pairs can be independently processed in parallel. For example, when the number of qubits of the quantum computer 200 is sufficient, the qubits can be divided into multiple groups, and the subcircuits can be executed for each group.

[0102] Next, the functions of the quantum computing system 300 for calculating the finite-temperature expectation value of a physical quantity will be described. FIG. 8 is a block diagram showing an example of the functions of a quantum computing system. The quantum computer 200 includes a quantum device 210 and a measurement device 220. The quantum device 210 is a circuit that constitutes a plurality of qubits. The quantum device 210 is, for example, a superconducting quantum device or an ion trap type quantum device. The quantum device 210 performs a gate operation on the qubits according to a gate operation instruction from the classical computer 100.

[0103] The measurement device 220 is a device that measures the state of the qubits in the quantum device 210. For example, the measurement device 220 performs a projective measurement of, for example, the computational basis (Z-basis) of the qubits. The classical computer 100 includes a quantum device control unit 110, a measurement result statistical processing unit 120, a calculation result storage unit 130, and a weighted average calculation unit 140.

[0104] The quantum device control unit 110 controls the quantum device 210 based on a quantum circuit showing the procedure of quantum calculation. For example, the quantum device control unit 110 controls the quantum device 210 according to a quantum circuit (a sub-circuit for each set of degrees obtained by polynomial expansion of imaginary time evolution) for calculating the finite temperature expectation value of a physical quantity.

[0105] The measurement result statistical processing unit 120 performs statistical processing of the measurement results of the quantum circuit. For example, the measurement result statistical processing unit 120 calculates the average or standard deviation of the measurement results by multiple measurements and determines whether the measurement results converge.

[0106] The calculation result storage unit 130 stores the calculation results of the physical quantities calculated for each sub-circuit. The calculation result storage unit 130 is, for example, a part of the storage area of the memory 102 or the storage device 103.

[0107] The weighted average calculation unit 140 calculates the weighted average of the calculation results for each sub-circuit. The weighted average calculation unit 140 outputs the calculated weighted average as the finite temperature expectation value. With the quantum computing system 300 having such functions, an efficient calculation of the finite temperature expectation value is performed.

[0108] The functions of the respective elements shown in FIG. 8 can be realized, for example, by causing the processor 101 to execute a program module corresponding to the element. FIG. 9 is a flowchart showing an example of a calculation procedure for the finite temperature expectation value. Hereinafter, the processing shown in FIG. 9 will be described along the step numbers.

[0109] [Step S101] The quantum device control unit 110 of the classical computer 100 receives an input of a set of the number of qubits N of the system of interest, the Hamiltonian H, the inverse temperature β, the cut-off order M of the polynomial expansion, and the number M (with tilde) of sets of degrees of the polynomial to be extracted.

[0110] [Step S102] The classical computer 100 repeats the processing of steps S103 to S111 while incrementing the value of the loop variable m from "0" to "M - 1" (M with tilde).

[0111] [Step S103] The quantum device control unit 110 generates a partial circuit corresponding to a set of two degrees extracted from a plurality of degrees for generating a polynomial by polynomial expansion of imaginary time evolution. For example, the quantum device control unit 110 expands the equation of imaginary time evolution into a polynomial up to the cut-off order M. As a result, polynomials from the 0th order to the Mth order are generated. The quantum device control unit 110 extracts an unextracted set from among the sets of two degrees that can be extracted from the degrees from the 0th order to the Mth order. By repeating this process M (with tilde) times, M (with tilde) sets of degrees are generated.

[0112] The quantum device control unit 110 generates a partial circuit corresponding to the extracted set of degrees. For example, the quantum device control unit 110 generates a unitary gate indicating the quantum calculation procedure of the polynomial corresponding to each degree included in the set. The number of qubits to be operated with the generated unitary gate is N. Then, the quantum device control unit 110 generates a partial circuit including two unitary gates corresponding to the set of degrees as shown in FIG. 6.

[0113] [Step S104] The quantum device control unit 110 repeatedly executes the process of step S105 a predetermined number of shots (number of repetitions) for the generated partial circuit. [Step S105] The quantum device control unit 110 instructs the quantum computer 200 to perform gate operations and measurements according to the generated partial circuit. For example, the quantum device control unit 110 transmits control signals for gate operations and measurements to the quantum device 210 in the order shown in the partial circuit.

