Quantum computing support program, quantum computing support method, and information processing device.

Abstract

To improve the computational efficiency of the thermal equilibrium expectation value of physical quantities at finite temperatures. [Solution] The information processing device 10 expands the imaginary time evolution equation for calculating the thermal equilibrium expectation value into equations for multiple orders, and generates multiple sets of orders. For each of the multiple sets, the information processing device 10 generates a quantum circuit that shows the procedure for quantum computation of the value of a physical quantity obtained by partial imaginary time evolution. For each of the multiple sets, the information processing device 10 repeatedly causes the quantum computer 1 to perform quantum computation based on the quantum circuit. At that time, the information processing device 10 uses the state within the steady distribution of measurement results obtained by quantum computation based on the second quantum circuit 2b, which corresponds to a second set different from the first set, as the input state at the start of quantum computation based on the first quantum circuit 2a, which corresponds to the first set. Then, the information processing device 10 calculates the thermal equilibrium expectation value of the physical quantity at a finite temperature based on the converged value of the physical quantity for each set.

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 a qubit error correction technique (quantum error correction), information is redundantly encoded. To realize quantum error correction at a practical level, about one million qubits are used. 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, those of a small scale 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 meaningful 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 a canonical statistical population, the METTS (Minimally Entangled Typical Thermal State) algorithm is known.

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

[0006] The METTS algorithm uses an imaginary time evolution algorithm to realize Boltzmann weights (the same applies to QMETTS). One quantum imaginary time evolution method that realizes 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 calculates physical quantities by polynomial expansion of the imaginary time evolution, representing each degree polynomial with quantum circuits, and then performing a linear combination of these.

[0007] When LCU is implemented using early FTQC, the target physical quantity is calculated based on the computational results obtained by applying each of the multiple subcircuits extracted from the original quantum circuit to the qubits. Since the LCU is implemented using small-scale subcircuits, it is possible to perform the calculations even with early FTQC.

[0008] As for quantum computing techniques, for example, a method for finding excited states of the Hamiltonian has been proposed. Furthermore, a combinatorial optimization method has been proposed that can obtain feasible solutions even for problems where FALQON (Feedback-based ALgorithm for Quantum OptimizatioN) cannot. Quantum algorithms that improve quantum optimization by utilizing peripheral data have also been proposed. In addition, a finite-temperature quantum many-body simulation method that samples the series expansion of quantum imaginary time evolution has been proposed. This quantum many-body simulation method is called Markov-Chain Monte Carlo with Sampled Pairs of Unitaries (MCMC-SPU). [Prior art documents] [Patent Documents]

[0009] [Patent Document 1] International Publication No. 2020 / 090559 [Patent Document 2] Japanese Patent Publication No. 2023-43100 [Patent Document 3] U.S. Patent Application Publication No. 2020 / 0057957 [Non-patent literature]

[0010] [Non-Patent Document 1] Norifumi Matsumoto, Shoichiro Tsutsui, Yuya O. Nakagawa, Yuichiro Hidaka, Shota Kanasugi, Kazunori Maruyama, Hirotaka Oshima, Shintaro Sato, "Quantum many-body simulation of finite-temperature systems with sampling a series expansion of a quantum imaginary-time evolution", arXiv:2409.07070v1, quant-ph, 11 Sep 2024 [Overview of the project] [Problems that the invention aims to solve]

[0011] In MCMC-SPU, the thermal equilibrium expectation values ​​of physical quantities in a quantum system at finite temperatures are obtained by linearly combining the computation results of multiple simplified subcircuits using a classical computer. In the iterative computation of each subcircuit, the output state of the subcircuit at each step is projected onto the computational basis, and the resulting state is used as the input state for the next step. This efficiently generates an appropriate statistical ensemble.

[0012] In MCMC-SPU, the input state for the first step (the first iteration) of each subcircuit's calculation is randomly selected, for example, from the calculation basis. This is equivalent to having a uniform initial distribution. During the iterative calculation of the subcircuit, the distribution of states gradually changes from this initial distribution and converges to a stationary distribution. The stationary distribution at which it converges becomes the statistical population that should be realized.

[0013] However, if the initial state is randomly selected from a uniform distribution, there is a possibility that a state far removed from the stationary distribution will be selected. In this case, convergence to the stationary distribution will take a long time. This prolonged convergence time tends to be more pronounced in higher-order sets where the circuit structure is complex.

[0014] In one aspect, this project aims to improve the computational efficiency of the thermal equilibrium expectation values ​​of physical quantities at finite temperatures. [Means for solving the problem]

[0015] One proposal provides a quantum computing support program that instructs a computer to perform the following processes: The computer expands the imaginary time evolution equation for calculating the thermal equilibrium expectation value of a physical quantity of the system under computation at a finite temperature into equations of multiple orders. The computer generates multiple sets of orders by extracting two orders from the multiple sets. For each set, the computer generates a quantum circuit that shows the procedure for quantum computation of the value of the physical quantity obtained by a partial imaginary time evolution based on the first-order equation and the second-order equation included in the set. The computer uses a state within the stationary distribution of measurement results obtained by quantum computation based on a second quantum circuit corresponding to a second set other than the first set as the input state at the start of quantum computation based on the first quantum circuit corresponding to at least one of the multiple sets. For each of the multiple sets, the computer repeatedly executes quantum computation based on the corresponding quantum circuit until the value of the physical quantity obtained from the quantum computation results converges. Then, based on the converged values ​​of the physical quantity for each of the multiple sets, the computer calculates the thermal equilibrium expectation value of the physical quantity at a finite temperature. [Effects of the Invention]

[0016] According to one embodiment, the calculation efficiency of the thermal equilibrium expectation value of a physical quantity at a finite temperature is improved. [Brief explanation of the drawing]

[0017] [Figure 1] This figure shows an example of a quantum computing support method according to the first embodiment. [Figure 2] This figure shows an example of the configuration of a quantum computing system. [Figure 3] This is a diagram showing an example of computer hardware. [Figure 4] This figure shows a first example (first implementation method) of a quantum circuit for realizing an LCU. [Figure 5] This figure shows a second example (second implementation method) of a quantum circuit for realizing an LCU. [Figure 6] This figure shows an example of an MCMC-SPU. [Figure 7]This figure shows an example of an initial state set by a uniform distribution. [Figure 8] This figure shows an example of an initial state set by a stationary distribution of low-order sets. [Figure 9] This figure shows an example of a method for determining the initial state when executing subcircuits in order, starting with those corresponding to lower-order sets. [Figure 10] This figure shows an example of a subcircuit corresponding to a set of degrees. [Figure 11] This block diagram shows an example of the functionality of a quantum computing system. [Figure 12] This flowchart shows an example of the procedure for calculating the expectation value at a finite temperature. [Figure 13] This figure shows an example of the Ising model. [Figure 14] This figure shows an example of convergence to a stationary distribution obtained for a pair of quadratic and quadratic equations. [Figure 15] This figure shows an example of convergence to a stationary distribution obtained for a quadratic and cubic pair without using the stationary distributions of other pairs of degrees. [Figure 16] This figure shows an example of convergence to a stationary distribution obtained for a quadratic and cubic pair using the stationary distribution of a quadratic and quadratic pair. [Modes for carrying out the invention]

[0018] The following description of this embodiment will be made with reference to the drawings. Note that each embodiment can be implemented by combining multiple embodiments within a reasonable scope. [First Embodiment] The first embodiment is a quantum computing support method for improving the computational efficiency of the thermal equilibrium expectation value of a physical quantity at a finite temperature.

[0019] Figure 1 shows an example of a quantum computing support method according to the first embodiment. Figure 1 shows an information processing device 10 for implementing the quantum computing support method. The information processing device 10 can implement the quantum computing support method, for example, by executing a quantum computing support program.

[0020] The information processing device 10 includes a storage unit 11 and a processing unit 12. The storage unit 11 is, for example, a memory or storage device of the information processing device 10. The processing unit 12 is, for example, a processor of the information processing device 10. The information processing device 10 may have multiple processors. Some of the multiple processes performed by the information processing device 10 may be executed on different processors.

[0021] The processing unit 12 calculates the thermal equilibrium expectation value of a physical quantity at a finite temperature using the following procedure. The processing unit 12 expands (polynomial expands) the equation for the imaginary time evolution of the physical quantities of the system under calculation (the system of interest) at a finite temperature into multiple equations of different degrees. The specific equations for expanding the imaginary time evolution will be described later (see equation (2)). This yields multiple n-th degree equations (where n is the degree), such as a 0th-degree equation, a 1st-degree equation, and a 2nd-degree equation.

