Quantum calculation method, quantum calculation system, and program

Abstract

To efficiently solve a combinatorial optimization problem using a quantum device with a limited coherence time.SOLUTION: To solve a combinatorial optimization problem with constraints using an Ising model represented by a plurality of spin variables, a quantum device performs quantum annealing according to an annealing path, which is a parameter representing the strength of quantum fluctuations in the Ising model, to generate a quantum state of the plurality of spin variables (102), and generates a probability distribution of the quantum state (104). A classical computer performs post-processing to convert infeasible solutions that do not satisfy the constraints into feasible solutions that satisfy the constraints on the basis of a solution representing the quantum state (106), calculates a new probability distribution after the post-processing (108), calculates an energy expectation value of the Ising model on the basis of the new probability distribution (110), and updates the annealing path to a new annealing path to minimize the energy expectation value (112).SELECTED DRAWING: Figure 1

Description

Technical Field

[0001] The present invention relates to a quantum computing method, a quantum computing system, and a program for solving a constrained combinatorial optimization problem.

Background Art

[0002] A combinatorial optimization problem is a problem of obtaining a combination of variables that minimizes or maximizes an objective function under constraint conditions in order to select the best combination from a large number of combinations. A combinatorial optimization problem is generally converted into a problem of searching for the ground state of an Ising model on a quantum device such as a quantum annealer or a gate-type quantum computer. In recent years, various quantum algorithms for solving large-scale combinatorial optimization problems with constraints have been proposed (see, for example, Non-Patent Document 1, Non-Patent Document 2, and Non-Patent Document 3).

Prior Art Documents

Non-Patent Documents

[0003]

Non-Patent Document 1

Non-Patent Document 2

Non-Patent Document 3

Summary of the Invention

Problems to be Solved by the Invention

[0004] Due to the influence of noise, there is an upper limit to the time (coherence time) during which a quantum device can execute a quantum algorithm without errors. However, there is a problem that it takes a very long time to solve large-scale combinatorial optimization problems using conventional quantum algorithms.

[0005] The present invention has been made in view of the above problems, and an object thereof is to provide a quantum computing method, a quantum computing system, and a program for efficiently solving combinatorial optimization problems using a quantum device with a limited coherence time.

Means for Solving the Problems

[0006] The quantum computing method according to the present invention is a quantum computing method for solving a combinatorial optimization problem with constraints using an Ising model represented by a plurality of spin variables. By executing quantum annealing according to an annealing path, which is a parameter representing the strength of the quantum fluctuation of the Ising model, by a quantum device, a quantum state of a plurality of spin variables is generated. By a quantum device, a probability distribution of the quantum state is generated. By a classical computer, post-processing is executed to convert a non-executable solution that does not satisfy the constraints into an executable solution that satisfies the constraints based on the solution representing the quantum state, and a new probability distribution after the post-processing is calculated. By a classical computer, an energy expectation value of the Ising model is calculated based on the new probability distribution, and the annealing path is updated to a new annealing path so that the energy expectation value becomes smaller.

[0007] The quantum computing system according to the present invention is a quantum computing system for solving a combinatorial optimization problem with constraints using an Ising model represented by a plurality of spin variables. It includes a quantum device that generates a quantum state of a plurality of spin variables and generates a probability distribution of the quantum state by executing quantum annealing according to an annealing path, which is a parameter representing the strength of the quantum fluctuation of the Ising model, and a classical computer that executes post-processing to convert a non-executable solution that does not satisfy the constraints into an executable solution that satisfies the constraints based on the solution representing the quantum state, calculates a new probability distribution after the post-processing, calculates an energy expectation value of the Ising model based on the new probability distribution, and updates the annealing path to a new annealing path so that the energy expectation value becomes smaller.

[0008] The program according to the present invention is a program for causing a classical computer to execute in combination with a quantum device a quantum computing method for solving a combinatorial optimization problem with constraints using an Ising model represented by a plurality of spin variables. When a probability distribution of quantum states of a plurality of spin variables is generated by executing quantum annealing by a quantum device, based on a solution representing the quantum state, post-processing is executed to convert an infeasible solution that does not satisfy the constraints into a feasible solution that satisfies the constraints, a new probability distribution after the post-processing is calculated, an energy expectation value of the Ising model is calculated based on the new probability distribution, and an annealing path, which is a parameter of the quantum annealing, is updated to a new annealing path so that the energy expectation value becomes smaller.

Effect of the Invention

[0009] According to the present invention, after a quantum device executes quantum annealing according to an annealing path to generate a probability distribution of quantum states, a classical computer executes post-processing for converting an infeasible solution into a feasible solution, and updates the annealing path so that the energy expectation value becomes smaller based on a new probability distribution of the quantum states after the post-processing. As a result, the annealing path is optimized, and it becomes possible to efficiently solve a combinatorial optimization problem using a quantum device with a limited coherence time.