[0114] Note that the quantum device control unit 110 sets the input state at the time of the first physical quantity calculation by the generated partial circuit to a preset input state. In the case of the second and subsequent physical quantity calculations, the quantum device control unit 110 sets the state updated in step S111 described later as the input state of the generated partial circuit.

[0115] The quantum device 210 performs gate operations on the qubits according to the control signals from the quantum device control unit 110. When the gate operation corresponding to the partial circuit is completed, the measuring device 220 measures the states of the qubits of interest and the auxiliary qubits. The measuring device 220 transmits the measurement results to the classical computer 100. In the classical computer 100, the measurement result statistical processing unit 120 receives the measurement results.

[0116] [Step S106] When the quantum gate operation and measurement instruction in step S105 are completed for a predetermined number of shots, the quantum device control unit 110 advances the process to step S107.

[0117] [Step S107] The measurement result statistical processing unit 120, based on the measurement results for the number of shots, determines the physical quantity obtained from the measurement results of the generated partial circuit (in Equation (1) <o> ab k ) is calculated. The measurement result statistical processing unit 120 adds the calculated physical quantity to the array r m .

[0118] [Step S108] The measurement result statistical processing unit 120 calculates the average of the values in the array r m . [Step S109] The measurement result statistical processing unit 120 determines whether the physical quantity obtained by the quantum calculation of the generated sub-circuit has converged. For example, if the change in the average calculated in step S108 is equal to or less than a predetermined value, the measurement result statistical processing unit 120 determines that convergence has occurred. Also, the measurement result statistical processing unit 120 may determine that convergence has occurred when the standard deviation of the values in the array r m is equal to or less than a predetermined value. If the measurement result statistical processing unit 120 determines that convergence has occurred, the process proceeds to step S112. If the measurement result statistical processing unit 120 determines that convergence has not occurred, the process proceeds to step S110.

[0119] [Step S110] The quantum device control unit 110 instructs the quantum computer 200 to perform a quantum gate operation and a projective measurement of the computational basis based on the generated sub-circuit. The input state of the quantum bit of interest at this time is the input state during the quantum calculation (quantum gate operation in step S105) for the previous physical quantity calculation.

[0120] In the quantum computer 200, a gate operation on the quantum device 210 is performed according to the instruction from the quantum device control unit 110. Then, a projective measurement of the computational basis of the quantum bit of interest after the gate operation according to the sub-circuit is performed by the measurement device 220. The measurement device 220 transmits the measurement result to the classical computer 100.

[0121] [Step S111] Based on the measurement results obtained in step S110, the quantum device control unit 110 updates the input state. For example, the quantum device control unit 110 uses the measurement results (|0> or |1>) of each qubit in the system of interest as the input state for the next physical quantity calculation. Then, the quantum device control unit 110 proceeds with the process to step S104.

[0122] As a result, the computational basis by the projective measurement of the output state becomes the input state of the qubits in the system of interest for the quantum calculation for the next physical quantity calculation. And such update of the input state is repeated until the calculated physical quantity converges.

[0123] [Step S112] When the physical quantity converges, the measurement result statistical processing unit 120 stores the average value calculated in step S108 in the array Avg in the calculation result storage unit 130. [Step S113] For the M (with tilde) sets of order sets, when the processing of steps S103 to S112 is completed, the quantum device control unit 110 proceeds with the process to step S114.

[0124] [Step S114] The weighted average calculation unit 140 calculates the weighted average of the contributions of the subcircuits. For example, the weighted average calculation unit 140 follows Equation (3) <o> β Calculate it. The weighted average calculation unit 140 outputs the calculation result of the weighted average as the expected value at a finite temperature.

[0125] In this way, based on the calculation results of quantum calculations according to a plurality of sub-circuits by the quantum computer 200, the expected value of a physical quantity at a finite temperature can be calculated. Since the sub-circuit is smaller in scale than the quantum circuit 30 shown in FIG. 4, it can be executed even in early FTQC. Also, the sub-circuit has a shallower circuit depth.

[0126] Moreover, in the iterative calculation of the physical quantity until the physical quantity using the sub-circuit converges, the classical computer 100 uses the result of the projective measurement of the computational basis of the output state after the quantum calculation by the sub-circuit as the input state in the calculation of the next physical quantity. Thereby, the input state to the sub-circuit becomes a statistical ensemble suitable for that sub-circuit.