[0022] Next, the processing unit 12 generates multiple sets of orders obtained by extracting an order twice from multiple orders. For each generated set, the processing unit 12 generates a quantum circuit that shows the procedure for quantum computation of the value of a physical quantity obtained by partial imaginary time evolution based on the first order formula and the second order formula included in the set. Details of the formula for calculating the value of the physical quantity will be described later (see formula (1)). For example, the processing unit 12 generates a quantum circuit that has Hermitian properties under appropriate conditions for the auxiliary qubit.

[0023] Quantum circuits do not output the state after imaginary time evolution, but rather output a state that reflects only the contribution of the extracted set of orders. Therefore, each quantum circuit for each set of orders can be considered a subset of the quantum circuit that represents the entire imaginary time evolution.

[0024] The processing unit 12 repeatedly causes the quantum computer 1 to perform quantum calculations based on the quantum circuit for each set of generated orders until the value of the physical quantity obtained from the result of the corresponding quantum calculation converges. At this time, the processing unit 12 uses a state within the stationary distribution of measurement results obtained by the quantum calculation based on the second quantum circuit 2b corresponding to the second set as the input state at the start of the quantum calculation based on the first quantum circuit 2a corresponding to at least one of the first sets of generated orders. At this time, the processing unit 12 determines, for example, that the second set is a set of generated sets that combines orders of a lower order than the first set.

[0025] The input state at the start of a quantum computation based on a quantum circuit is the initial state of the qubits in the system of interest when the quantum circuit is first executed. The qubits in the system of interest are the qubits that the quantum circuit is operating on, excluding the auxiliary qubits, and are also called data qubits.

[0026] The processing unit 12 stores the converged values ​​of the physical quantities 3a, 3b, ... for each set of generated orders in, for example, the memory unit 11. Then, the processing unit 12 calculates the thermal equilibrium expectation value of the physical quantities at a finite temperature based on the converged values ​​of the physical quantities 3a, 3b, ... for each set of generated orders. For example, the processing unit 12 calculates a weighted average of the converged values ​​of the physical quantities 3a, 3b, ... for each set of generated orders. The resulting weighted average becomes the thermal equilibrium expectation value of the physical quantities at a finite temperature.

[0027] In this way, in calculating the thermal equilibrium expectation value of physical quantities at a finite temperature, the state within the stationary distribution obtained when the calculation results of the physical quantity calculations for the contributions of lower-order sets converge is used as the initial state for the calculation of the physical quantity contributions of higher-order sets. This prevents the initial state from deviating too much from the stationary distribution obtained when the calculation results based on the quantum circuits corresponding to the higher-order sets converge. As a result, the calculation results can be converged earlier, improving computational efficiency.

[0028] Furthermore, it is possible to use the state within the stationary distribution obtained when the calculation results of the physical quantity calculation for the contribution of a higher-order set converge to a stationary distribution as the initial state for the calculation of the physical quantity contribution of a lower-order set. In this case as well, the distribution of the values ​​of the calculated physical quantity can be converged earlier. However, the quantum circuit corresponding to the higher-order set has more quantum gates (greater depth) than the quantum circuit corresponding to the lower-order set. Therefore, using the stationary state of the calculation result of the lower-order set as the initial state of the quantum circuit of the higher-order set can yield a greater effect in terms of computational efficiency.

[0029] Therefore, the processing unit 12 may, for example, rank the generated sets in ascending order of degree, and when the rank of the first set is N+1 (where N is a natural number), it may determine the set with the Nth rank to be the second set. In this case, the processing unit 12 executes quantum computations based on the quantum circuits for each generated set in ascending order of rank. Note that an ascending order of a set means, for example, that the sum of the degrees included in the set is small.

[0030] By ranking each set of quantum machines in ascending order of order and using the stationary distribution of the previous set of quantum machines, the initial state of the next set of quantum machines can be determined to be closer to the stationary distribution of the set of quantum machines being computed. As a result, computational efficiency is improved.

[0031] Furthermore, when the processing unit 12 executes each quantum circuit of a set of orders, it repeatedly causes the quantum computer 1 to perform a quantum computation based on that quantum circuit. In such iterative computations, the processing unit 12 may use the output state after the quantum computation of the quantum circuit corresponding to the set of orders as the input state for the next quantum computation of that quantum circuit. For example, the processing unit 12 may use the projection measurement result of the computational basis of the output state of the quantum circuit based on the input state of the Mth (M is a natural number) quantum computation as the input state for the M+1th quantum computation.

[0032] This means that the input state when iteratively calculating the value of a physical quantity based on a quantum circuit becomes a statistical sect suitable for quantum circuits. A statistical sect suitable for quantum circuits is one that allows the value of a physical quantity calculated based on a quantum circuit to converge quickly. For example, it has been confirmed that generating a Hermitian quantum circuit under appropriate conditions for auxiliary qubits results in a statistical sect that allows the value of a physical quantity calculated based on the quantum circuit to converge quickly. Because an appropriate statistical sect is automatically generated, the value of the physical quantity obtained by quantum computation converges quickly, and processing becomes more efficient.

[0033] The processing unit 12, for example, randomly selects a state from within the stationary distribution corresponding to a lower-order set, and uses this state as the initial state of the first quantum circuit 2a of the higher-order set. Specifically, the processing unit 12 selects one state from among several states shown in the results of multiple quantum calculations performed by the second quantum circuit 2b after the measurement results of the quantum calculation based on the second quantum circuit 2b corresponding to the lower-order set have become a stationary distribution. The processing unit 12 then uses the selected state as the input state at the start of the quantum calculation based on the first quantum circuit 2a. In this way, the initial state of the first quantum circuit 2a of the higher-order set can be reliably selected from within the stationary distribution corresponding to the lower-order set.

[0034] [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.

[0035] Figure 2 shows an example of the configuration of a quantum computing system. The quantum computing system 300 is a hybrid computer system in which a classical computer 100 and a quantum computer 200 work in conjunction. The classical computer 100 is also called a von Neumann architecture computer.

[0036] A terminal device 400 is connected to the classical computer 100 via a network 20. The terminal device 400 is a computer used by users who request quantum computations from the quantum computing system 300. The classical computer 100 receives quantum circuits 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 the "0" state and the "1" state.

[0037] The classical computer 100, following the quantum circuit received from the terminal device 400, issues instructions to the quantum computer 200 for controlling the qubits. The classical computer 100 also obtains the measurement results for each qubit from the quantum computer 200.

[0038] Quantum computer 200 has multiple qubits and devices for manipulating each of these qubits. The multiple qubits of quantum computer 200 are realized using, for example, superconducting quantum devices. Alternatively, the qubits may be realized using other types of quantum devices, such as ion trap devices.

[0039] Figure 3 shows an example of computer hardware. In the classical computer 100, the entire device is controlled by a processor 101. The processor 101 is connected to memory 102 and several peripheral devices via bus 100a.

[0040] The classical computer 100 may be a multiprocessor system having multiple processors. The set of multiple processors in a multiprocessor system can be called a processor 101. A processor 101 may also be called a processor circuitry. Each of the multiple processors can execute some or all of the multiple processes that are executed on the classical computer 100. When there are multiple related processes, the processor that executes one process may be different from the processor that executes a different process.

[0041] The processor 101 is, for example, a CPU (Central Processing Unit), an MPU (Micro Processing Unit), or a DSP (Digital Signal Processor). At least some of the functions that the processor 101 implements by executing a program may be implemented by electronic circuits such as an ASIC (Application Specific Integrated Circuit) or a PLD (Programmable Logic Device).

[0042] 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 (Random Access Memory) is used.

[0043] Peripheral devices connected to bus 100a include a storage device 103, a graphics controller 104, an input interface 105, an optical drive device 106, a device connection interface 107, a network interface 108, and a communication interface 109.

[0044] 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, the storage device 103 can be an HDD (Hard Disk Drive) or an SSD (Solid State Drive).

[0045] The graphics controller 104 is an arithmetic unit that performs image processing. The graphics controller 104 is, for example, a GPU (Graphics Processing Unit). A monitor 21 is connected to the graphics controller 104. The graphics controller 104 displays images on the screen of the monitor 21 according to instructions from the processor 101. The monitor 21 can be an OLED (Electroluminescence) display device or a liquid crystal display device. If a GPU is used as the graphics controller 104, the graphics controller 104 can also perform complex numerical calculations such as matrix calculations.