[0127] Obtaining a statistical ensemble suitable for the sub-circuit specifically means that "|Φ ab ik >" is obtained according to the probability distribution "W ab ik / Σ j W ab jk ". In order to efficiently obtain such a statistical ensemble, the state obtained as the result of the projective measurement of the computational basis of the output state of the sub-circuit is used as the next input state. This means that in the repeated cycle of "1. Selection of input state", "2. Imaginary time evolution", "3. Measurement of physical quantity", and "4. Measurement of output state", the measurement result of "4. Measurement of output state" becomes the input state of the next cycle. This is the same process as the METTS algorithm, and the input state in the repeated cycle probabilistically transitions, and the stationary distribution obtained as the convergence destination becomes a statistical ensemble suitable for the sub-circuit.

[0128] Note that the operation of each partial circuit is a partial contribution to the imaginary-time evolution represented by the polynomial corresponding to the extracted degree set. And the physical quantity after convergence of each partial circuit will indicate the contribution of the partial circuit to the thermal equilibrium expectation value of the physical quantity of the target system. Therefore, the thermal equilibrium expectation value of the final target system's physical quantity is obtained by the weighted average of the calculation results of the physical quantities by each of the plurality of partial circuits.

[0129] Next, a specific example of calculating the thermal equilibrium expectation value of a physical quantity using the transverse-field Ising model will be described. FIG. 10 is a diagram showing an example of the Ising model. The Ising model 70 is a theoretical model that describes the quantum mechanical behavior of a magnetic material. In the Ising model 70, sites 71 to 73 are provided on the lattice points. Spins are defined at sites 71 to 73. An interaction acts between adjacent spins. Using the Ising model 70, the directions of the spins at sites 71 to 73 when a transverse magnetic field is applied can be calculated by numerical simulation.

[0130] For example, the Hamiltonian H of the Ising model 70 is represented by the following equation (5).

[0131]

Equation

[0132] X i is the Pauli operator that describes the X-component of the spin at the i-th (i is a natural number) site. X j is the Pauli operator that describes the X-component of the spin at the j-th (j is a natural number) site. Z i is the Pauli operator that describes the Z-component of the spin at the i-th site. J is a parameter (real number) indicating the Ising interaction. h is a parameter (real number) indicating the transverse magnetic field.

[0133] By representing the target system with the Ising model 70, the thermal equilibrium expected value of the energy of the target system can be obtained. For example, the thermal equilibrium expected value of the energy can be obtained as the canonical average of the eigenvalues of the Hamiltonian H of the Ising model 70.

[0134] Here, the target system is a two-qubit system. The coefficients of the Hamiltonian are set as follows. J = 0.96 h = 0.02 The truncation order of the Chebyshev polynomial expansion is set to 3. The Chebyshev polynomial expansion of the Hamiltonian H of the transverse-field Ising model up to the 3rd order is represented by the following equation (6).

[0135]

Equation

[0136] The first term of the polynomial after the expansion of equation (6) is a 0th-order equation, the second term is a 1st-order equation, the third term is a 2nd-order equation, and the fourth term is a 3rd-order equation. Here, the expansion coefficients are represented using the Bessel function of the first kind J n (x) (n is the order). Specifically, the expansion coefficients take the following values.

[0137]

Equation

[0138] The “(-i) n J n (-iβ)” shown on the left side of equation (7) is called the modified Bessel function of the first kind. As an example of the method for extracting pairs of two orders, a method of comprehensively selecting all combinations can be considered. When the truncation order is 3, the possible combinations of orders are as follows 16 cases. (0, 0), (0, 1), (0, 2), (0, 3) (1, 0), (1, 1), (1, 2), (1, 3) (2, 0), (2, 1), (2, 2), (2, 3) (3,0),(3,1),(3,2),(3,3) The numerical values within the above parentheses indicate the extracted order. In generating the sub-circuits corresponding to the order pairs, for example, the unitary gate corresponding to the order shown on the left side within the parentheses is arranged first (the unitary gate 61 in FIG. 7), and the unitary gate corresponding to the order shown on the right side within the parentheses is arranged later (the unitary gate 62 in FIG. 7).