[0046] The input interface 105 is connected to a keyboard 22 and a mouse 23. The input interface 105 transmits signals from the keyboard 22 and mouse 23 to the processor 101. Note that the mouse 23 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.

[0047] The optical drive device 106 uses laser light or the like to read data recorded on the optical disc 24 or write data to the optical disc 24. The optical disc 24 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 24 include DVD (Digital Versatile Disc), DVD-RAM, CD-ROM (Compact Disc Read Only Memory), and CD-R (Recordable) / RW (ReWritable).

[0048] The device connection interface 107 is a communication interface for connecting peripheral devices to the classical 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 or reads data from the memory card 27. The memory card 27 is a card-type recording medium.

[0049] The network interface 108 is connected to the network 20. The network interface 108 transmits and receives data to and from other computers or communication devices via the network 20. The network interface 108 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 108 may be a wireless communication interface, connected by radio waves to a wireless communication device such as a base station or access point.

[0050] The communication interface 109 is connected to the quantum computer 200. The communication interface 109 communicates with the quantum computer 200. For example, the communication interface 109 sends quantum gate operation instructions based on quantum circuits to the quantum computer 200. The communication interface 109 also receives the execution results of quantum circuits from the quantum computer 200.

[0051] 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.

[0052] 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 24, a memory device 25, or a memory card 27. 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.

[0053] 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 instructions from the classical computer 100. For example, the control device 201 performs a gate operation on a qubit by irradiating it with microwaves of a predetermined frequency.

[0054] The qubit device 202 has multiple qubits. The qubit device 202 may have qubits of various types, such as superconducting, ion trapping, or cold atom architectures. The qubit device 202 is sometimes also called a QPU (Quantum Processing Unit).

[0055] A user who uses the quantum computing system 300 uses the terminal device 400 to generate, for example, a quantum circuit for solving a problem to be solved by quantum computing. When the user instructs the terminal device 400 to execute quantum computing, a quantum computing request including the generated quantum circuit is transmitted from the terminal device 400 to the quantum computing system 300.

[0056] In the quantum computing system 300, the classical computer 100 causes the quantum computer 200 to execute quantum computing based on the quantum circuit in response to the quantum computing request. At this time, the classical computer 100 converts the quantum circuit to be executed into a quantum circuit using executable quantum gates according to the specifications of the hardware of the quantum computer 200 (such as native gates corresponding to quantum bit devices).

[0057] 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 fields 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.

[0058] 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 representing a thermal equilibrium state is calculated. As a population of quantum states representing a thermal equilibrium state at a finite temperature, there is a canonical statistical population. The canonical statistical population 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.

[0059] When the canonical statistical population ρ is expressed 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 population <o>teeth," <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).

[0060] Canonical statistical populations can be efficiently generated using the METTS algorithm. When classical computer 100 generates a canonical statistical population using METTS, the process is carried out in the following steps. 1. Classical computer 100 selects an input state from the computational base. 2. Classical computer 100 executes an imaginary time evolution algorithm to realize Boltzmann weights. 3. Classical computer 100 calculates the expected value of the physical quantity to be found. 4. Classical computer 100 computes a probability distribution corresponding to a projection measurement on the computational basis of the output state in order to generate a probability distribution that follows Boltzmann weights. 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.

[0061] The set of quantum states used as input states in iterations 2-4 constitutes the canonical statistical set. In other words, with the METTS algorithm, the canonical statistical set is automatically generated during the process of calculating the expectation value of a physical quantity.

[0062] While the METTS algorithm is a computational method intended to be executed only on classical computers (100), a quantum computing system (300) can execute a method called QMETTS, which is equivalent to METTS.

[0063] In QMETTS as well, imaginary time evolution is required to realize Boltzmann weights. There are various methods for quantum imaginary time evolution to be implemented in the quantum computer 200.

[0064] Effective quantum imaginary time evolution methods in NISQ include variational imaginary time evolution, strict imaginary time evolution, and stochastic imaginary time evolution. Variational imaginary time evolution is a method that variationally optimizes the parameters of a quantum circuit to reproduce imaginary time evolution. Strict imaginary time evolution is a method that determines the coefficients of a simple quantum gate from equations to reproduce imaginary time evolution. Stochastic imaginary time evolution is a method that retrospectively selects only events that satisfy specific conditions for the added auxiliary qubit.

[0065] Effective quantum imaginary time evolution methods in FTQC include methods based on quantum singular value transformation algorithms and methods based on LCU. Methods based on quantum singular value transformation algorithms approximate imaginary time evolution by transforming the eigenvalues ​​of the Hamiltonian into polynomials. LCU is a method that polynomializes the imaginary time evolution, represents each polynomial of degree with quantum circuits, and finds a linear combination of all of them. LCU can be implemented even in relatively small-scale early FTQC.

[0066] In the quantum computing system 300 according to the second embodiment, the thermal equilibrium expectation values ​​of the physical quantities of the quantum system at finite temperatures are calculated by QMETTS accompanied by imaginary time evolution by LCU. The quantum circuit for imaginary time evolution can be executed by dividing it into multiple subcircuits with a small number of qubits, and is well compatible with early FTQC.

[0067] Figure 4 shows a first example (first implementation method) of a quantum circuit for realizing an LCU. The quantum circuit 30 shows multiple qubits representing the state |ψ> of the system of interest, and gate operations on auxiliary qubits (each qubit with an initial state of |0>).

[0068] The qubits of the system of interest are equipped with unitary gates 33a, 33b, ..., 33k, corresponding to the polynomials of each order obtained when the imaginary time evolution is polynomialized. The auxiliary qubits are subjected to gate operations on a predetermined unitary gate 31.

[0069] Each unitary gate 33a, 33b, ..., 33k in the system of interest uses an auxiliary qubit as a control qubit, and is activated when the state of the control qubit satisfies a predetermined condition. The auxiliary qubits that serve as the control qubits for each unitary gate 33a, 33b, ..., 33k are indicated by white or black circles. White circles indicate negative polarity, meaning that the gate operation is applied to the target qubit (the qubit in the system of interest) when its state is "0". Black circles indicate positive polarity, meaning that the gate operation is applied to the target qubit (the qubit in the system of interest) when its state is "1".

[0070] For each of the unitary gates 33a, 33b, ..., 33k, if all the control qubits are in a state where a gate operation can be applied, the gate operation is performed on the qubits of the system of interest according to the corresponding unitary gate.

[0071] After the gate operations on unitary gates 33a, 33b, ..., 33k are performed, the gate operations on unitary gate 32 are performed on the auxiliary qubits. The quantum circuit 30 enables computation by sequentially acting on the quantum circuits of each component after polynomial expansion in a controlled unitary form to perform linear combination. In quantum circuit 30, in addition to the qubits of the system of interest that are the target of the operation on the unitary gates 33a, 33b, ..., 33k, a large number of auxiliary qubits are used. Therefore, the number of qubits used increases.

[0072] Figure 5 shows a second example (second implementation method) of a quantum circuit for realizing an LCU. Subcircuit 40 is a simplified quantum circuit obtained by extracting gate operations corresponding to a pair of degrees from the degrees obtained when the imaginary time evolution equation is polynomialized. Subcircuit 40 uses one auxiliary qubit. First, the auxiliary qubit is subjected to a gate operation of the unitary gate 41.

[0073] When the state of the auxiliary qubit in the superposed state by the unitary gate 41 is |0>, the gate operation of the unitary gate 43a corresponding to a polynomial of one of the two selected orders is performed on the qubit of the target system. Next, when the state of the auxiliary qubit is |1>, the gate operation of the unitary gate 43b corresponding to the polynomial of the other of the two selected orders is performed on the qubit of the target system. After the two unitary gates 43a and 43b, the gate operation of the unitary gate 42 is performed on the auxiliary qubit. When the gate operation of the unitary gate 41 is represented by the matrix "V R ", the matrix representing the gate operation of the unitary gate 42 is represented as "V R † ". "V R † " is called the adjoint matrix and is a matrix obtained by transposing "V R " and taking the complex conjugate of the components.

[0074] By using the subcircuit 40, it becomes possible to extract two-component quantum circuits, operate on the simplified subcircuits obtained by simplification, calculate physical quantities, and then calculate the linear combination of measurement results with a classical computer 100 later.

[0075] In addition, 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, it becomes necessary to randomly sample a huge number from an exponentially large number of states with respect to the size of the system and calculate for each sample, resulting in poor calculation efficiency.

[0076] Therefore, the second implementation method can improve the calculation efficiency by applying MCMC-SPU (see Non-Patent Document 1). MCMC-SPU is a technique for obtaining a value indicating the contribution of a subcircuit by efficiently generating a statistical population suitable for the subcircuit for each subcircuit from which the contributions of two sets of orders are extracted.

[0077] Figure 6 shows an example of an MCMC-SPU. When the imaginary time evolution is expanded into a polynomial, polynomials for each degree are obtained. These polynomials for each degree are represented by unitary matrices U1, U2, U3, U4, U5, U6, ... In an MCMC-SPU, multiple pairs of two degrees are extracted from the multiple degrees obtained by the polynomial expansion. Then, in an MCMC-SPU, subcircuits 51, 52, 53, ... are generated for each pair of degrees.

[0078] Subcircuit 51 shows gate operations on multiple qubits representing the system of interest and one auxiliary qubit. The input state of the multiple qubits representing the system of interest is |ψ>, and the input state of the auxiliary qubit is |0>.

[0079] In subcircuit 51, first, a unitary gate 51a (gate operation "V") is used to connect to the auxiliary qubit. R Next, a unitary gate 51b, controlled by an auxiliary qubit as a negative-polarity control qubit, is placed in the qubit of the system of interest. Furthermore, a unitary gate 51c, controlled by an auxiliary qubit as a positive-polarity control qubit, is placed in the qubit of the system of interest. The two unitary gates 51b and 51c correspond to the respective orders of the two extracted orders and are quantum circuits that cause the quantum computer 200 to compute the polynomial of the corresponding order.

[0080] Next to the unitary gate 51c, the auxiliary qubit is given a unitary gate 51d (gate operation "V"). R † A qubit is positioned there. The subcircuit 51 shows measurements 51e and 51f of the states of the auxiliary qubit and the qubit of the system of interest, respectively.

[0081] Regarding the subcircuit 52, a unitary gate 52a (gate operation "V") is also used to connect to the auxiliary qubit. R "), two unitary gates 52b, 52c to the qubit representing the state of the system of interest, and a unitary gate 52d to the auxiliary qubit (gate operation "V" R † A qubit is positioned there. And in the subcircuit 52, measurements 52e and 52f of the states of the auxiliary qubit and the qubit of the system of interest are shown.

[0082] Regarding the subcircuit 53, a unitary gate 53a (gate operation "V") to the auxiliary qubit is also used. R "), two unitary gates 53b, 53c to the qubit representing the state of the system of interest, and a unitary gate 53d to the auxiliary qubit (gate operation "V" R † A qubit is positioned there. And in the subcircuit 53, measurements 53e and 53f of the states of the auxiliary qubit and the qubit of the system of interest are shown.

[0083] The classical computer 100 instructs the quantum computer 200 to repeatedly perform quantum computations corresponding to the subcircuits 51, 52, 53, ... that correspond to each set of orders, until the physical quantity obtained from the measurement results converges. If the physical quantity does not converge, the classical computer 100 instructs the quantum computer 200 to perform a projection measurement of the output state of the qubits of the system of interest on the computational basis, and uses the measurement result as the input state for the quantum computation in the next iteration step.

[0084] By using the results of the projection measurement of the output state as the input state in the next step, a statistical sect suitable for quantum computation corresponding to the extracted set of orders is generated as the input state. For example, the input state of subcircuit 51 becomes a statistical sect suitable for calculating the unitary matrices U1 and U2 corresponding to each of the extracted orders. The input state of subcircuit 52 becomes a statistical sect suitable for calculating the unitary matrices U3 and U4 corresponding to each of the extracted orders. The input state of subcircuit 53 becomes a statistical sect suitable for calculating the unitary matrices U5 and U6 corresponding to each of the extracted orders.

[0085] Based on the physical quantities at the convergence of each of these subcircuits 51, 52, 53, ..., the thermal equilibrium expectation value at a finite temperature is calculated. In other words, the thermal equilibrium expectation value at a finite temperature is calculated by efficiently and automatically generating input states that form an appropriate statistical sect.

[0086] The reason why computational efficiency is improved by MCMC-SPU as shown in Figure 6 is as follows: The contribution of a particular subcircuit to the thermal equilibrium expectation value of a physical quantity can be obtained by calculating the statistical population mean corresponding to that subcircuit. Specifically, the statistical population |φ ab ik > is the probability distribution "W ab ik / Σ j w ab jk This is obtained according to the formula. To efficiently obtain such a statistical ensemble, projection measurements can be performed on the output state of the subcircuit using a computational basis, and the state obtained as a result of these measurements can be used as the input state for the next step. Through this operation, the input state transitions probabilistically, and the stationary distribution obtained as the convergence point coincides with the statistical ensemble to be realized.

[0087] The statistical ensemble obtained from the subcircuit will be explained in more detail below. A unitary gate U that represents the contributions of two degrees. a and U b The contribution of a subcircuit containing this subcircuit can be obtained by calculating the following statistical ensemble mean of the physical quantity O.

[0088]

number

[0089] Equation (1) shows the unitary gates U corresponding to the two degrees a and b (where a and b are non-negative integers). a ,U b This represents the contribution of the subcircuit that includes |Φ ab ik > represents the normalized state (the state of the system of interest) obtained by applying a subcircuit to the input state of the system of interest, taking |i> as the input state, and then projecting the auxiliary qubits with a computational basis to select an event that yields k∈{0,1}. ab ik is, |Φ ab ik This is the probability obtained as a result of the measurement.

[0090] The mean of the statistical ensemble obtained from the calculation of the subcircuit is |Φ ab ik > is probability "W ab ik / Σ j W ab jk This is the expected value for a statistical population, as obtained from "[...]." Next, we will explain in detail the polynomial expansion of imaginary time evolution.

[0091] For example, the development of imaginary time evolution involves 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, which approximates the function with the best accuracy for arguments within a finite interval. The specific form of the expansion is given by equation (2) below.

[0092]

number

[0093] Here, the expansion coefficient c n (β) is a modified Bessel function of the first kind. Finite temperature expectation value of a physical quantity <o> β This is expanded as shown in equation (3).

[0094]

number

[0095] m and n are integers representing the degree. In this way, the polynomial expansion of imaginary time evolution becomes possible. Here, among the two selected degrees (m,n), let Ua be the unitary gate corresponding to the m-th degree polynomial, and let Ua be the unitary gate corresponding to the n-th degree polynomial. b In this case, the measurement results of the subcircuits corresponding to the two selected pairs of orders (m,n) are shown in equation (1). <o> ab k You can obtain this. <o> ab k The relationship between equation (3) and equation (3) is as follows:

[0096]

number

[0097] Extracting pairs of degrees from a polynomial expansion can be done, for example, as follows: In the polynomial expansion of the finite temperature expectation value of a physical quantity, pairs of degrees (m,n) are extracted in order from the lowest-degree pair with the largest contribution.

[0098] If the truncation degree of the expansion is small, the total number of possible degree combinations is limited. In this case, all possible degree combinations may be extracted exhaustively. If the truncation order of the expansion is large, the total number of possible sets of orders becomes enormous. In this case, it is sufficient to extract only the low-order sets with large contributions. In particular, the expansion coefficient "c m (β / 2)c n By probabilistically extracting sets of degrees according to a probability distribution proportional to the absolute value of (β / 2), the correct thermal equilibrium expectation value can be obtained.

[0099] Each subcircuit representing a set of extracted order values ​​can be processed independently in parallel. For example, if the quantum computer 200 has a sufficient number of qubits, the qubits can be divided into multiple groups, and a subcircuit can be executed for each group.

[0100] Here, we compare QMETTS and MCMC-SPU. QMETTS is assumed to be implemented using the first implementation method shown in Figure 4 (coherent superposition of all orders). These implementation methods have their advantages and disadvantages in terms of performance and the load on implementation resources. Therefore, the appropriate implementation method is used by considering the advantages and disadvantages. In particular, there are the following differences between QMETTS and MCMC-SPU when it comes to implementing quantum imaginary time evolution.

[0101] [Implementation of Imaginary Time Evolution] QMETTS is implemented using quantum circuits that perform coherent superposition of all orders. On the other hand, MCMC-SPU is implemented using subcircuits that extract the contributions of pairs of orders.

[0102] [Suitable quantum computer] QMETTS requires execution on a large-scale FTQC. On the other hand, MCMC-SPU can be implemented even on a relatively small-scale FTQC (early FTQC).