[0139] Among the extractable order pairs, the pair (0,0) corresponds to the case where nothing is operated, so the generation of the corresponding sub-circuit is unnecessary. For pairs with the order reversed like (m,n) and (n,m), in an ideal quantum computer 200 where errors are completely removed, the respective sub-circuits would be equivalent. Therefore, if errors can be completely removed, only one of the pairs (m,n) and (n,m) needs to be extracted and the corresponding sub-circuit generated. In the following examples, only one of the pairs (m,n) and (n,m) will be extracted. As a result, the order pairs to be extracted are the following nine cases. (0,1),(0,2),(0,3) (1,1),(1,2),(1,3) (2,2),(2,3) (3,3) The precision for convergence determination is set to "0.5". That is, the classical computer 100 determines that it has converged when the standard deviation of the statistical population average of the measured values of the physical quantity falls below "0.5". Finally, the classical computer 100 calculates the finite temperature expectation value by calculating the linear combination using the expansion coefficients.

[0140] The calculation results of the thermal equilibrium expectation value of energy under the above conditions will be described with reference to FIGS. 11 and 12. Figure 11 is a diagram showing an example of the comparison result of the calculation accuracy of the thermal equilibrium expectation value of energy. The graph 80 shown in Figure 11 shows the error from the exact value of the calculation result of the thermal equilibrium expectation value of energy corresponding to the inverse temperature. The horizontal axis of the graph 80 is the inverse temperature, and the vertical axis is the error from the exact value. The broken line 81 shows the thermal equilibrium expectation value of energy calculated by the proposed method of this case. The broken line 82 shows the thermal equilibrium expectation value of energy calculated by the QMETTS algorithm. The broken line 83 shows the thermal equilibrium expectation value of energy calculated by uniformly sampling the input state for the partial circuit without using the proposed method.

[0141] The canonical average showing the exact value is a value obtained by substituting the energy eigenvalue obtained by exact diagonalization after expressing the Hamiltonian in matrix form into the expression of the canonical average. The proposed method shown by the broken line 81 has achieved the same level of accuracy as the QMETTS algorithm shown by the broken line 82. Note that compared with the QMETTS algorithm, the circuit depth of the proposed method is 1 / 3, and the non-Clifford gates are minimized. Moreover, with the proposed method, a calculation result with an accuracy one digit or more higher than that in the case of uniformly sampling the input state (broken line 83) can be obtained.

[0142] Figure 12 is a diagram showing an example of the comparison result of the calculation efficiency of the thermal equilibrium expectation value of energy. The graph 90 shown in Figure 12 shows the calculation efficiency of the thermal equilibrium expectation value of energy corresponding to the inverse temperature. The horizontal axis of the graph 90 is the inverse temperature, and the vertical axis is the number of samples of the quantum state (input state to the partial circuit) until the thermal equilibrium expectation value of energy converges.

[0143] The broken line 91 shows the total number of samples for each partial circuit when calculating the expected value of the thermal equilibrium of energy by the proposed method of this case. The broken line 92 shows the maximum value among the number of samples for each partial circuit when calculating the expected value of the thermal equilibrium of energy by the proposed method of this case. The broken line 93 shows the number of samples when calculating the expected value of the thermal equilibrium of energy by the QMETTS algorithm. The broken line 94 shows the number of samples when calculating the expected value of the thermal equilibrium of energy by uniformly sampling the input states for the partial circuits without using the proposed method.

[0144] The total number of samples for each partial circuit (broken line 91) by the proposed method of this case is nearly two digits less than that when uniformly sampling the input states for each partial circuit (broken line 94). Also, the total number of samples for each partial circuit (broken line 91) by the proposed method of this case has an increase in the number of samples suppressed to about one digit compared to the QMETTS algorithm (broken line 93) using the first implementation method of LCU (see Figure 4) for the implementation of imaginary time evolution.

[0145] In addition, in the examples shown in Figures 11 and 12, an ideal situation where errors are completely removed is assumed. However, when the error removal is incomplete, in the extraction of the degree pairs, both pairs where the extraction order of (m, n) and (n, m) is reversed may be extracted, and the corresponding partial circuits may be generated for each. This further improves the calculation accuracy.

[0146] As described above, according to the second embodiment, the expected value of the thermal equilibrium at a finite temperature of a physical quantity can be calculated efficiently with high precision. Moreover, since the quantum circuit is calculated by being divided into partial circuits, it can be implemented even with a small-scale quantum computer 200. Furthermore, by executing multiple partial circuits in parallel, the calculation efficiency can also be improved.

[0147] For example, in the calculation of the thermal equilibrium expectation value of energy at finite temperature in the transverse magnetic field Ising model, it is possible to suppress the increase in the number of samples until the expectation value converges while reducing the depth of the quantum circuit to less than half. Therefore, without degrading the efficiency, the implementation resources such as the circuit depth can be reduced compared to the existing methods.