[0103] [Circuit depth] In QMETTS, the circuit depth is the sum of the depths of all unitary gates of all orders, resulting in a deep overall circuit. On the other hand, in MCMC-SPU, the circuit depth is the sum of the depths of the two extracted unitary gates of different orders, resulting in a shallower circuit compared to the first implementation method.

[0104] [Implementation cost] QMETTS has high implementation costs because it uses a large number of non-Clifford gates. On the other hand, MCMC-SPU minimizes the number of non-Clifford gates, resulting in relatively low implementation costs.

[0105] [Number of quantum circuits] QMETTS uses one quantum circuit, while MCMC-SPU uses as many quantum circuits as there are sets of order values ​​to be extracted. When multiple quantum circuits are used, as in MCMC-SPU, it becomes possible to perform quantum computations based on quantum circuits in parallel.

[0106] [Method for realizing linear combinations] In QMETTS, linear combinations are achieved through coherent superposition. On the other hand, in MCMC-SPU, linear combinations are achieved by summing the results of each subcircuit using the classical computer 100.

[0107] [Output status] QMETTS outputs the state after imaginary time evolution. On the other hand, MCMC-SPU outputs the state after a portion of the imaginary time evolution has been applied.

[0108] [Statistical population obtained by projection measurements on the output state] QMETTS yields a canonical statistical population. On the other hand, MCMC-SPU yields a statistical population (different from the canonical statistical population) corresponding to the extracted subcircuits.

[0109] The differences between QMETTS and MCMC-SPU are as described above. As mentioned above, one of the advantages of MCMC-SPU is that it can be implemented in medium-scale FTQC. Therefore, to calculate the thermal equilibrium expectation values ​​of physical quantities in a quantum system at finite temperatures, it is practical to first implement it using MCMC-SPU. Below, we will examine the characteristics of MCMC-SPU in more detail.

[0110] In MCMC-SPU, the output state of the circuit is projected onto a computational basis, and the resulting state is used as the input state for the next step. The initial input state is usually randomly selected from the computational basis. That is, the distribution of the measured states gradually changes from a uniform initial distribution and converges to a stationary distribution. This stationary distribution becomes the statistical population that should be realized.

[0111] Figure 7 shows an example of an initial state set by a uniform distribution. When the initial state is uniformly distributed, the initial state 60a is a state at a randomly selected position within the state space 60 of the states represented by the qubits of the subcircuit. When the initial state 60a is randomly selected from a uniform distribution in this way, there is a possibility that a state far removed from the stationary distribution 60b may be selected. In this case, it will take a long time for the output of the subcircuit to converge to the stationary distribution 60b. This prolongation of computation time tends to be more pronounced in higher-order sets where the circuit structure becomes more complex.

[0112] Here, as a method for calculating the thermal equilibrium expectation value of physical quantities in a quantum system other than MCMC-SPU at a finite temperature, the annealing method, which simulates a high-temperature state and then gradually lowers the temperature, is expected to be effective. However, since the subcircuits in MCMC-SPU do not have temperature dependence, the annealing method cannot be directly applied to MCMC-SPU.

[0113] In MCMC-SPU, it is necessary to incorporate higher-order contributions as the temperature decreases. Therefore, the operation of lowering the temperature can be considered equivalent to the operation of adding higher-order contributions to lower-order contributions. This procedure of calculating contributions sequentially from lower-order to higher-order contributions can be applied to MCMC-SPU. In this case, if the steady distribution of lower-order factors can be effectively used in the calculation of the steady distribution of higher-order factors, the efficiency of state convergence can be expected to improve.

[0114] Therefore, the quantum computing system 300 according to the second embodiment determines the initial input state of the subcircuit to be executed from a stationary distribution obtained by a subcircuit corresponding to a set of orders lower in order than the set of orders corresponding to the subcircuit to be executed.

[0115] Figure 8 shows an example of an initial state set by a stationary distribution of low-order sets. For example, the quantum computing system 300 performs a polynomial expansion of physical quantities and generates multiple sets of orders. The quantum computing system 300 then executes the subcircuits 61a and 61b corresponding to each set of orders in ascending order of order.

[0116] When a quantum computation of subcircuit 61a corresponding to a low-order set is performed, a stationary distribution 60c of the low-order set is obtained. Next, when a quantum computation of subcircuit 61b corresponding to a high-order set is performed, the initial state 60d is determined from within the stationary distribution 60c of the low-order set. By repeatedly performing the quantum computation of subcircuit 61b with this initial state 60d as the first input state, the stationary distribution 60b of the high-order set is obtained. It is generally expected that the range containing the main contributions of the stationary distribution 60b of the high-order set is narrower than the range containing the main contributions of the stationary distribution 60c of the low-order set.

[0117] Thus, by adopting a stationary distribution 60c for lower-order sets instead of a uniform distribution as the initial distribution, the possibility of selecting a state far removed from the stationary distribution as the initial state 60d when executing the subcircuit 61b for higher-order sets is reduced.

[0118] In other words, the stationary distributions 60b and 60c of the low-order set and the high-order set are expected to have some overlap. Therefore, if the initial state 60d of the quantum computation corresponding to the high-order set is selected from within the stationary distribution 60c of the low-order set, it is suppressed that the initial state 60d will be a state far removed from the stationary distribution 60b of the high-order set. As a result, the state converges in a short time, and computational efficiency is improved.

[0119] Figure 9 shows an example of how to determine the initial state when executing subcircuits in order, starting with those corresponding to lower-order sets. In the quantum computing system 300, subcircuits 62a, 62b, 62c, ... for each set of orders are generated in the classical computer 100, and these subcircuits 62a, 62b, 62c, ... are executed in the quantum computer 200 according to instructions from the classical computer 100.

[0120] [Step 1] The classical computer 100 first determines the number of sets of degrees to be used in the calculation. For example, the classical computer 100 uses the Hefting inequality to determine the number of sets of degrees to be used in the calculation based on the required precision, etc.

[0121] [Step 2] The classical computer 100 extracts the number of degree sets obtained in Step 1, according to a probability distribution proportional to the weights of the expansion coefficients of the imaginary time evolution. [Step 3] Classical computer 100 assigns order to the degree pairs extracted in Step 2. Specifically, for degree pairs (m,n) and (m',n') [where m≦n and m'≦n'], classical computer 100 defines the order of degrees of the degree pairs, for example, if m≦m' or (m=m' and n≦n'), then (m,n)≦(m',n'). Then classical computer 100 arranges the degree pairs in ascending order of degree. This type of order is called lexicographical order.

[0122] [Step 4] The classical computer 100 generates subcircuits 62a, 62b, 62c, ... corresponding to the order assigned in Step 3. Furthermore, the classical computer 100 repeatedly causes the quantum computer 200 to execute the generated subcircuits 62a, 62b, 62c, ... and perform projection measurements on the output states, and generates a statistical ensemble based on the results of the projection measurements.

[0123] In step 4, the classical computer 100 assumes that the initial distribution of the input state (initial state) for subcircuit 62a of the first order of degree sets is a uniform distribution. For subcircuits 62b, 62c, ... of the second and subsequent order of degree sets, the classical computer 100 assumes that the initial distribution of the input state is the stationary distribution of the immediately preceding order set.

[0124] In this way, for subcircuits 62b, 62c, ... corresponding to the second and subsequent order sets, the calculation starts with the state within the steady-state distribution of the immediately preceding order set set as the initial state. As a result, the initial state is prevented from deviating too far from the steady-state distribution, and the time until the state converges is shortened.

[0125] Furthermore, by using the stationary distribution of the immediately preceding order of order, a stationary distribution that is expected to have a large overlap with the stationary distribution of the order of order being calculated is used as the initial state. The large overlap between the stationary distributions of the immediately preceding order of order and the order of order being calculated can be understood from the similarity of the subcircuits corresponding to the order of order.

[0126] Figure 10 shows an example of a subcircuit corresponding to a set of orders. Figure 10 shows a subcircuit 63 corresponding to a set of first-order and second-order orders, and a subcircuit 64 corresponding to a set of second-order and second-order orders.

[0127] In the subcircuit 63, an Adamard gate is initially placed in the auxiliary qubit 63a. Subsequently, unitary gates 63b to 63d, representing the basic structure by matrix "W", are placed in the qubits of the system of interest. Unitary gate 63b is negative polarity (acting when auxiliary qubit 63a is "0"), while unitary gates 63c and 63d are positive polarity (acting when auxiliary qubit 63a is "1"). Finally, an Adamard gate 63e is placed in the auxiliary qubit.

[0128] In the subcircuit 64, an Adamard gate is initially placed in the auxiliary qubit 64a. Subsequently, unitary gates 64b to 64e, represented by a predetermined matrix "W", are placed in the qubits of the system of interest. Unitary gates 64b and 64c are negative polarity, while unitary gates 64d and 64e are positive polarity. Finally, an Adamard gate 64f is placed in the auxiliary qubit.

[0129] Thus, the subcircuits corresponding to the m-th and n-th order pairs operate a predetermined unitary gate m times under the condition that the auxiliary qubit is in the "0" state, and operate a predetermined unitary gate n times under the condition that the auxiliary qubit is in the "1" state. Therefore, when comparing any two subcircuits, the number of unitary gates in the basic structure differs by the difference in the value of m+n in the order pair.

[0130] Comparing the sum of the corresponding order pairs (m+n) of subcircuit 63 and subcircuit 64, the difference is "1". Therefore, subcircuit 63 and subcircuit 64 differ by only one in the number of control gates in their basic structure.

[0131] When there are two subcircuits corresponding to pairs of degrees, the smaller the difference in their degrees, the smaller the difference between the subcircuits. If the circuit structures of the subcircuits are similar, the stationary distributions obtained from them will also be similar. In other words, when there are two pairs of degrees, if the difference in their degrees is small, the region in which their stationary distributions overlap will be larger.

[0132] Next, we will specifically describe the functions of the quantum computing system 300 for calculating the thermal equilibrium expectation value at a finite temperature using an MCMC-SPU to which the initial state determination method shown in Figure 9 is applied.

[0133] Figure 11 is a block diagram showing an example of the functions of a quantum computing system. The quantum computer 200 has a gate operation unit 210 that performs gate operations on qubits in a qubit device 202, and a measurement unit 220 that performs projection measurements of the qubit state.

[0134] The gate operation unit 210 performs gate operations on the qubit 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 operated on.

[0135] The measurement unit 220 is a device that measures the state of qubits within the qubit device 202. For example, the measurement unit 220 irradiates the qubit to be measured with microwaves and performs a projection measurement of the computational basis (Z basis).

[0136] The classical computer 100 includes a quantum device control unit 110, a measurement result statistics processing unit 120, a calculation result storage unit 130, and a weighted average calculation unit 140. The quantum device control unit 110 controls the qubit device 202 based on a quantum circuit that represents the procedure for quantum computation. For example, the quantum device control unit 110 controls the qubit device 202 according to a quantum circuit (a subcircuit for each set of degrees obtained by polynomial expansion of the imaginary time evolution) for calculating the finite temperature expectation value of a physical quantity.

[0137] The measurement result statistical processing unit 120 performs statistical processing on the measurement results of the quantum circuit. For example, the measurement result statistical processing unit 120 calculates the mean or standard deviation of the measurement results from multiple measurements and determines whether or not the measurement results have converged.

[0138] The calculation result storage unit 130 stores the calculation results of physical quantities calculated for each subcircuit. The calculation result storage unit 130 is, for example, part of the storage area of ​​memory 102 or storage device 103.

[0139] The weighted average calculation unit 140 calculates the weighted average of the calculation results for each subcircuit. The weighted average calculation unit 140 outputs the calculated weighted average as the finite temperature expected value. The quantum computing system 300, which possesses these capabilities, enables efficient calculation of finite temperature expectation values.

[0140] The functions of each element shown in Figure 11 can be realized, for example, by having the processor 101 execute the program module corresponding to that element. Figure 12 is a flowchart showing an example of the procedure for calculating the finite temperature expectation value. The process shown in Figure 12 will be explained below according to the step numbers.

[0141] [Step S101] The quantum device control unit 110 of the classical computer 100 accepts inputs of the number of qubits N of the system of interest, the Hamiltonian H, the inverse temperature β, the truncation order M of the polynomial expansion, and the number of sets of polynomial orders M (with ~).

[0142] [Step S102] The quantum device control unit 110 performs a polynomial expansion of the imaginary time evolution and extracts M pairs of degrees from the degrees of the terms included in the expanded expression. For example, the quantum device control unit 110 performs a polynomial expansion of the imaginary time evolution expression up to the truncation degree M. This generates polynomials from degree 0 to degree M. The quantum device control unit 110 extracts pairs of degrees from the degrees from degree 0 to degree M, allowing for duplicate extraction of already extracted degrees. This process is repeated M times, resulting in the generation of M pairs of degrees. The quantum device control unit 110 also ranks the extracted pairs of degrees in descending order of the sum of their degrees.

[0143] [Step S103] The classical computer 100 repeats the process from steps S104 to S114, counting up the value of the loop variable m from "0" to "M-1" (where M is preceded by ~).

[0144] [Step S104] The quantum device control unit 110 generates a subcircuit corresponding to the extracted m-th order set. For example, the quantum device control unit 110 generates a unitary gate that represents the procedure for quantum computation of the polynomial corresponding to each order included in the set. The number of qubits operated by the generated unitary gate is N. Then, the quantum device control unit 110 generates a subcircuit containing two unitary gates corresponding to the order set, as shown in Figure 5.

[0145] The quantum device control unit 110 then arranges the input states of the generated subcircuits into an array In m-1 Select from the following. For example, the quantum device control unit 110 selects the data sequence (array In) after the stationary distribution has been reached for a set of low-order numbers. m-1 When you obtain the data, the number of data points in the obtained data column is N. data as N data The following natural numbers are randomly selected according to a uniform distribution. The quantum device control unit 110 then selects a state labeled with the randomly selected natural number element from the computational basis and uses it as the initial state to input to the quantum circuit corresponding to the higher-order set. Through this procedure, an initial state according to a stationary distribution corresponding to the lower-order set is selected.

[0146] [Step S105] The quantum device control unit 110 repeatedly executes the process in step S106 for a predetermined number of shots (number of repetitions) for the generated subcircuit. [Step S106] The quantum device control unit 110 instructs the quantum computer 200 to perform gate operations and measurements according to the generated subcircuits. For example, the quantum device control unit 110 transmits control signals for gate operations or measurements to the gate operation unit 210 or the measurement unit 220 in the order shown in the subcircuits. As a result, the gate operation unit 210 performs gate operations on the qubits, and the measurement unit 220 measures the state of the qubits after the gate operations.

[0147] The quantum device control unit 110 sets the input state selected in step S104 as the input state to the subcircuit generated during the first physical quantity calculation. For subsequent physical quantity calculations (repeated processing after steps S111 to S113), the quantum device control unit 110 sets the state updated in step S112 (described later) as the input state to the generated subcircuit.

[0148] The gate operation unit 210 performs gate operations on the qubits according to the control signals from the quantum device control unit 110. Once the gate operations corresponding to the subcircuit are completed, the measurement unit 220 measures the state of the qubits and auxiliary qubits of the system of interest. The measurement unit 220 transmits the measurement results to the classical computer 100. In the classical computer 100, the measurement result statistics processing unit 120 receives the measurement results.

[0149] [Step S107] When the quantum device control unit 110 has completed the quantum gate operation and measurement instructions in step S106 for a predetermined number of shots, it proceeds to step S108.

[0150] [Step S108] The measurement result statistical processing unit 120 calculates the physical quantity obtained from the measurement results of the generated subcircuit (shown on the right side of equation (1) <φ) based on the measurement results for the number of shots. ab ik |O|φ ab ik >) is calculated. The measurement result statistical processing unit 120 processes the calculated physical quantities into array r m Add to it.

[0151] [Step S109] The measurement result statistical processing unit 120 processes the sequence r m Calculate the average of the values ​​within. [Step S110] The measurement result statistical processing unit 120 determines whether the physical quantities obtained by the quantum computation of the generated subcircuits have converged. For example, the measurement result statistical processing unit 120 determines that convergence has occurred if the change in the average calculated in step S109 is less than or equal to a predetermined value. The measurement result statistical processing unit 120 also determines the array r m The system may determine that convergence has occurred when the standard deviation of the values ​​falls below a predetermined value. If convergence has occurred, the measurement result statistical processing unit 120 proceeds to step S114. If convergence has not occurred, the measurement result statistical processing unit 120 proceeds to step S111.