[0148] 〔Other Embodiments〕 In the second embodiment, as an example, a calculation example of the thermal equilibrium expectation value of energy by the transverse magnetic field Ising model was shown, but the calculation of the thermal equilibrium expectation value of other physical quantities at finite temperature can also be efficiently performed.

[0149] As described above, the embodiments have been illustrated, but the configurations of each part shown in the embodiments can be replaced with other ones having the same functions. Also, any other components or processes may be added. Further, a combination of any two or more configurations (features) among the above-described embodiments may be used.

Description of Reference Numerals

[0150] 1 Quantum computer 2 Quantum circuit 3a, 3b, ··· Values of physical quantities 10 Information processing apparatus 11 Storage unit 12 Processing unit< / o> < / o> < / o> < / o> < / o> < / o> < / o>

Claims

1. The formula for imaginary-time evolution for calculating the thermal equilibrium expectation value of a physical quantity of a system to be calculated at a finite temperature is expanded into formulas for each of a plurality of orders, a plurality of combinations of orders obtained by extracting the order twice from the plurality of orders are generated, for each of the generated combinations, a quantum circuit showing a procedure for quantum calculation of the value of the physical quantity obtained by partial imaginary-time evolution based on the formula of the first order included in the combination and the formula of the second order included in the combination is generated, for each of the generated combinations, using the output state after the quantum calculation according to the quantum circuit corresponding to the combination as the input state in the subsequent quantum calculation, the quantum calculation based on the quantum circuit is repeatedly executed on a quantum computer until the value of the physical quantity obtained from the result of the quantum calculation converges, based on the value of the physical quantity after convergence for each of the generated combinations, the thermal equilibrium expectation value of the physical quantity at a finite temperature is calculated, A quantum calculation support program for causing a computer to execute the process.

2. In the process of repeatedly executing the quantum calculation on the quantum computer, the projective measurement result of the computational basis of the output state after the quantum calculation based on the input state in the N-th (N is a natural number) quantum calculation is used as the input state in the (N + 1)-th quantum calculation, The quantum calculation support program according to Claim 1.

3. In the process of generating the quantum circuit, the process includes a first unitary gate showing the calculation of the unitary matrix corresponding to the formula of the first order and a second unitary gate showing the calculation of the unitary matrix corresponding to the formula of the second order, and generates the quantum circuit having Hermiticity, The quantum calculation support program according to Claim 1.

4. In the process of generating a plurality of the combinations, the combinations including the orders obtained by extracting the order twice from the plurality of orders allowing duplication are generated, The quantum calculation support program according to Claim 1.

5. The formula for imaginary-time evolution for calculating the thermal equilibrium expectation value of a physical quantity of a system to be calculated at a finite temperature is expanded into formulas for each of a plurality of orders, a plurality of combinations of orders obtained by extracting the order twice from the plurality of orders are generated, for each of the generated combinations, a quantum circuit showing a procedure for quantum calculation of the value of the physical quantity obtained by partial imaginary-time evolution based on the formula of the first order included in the combination and the formula of the second order included in the combination is generated, For each of the generated groups, using the output state after the quantum calculation according to the quantum circuit corresponding to the group as the input state in the subsequent quantum calculation, repeatedly execute the quantum calculation based on the quantum circuit on a quantum computer until the value of the physical quantity obtained from the result of the quantum calculation converges. Calculate the thermal equilibrium expectation value of the physical quantity at a finite temperature based on the value of the physical quantity after convergence for each of the generated groups. A quantum calculation support method in which a computer executes a process.

6. Expand the formula for imaginary time evolution for calculating the thermal equilibrium expectation value of a physical quantity of a system to be calculated at a finite temperature into formulas for each of a plurality of orders, generate a plurality of sets of orders obtained by extracting the order twice from the plurality of orders, and for each of the generated sets, generate a quantum circuit showing the procedure of quantum calculation of the value of the physical quantity obtained by partial imaginary time evolution based on the formula of the first order included in the set and the formula of the second order included in the set. For each of the generated sets, using the output state after the quantum calculation according to the quantum circuit corresponding to the set as the input state in the subsequent quantum calculation, repeatedly execute the quantum calculation based on the quantum circuit on a quantum computer until the value of the physical quantity obtained from the result of the quantum calculation converges, and calculate the thermal equilibrium expectation value of the physical quantity at a finite temperature based on the value of the physical quantity after convergence for each of the generated sets, a processing unit. An information processing apparatus having the above.

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