[0152] [Step S111] The quantum device control unit 110 instructs the quantum computer 200 to perform quantum gate operations and projection measurements of the computational basis based on the generated subcircuits. At this time, the input state of the qubits of the system of interest is the same as the input state during the previous quantum computation for calculating physical quantities (quantum gate operations in step S106).

[0153] In the quantum computer 200, gate operations are performed on the qubit device 202 according to instructions from the quantum device control unit 110. Then, the measurement unit 220 performs projection measurements of the computational basis of the qubits of the system of interest after the gate operations according to the subcircuit. The gate operations and projection measurements according to the decomposition circuit are repeated for a predetermined number of shots. The result of one projection measurement is represented by a bit string consisting of the measurement results (|0> or |1>) of the number of qubits in the system of interest. This bit string is the unit data, and a data string showing the result for each repeatedly performed projection measurement is output from the quantum computer 200 as the measurement result.

[0154] [Step S112] The quantum device control unit 110 updates the input state for the next quantum computation based on the measurement results obtained in step S111. For example, the quantum device control unit 110 uses one data (a bit sequence of |0> or |1>) from the data sequences shown in the measurement results of each qubit of the system of interest as the input state for the next physical quantity calculation.

[0155] As a result, the computational basis obtained from the projection measurement of the output state becomes the input state of the qubit in the system of interest for the next physical quantity calculation. This updating of the input state is then repeated until the calculated physical quantity converges.

[0156] [Step S113] The quantum device control unit 110 processes the measurement results (data sequence) obtained in step S111 into the array In m Recorded in step S112. If the quantum computation in steps S105 to S107 using the input state updated in step S112 converges, the array In recorded in step S113 is recorded. m From there, it is selected as the input state for the quantum computation of the subcircuit corresponding to the m+1th order set.

[0157] After step S113, the quantum device control unit 110 proceeds to step S105. [Step S114] When the physical quantity converges, the measurement result statistical processing unit 120 stores the average value calculated in step S109 in the array Avg in the calculation result storage unit 130.

[0158] [Step S115] If the quantum device control unit 110 has completed the processing in steps S104 to S114 for a set of order M(with ~), it proceeds to step S116.

[0159] [Step S116] The weighted average calculation unit 140 calculates the weighted average of the contributions of the subcircuits. For example, the weighted average calculation unit 140 calculates according to equation (3) <o> β The weighted average calculation unit 140 calculates the weighted average and outputs the result as the expected value at a finite temperature.

[0160] In this way, the expected value of a physical quantity at a finite temperature can be calculated based on the results of quantum computations performed by the quantum computer 200 according to multiple subcircuits. At this time, the stationary distribution obtained as a result of the quantum computation of the subcircuits corresponding to lower-order sets is used as the initial distribution in the quantum computation of the subcircuits corresponding to higher-order sets. As a result, the computation results in the quantum computation of the subcircuits corresponding to higher-order sets converge efficiently to the stationary distribution. Note that the subcircuits are smaller in scale and have shallower circuit depths compared to the quantum circuit 30 shown in Figure 4. Therefore, if they are subcircuits, they can be executed even in early FTQC.

[0161] Furthermore, in the iterative calculations of a physical quantity obtained using a certain subcircuit until it converges, the classical computer 100 uses the result of a projection measurement of the computational basis of the output state after quantum computation by that subcircuit as the input state for the next physical quantity calculation. As a result, the input state to the subcircuit becomes a statistical sect suitable for that subcircuit.

[0162] Specifically, obtaining a statistical ensemble suitable for a subcircuit means "|Φ ab ik >" is the probability distribution "W ab ik / Σ j W ab jk This is obtained according to the above. In order to efficiently obtain such a statistical ensemble, the state obtained as a result of the projection measurement of the computational basis of the output state of the subcircuit is used as the next input state. This means that in the repeated cycle of "1. Selection of input state", "2. Part of imaginary time evolution (contribution of components of the degree set)", "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 process is similar to the METTS algorithm, and the input state in the repeated cycle transitions probabilistically, and the stationary distribution obtained as the convergence destination becomes a statistical ensemble suitable for the subcircuit.

[0163] The action of each subcircuit is a partial contribution to the imaginary time evolution represented by the polynomial corresponding to the extracted set of degrees. The physical quantities obtained after convergence of each subcircuit represent the subcircuit's contribution to the thermal equilibrium expectation value of the physical quantities of the system of interest. Therefore, the final thermal equilibrium expectation value of the physical quantities of the system of interest can be obtained by taking a weighted average of the calculation results of the physical quantities by each of the multiple subcircuits.

[0164] Next, we will specifically explain an example of calculating the thermal equilibrium expectation value of a physical quantity using the transverse magnetic field Ising model. Figure 13 shows 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 defined on the lattice points. Spins are defined at sites 71 to 73. Interactions act between adjacent spins. Using the Ising model 70, the orientation of the spins at sites 71 to 73 when a transverse magnetic field is applied can be calculated by numerical simulation.

[0165] For example, the Hamiltonian H of the Ising model 70 is given by the following equation (5).

[0166]

number

[0167] X i This is the Pauli operator that describes the X-direction component of the spin at the i-th (where i is a natural number) site. j This is the Pauli operator that describes the X-direction component of the spin at the j-th (where j is a natural number) site. i is the Pauli operator describing the Z-direction 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.

[0168] By representing the system of interest using the Ising Model 70, we can determine the expected value of the energy in thermal equilibrium of the system of interest. For example, the expected value of the energy in thermal equilibrium can be obtained as the canonical average of the eigenvalues ​​of the Hamiltonian H of the Ising Model 70.

[0169] Here, we consider the system of interest to be an 8-qubit system. The coefficients of the Hamiltonian are set as follows.

[0170]

number

[0171]

number

[0172]

number

[0173] The criterion for convergence to a stationary distribution is when the Gelman-Rubin statistic calculated for data sequences generated from different initial states falls below "1.1". The order of the Chebyshev polynomial expansion is assumed to be 3.

[0174] The quantum computing system 300 first determines the stationary distribution for a combination of two quadratic states. Assuming that the initial distribution follows a uniform distribution, the quantum computing system 300 selects two typical states as initial states.

[0175] The quantum computing system 300 then calculates stationary distributions for combinations of second-order and third-order systems. In doing so, the quantum computing system 300 is instructed to perform calculations using the stationary distribution for the second-order and second-order combination obtained above as the initial distribution, and to perform calculations using a uniform distribution as the initial distribution, and the results of these calculations are compared.

[0176] The following shows a comparison of the number of steps required to converge to a stationary distribution when the quantum computing system 300 is subjected to calculations under these conditions. Note that one step is equivalent to one iteration of steps S105 to S113 shown in Figure 12.

[0177] Figure 14 shows an example of convergence to a steady-state distribution obtained for a pair of quadratic and quadratic circuits. Graph 81 shows the change in energy obtained from measurement results when the execution of the subcircuit and projection measurement are repeated. In Graph 81, the horizontal axis is the number of iterations and the vertical axis is energy. The initial input state is selected from the following two computational basis states. |00000000>State |01010101>Status The circular plot 81a shows the average (expected) energy obtained from the execution of the subcircuit up to the number of iterations indicated on the horizontal axis, starting with the input state |00000000>. The solid vertical line 81b shows the variance of energy obtained from the execution of the subcircuit up to the number of iterations indicated on the horizontal axis, starting with the input state |00000000>.

[0178] The rectangular plot 81c shows the average (expected) energy obtained from the execution of the subcircuit up to the number of iterations indicated on the horizontal axis, starting with the input state |01010101>. The dashed vertical line 81d shows the variance of energy obtained from the execution of the subcircuit up to the number of iterations indicated on the horizontal axis, starting with the input state |01010101>.

[0179] By starting calculations from different initial states, the stationary distribution can be determined to have converged when the expected value and variance become statistically close. In the example shown in Graph 81, the numerical calculation results indicate that convergence occurred in 30 steps.

[0180] Figure 15 shows an example of convergence to a steady distribution obtained for the quadratic and cubic pairs without using the steady distributions of other order pairs. Graph 82 shows the change in energy obtained from measurement results when the execution of the subcircuit and projection measurement are repeated. In Graph 82, the horizontal axis is the number of iterations and the vertical axis is energy. The initial input state is selected from the following two computational basis sets. |00000000>State |01010101>Status The circular plot 82a shows the average (expected) energy obtained from the execution of the subcircuit up to the number of iterations indicated on the horizontal axis, starting with the input state |00000000>. The solid vertical line 82b shows the variance of energy obtained from the execution of the subcircuit up to the number of iterations indicated on the horizontal axis, starting with the input state |00000000>.

[0181] The rectangular plot 82c shows the average (expected) energy obtained from the execution of the subcircuit up to the number of iterations indicated on the horizontal axis, starting with the input state |01010101>. The dashed vertical line 82d shows the variance of energy obtained from the execution of the subcircuit up to the number of iterations indicated on the horizontal axis, starting with the input state |01010101>.

[0182] The calculation results shown in Graph 82 were judged to have converged in 90 steps under the same conditions as the example in Figure 14. Figure 16 shows an example of convergence to a steady-state distribution obtained for a quadratic and cubic pair using the steady-state distribution of a quadratic and quadratic pair. Graph 83 shows the change in energy obtained from measurement results when the execution of the subcircuit and projection measurement are repeated. In Graph 83, the horizontal axis is the number of iterations and the vertical axis is energy. In the example in Figure 16, the following states are selected as input states from the steady-state distribution of the quadratic and quadratic pair. |11111001>Status Graph 83 also shows the calculation results when the following two states are used as input states for determining convergence. |00000000>State |01010101>Status The star-shaped plot 83a shows the average (expected) energy obtained from the execution of the subcircuit up to the number of iterations indicated on the horizontal axis, starting with the input state |11111001>. The solid vertical line 83b shows the variance of energy obtained from the execution of the subcircuit up to the number of iterations indicated on the horizontal axis, starting with the input state |11111001>.

[0183] The circular plot 83c shows the average (expected) energy obtained from the execution of the subcircuit up to the number of iterations indicated on the horizontal axis, starting with the input state |00000000>. The dashed vertical line 83d shows the variance of energy obtained from the execution of the subcircuit up to the number of iterations indicated on the horizontal axis, starting with the input state |00000000>.

[0184] The rectangular plot 83e shows the average (expected) energy obtained from the execution of the subcircuit up to the number of iterations indicated by the value on the horizontal axis, starting from when the calculation was initiated with the input state |01010101>.

[0185] The calculation results shown in Graph 83 were evaluated for convergence under the same conditions as the example in Figure 14, and it was determined that convergence occurred in 40 steps. In other words, by selecting the initial input state from within a stationary distribution consisting of a pair of quadratic and quadratic distributions, convergence was approximately twice as fast compared to selecting from a uniform distribution as shown in Figure 16.

[0186] As explained above, in the calculation process based on the dispersion processing of the finite temperature expectation value of a physical quantity, the process becomes more efficient by selecting an initial distribution from a set of stationary distributions of low order to efficiently generate a statistical sect corresponding to the subcircuit from which higher-order components have been extracted. Moreover, the subcircuit can be implemented even on a medium-scale quantum computer, and calculations in the low-temperature region where the contribution of higher-order components is important can be performed efficiently.

[0187] For example, when calculating the thermal equilibrium expectation value of energy at a finite temperature in a transverse field Ising model, using the steady-state distribution for a pair of second-order and second-order energy factors as the initial distribution for finding the steady-state distribution for a pair of second-order and third-order energy factors results in convergence twice as fast compared to not using it. Therefore, when calculating the energy expectation value at a certain temperature, it is possible to reduce the number of samples used without degrading accuracy.

[0188] [Other embodiments] In the second embodiment, an example of calculating the thermal equilibrium expectation value of energy using the transverse magnetic field Ising model was shown, but the calculation of the thermal equilibrium expectation value of other physical quantities at finite temperatures can also be performed efficiently.

[0189] Although embodiments have been illustrated above, the configurations of each part shown in the embodiments can be replaced with others having similar functions. Furthermore, other arbitrary components or processes may be added. Moreover, any two or more configurations (features) from the embodiments described above may be combined. [Explanation of Symbols]

[0190] 1. Quantum computer 2a The first quantum circuit 2b The second quantum circuit 3a, 3b, ... Values ​​of physical quantities 10 Information Processing Devices 11 Storage section 12 Processing Units< / o> < / o> < / o> < / o> < / o> < / o>

Claims

1. The imaginary time evolution equation for calculating the thermal equilibrium expectation value of the physical quantities of the system under investigation at a finite temperature is expanded into equations of multiple orders. Multiple pairs of degrees are generated by extracting two degrees from the aforementioned multiple degrees. For each of the aforementioned sets, a quantum circuit is generated that shows the procedure for quantum computation of the value of the physical quantity obtained by a partial imaginary time evolution based on a first-order equation and a second-order equation included in the set. As the input state at the start of the quantum computation based on the first quantum circuit corresponding to at least one of the plurality of sets, the state within the stationary distribution of the measurement results obtained by the quantum computation based on the second quantum circuit corresponding to a second set other than the first set among the plurality of sets is used, thereby causing the quantum computer to repeatedly execute the quantum computation based on the corresponding quantum circuit for each of the plurality of sets until the value of the physical quantity obtained from the results of the quantum computation converges. Based on the converged values ​​of the physical quantities in each of the multiple sets, the thermal equilibrium expectation value of the physical quantity at a finite temperature is calculated. A quantum computing support program that allows a computer to perform a process.

2. In the process of having the quantum computer perform the quantum computation based on the quantum circuit for each of the multiple sets, the set of the multiple sets that combines a lower order than the first set is determined to be the second set. The quantum computing support program according to claim 1.

3. In the process of having the quantum computer execute the quantum computation based on the quantum circuit for each of the multiple sets, the multiple sets are ranked in ascending order of order of degree, the quantum computation based on the quantum circuit for each of the multiple sets is executed in ascending order of rank, and when the rank of the first set is N+1 (where N is a natural number), the set with rank N is determined to be the second set. The quantum computing support program according to claim 1.

4. In the process of causing the quantum computer to perform the quantum computation based on the quantum circuit for each of the multiple sets, the output state after the quantum computation according to the quantum circuit corresponding to the set is used as the input state in the subsequent quantum computation, and the quantum computer is repeatedly caused to perform the quantum computation based on the quantum circuit until the value of the physical quantity obtained from the result of the quantum computation converges. The quantum computing support program according to claim 1.

5. In the process of causing the quantum computer to execute the quantum computation based on the quantum circuit for each of the multiple sets, one state is selected from a plurality of states shown in the calculation results of multiple quantum computations based on the second quantum circuit after the measurement results of the quantum computation based on the second quantum circuit have become a stationary distribution, and the selected state is set as the input state at the start of the quantum computation based on the first quantum circuit. The quantum computing support program according to claim 1.

6. The imaginary time evolution equation for calculating the thermal equilibrium expectation value of the physical quantities of the system under investigation at a finite temperature is expanded into equations of multiple orders. Multiple pairs of degrees are generated by extracting two degrees from the aforementioned multiple degrees. For each of the aforementioned sets, a quantum circuit is generated that shows the procedure for quantum computation of the value of the physical quantity obtained by a partial imaginary time evolution based on a first-order equation and a second-order equation included in the set. As the input state at the start of the quantum computation based on the first quantum circuit corresponding to at least one of the plurality of sets, the state within the stationary distribution of the measurement results obtained by the quantum computation based on the second quantum circuit corresponding to a second set other than the first set among the plurality of sets is used, thereby causing the quantum computer to repeatedly execute the quantum computation based on the corresponding quantum circuit for each of the plurality of sets until the value of the physical quantity obtained from the results of the quantum computation converges. Based on the converged values ​​of the physical quantities in each of the multiple sets, the thermal equilibrium expectation value of the physical quantity at a finite temperature is calculated. A quantum computing support method that allows a computer to perform a process.

7. The process involves expanding an imaginary time evolution equation for calculating the thermal equilibrium expectation value of a physical quantity of a system under calculation at a finite temperature into equations of multiple orders, generating multiple sets of orders by extracting two orders from the multiple orders, generating a quantum circuit for each of the multiple sets that shows the procedure for quantum computation of the value of the physical quantity obtained by partial imaginary time evolution based on the first order equation and the second order equation included in the set, using a state within the steady distribution of measurement results obtained by the quantum computation based on a second quantum circuit, which is different from the first set among the multiple sets, as the input state at the start of the quantum computation based on the first quantum circuit, which is corresponding to at least one of the multiple sets, causing the quantum computer to repeatedly execute the quantum computation based on the corresponding quantum circuit until the value of the physical quantity obtained from the quantum computation results converges, and calculating the thermal equilibrium expectation value of the physical quantity at a finite temperature based on the converged value of the physical quantity for each of the multiple sets, An information processing device having

Images