Brief Description of the Drawings

[0010]

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Embodiments for Carrying Out the Invention

[0011] Hereinafter, embodiments of the present invention will be described with reference to the drawings.

[0012] First, the Ising model and quantum annealing will be briefly described. The combinatorial optimization problem (hereinafter referred to as COP) is generally converted into a ground state search problem of the Ising model. The Ising model is defined as in Equation (1) on an undirected graph G = (V, E).

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[0013] Here, V is a set of vertices, and E is a set of edges. σ i x 、σ i y 、σ i z are, respectively, the x - component, y - component, and z - component of the Pauli matrix acting on site i, J ij is the spin variable σ i z and σ j z represents the interaction coefficient between, and hi is the spin variable σ i z represents the external magnetic field coefficient above, and H0 is a constant.

[0014] The Hamiltonian H of the constrained COP Ising model Ising is given as in Equation (2). H obj , H cst are the objective function and the constraint term, respectively.

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[0015] A solution that satisfies the constraints is called a feasible solution, and a solution that does not satisfy the constraints is called an infeasible solution.

[0016] Here, the cost function C(σ) is given as in Equation (3), and for i ∈ V, |σ i 〉 is the eigenbasis of σ i z . That is, for σ i ∈ {-1, 1}, σ i z |σ i 〉 = σ i |σ i 〉.

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[0017] The constrained COP can be solved by converting it into a quadratic unconstrained binary optimization (QUBO) model. The QUBO model is an energy function represented by a quadratic expression of a spin variable x i that takes binary values of 0 and 1. The Hamiltonian of the QUBO defined on an undirected graph G q =(V q , E q ) is given as in Equation (4).

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[0018] The spin variable x in QUBO i ∈ {0, 1} is replaced by (σ i z + 1) / 2, and the Ising model defined by Equation (1) is obtained.

[0019] Quantum annealing is a method for exploring the ground state of the Ising model using quantum fluctuations. In quantum annealing, as given by Equation (5), the quantum state |ψ(t)〉 evolves in time according to the time-dependent Schrödinger equation. Here, the Planck constant h is set to h / 2π = 1.

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[0020] H(t) in Equation (5) is represented by the Hamiltonian H of the Ising model Ising and the transverse magnetic field Hamiltonian H corresponding to quantum fluctuations q as shown in Equation (6).

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[0021] s(t) is a parameter representing the strength of quantum fluctuations at time t and is called the annealing path. s(t) takes values between 0 and 1 due to the hardware constraints of the quantum device that executes quantum annealing. The scheduling of the annealing path s(t) will be described later (see Figures 3 and 4).

[0022] As shown in Equation (1), the Ising Hamiltonian H Ising is represented by the z-component σ of the Pauli matrix i z while the transverse magnetic field Hamiltonian H q is represented by the x-component σ of the Pauli matrix as shown in Equation (7) i x

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[0023] The initial state |ψ(0)〉 is set to the ground state of the transverse magnetic field Hamiltonian H q . Denoting the annealing time as T, the output state |ψ(T)〉 is obtained by solving Equation (5) from t = 0 to t = T.

[0024] In the quantum computing method according to this embodiment, the optimal solution (ground state) of the constrained COP is searched for using the variational quantum algorithm (VQA) and post-processing. VQA is a method for finding an optimal annealing path that leads to a (quasi-)optimal solution in a short time by the variational method. Post-processing is a method for converting the output solution of the quantum device into an executable solution that satisfies the constraints of the COP. In this embodiment, the method that combines VQA and post-processing is called the post-processing variational quantum algorithm (pVQA).

[0025] Figure 1 is a flowchart showing the quantum computing method of this embodiment. First, the Ising model of the COP and the initial value s (1) (t) of the annealing path at t ∈ [0, T] are input into the quantum device. Then, the annealing path is updated by repeating Steps 102 to 112 by the quantum device and the classical computer. The annealing path used in the m-th quantum annealing is denoted as s (m) (t), and the annealing path obtained after m updates is denoted as s (m+1) (t).

[0026] The quantum device generates the quantum state |ψ(T)〉 by performing quantum annealing along the current annealing path s (m) (t) (Step 102).

[0027] Next, the quantum device generates the probability distribution p′(σ) of the quantum state |ψ(T)〉 generated in Step 102 (Step 104). p′(σ) is defined as in Equation (8). In practice, the probability distribution is approximately obtained by repeating the measurement of the quantum state |ψ(T)〉.

Equation

[0028] Next, the classical computer performs post-processing to convert the non-executable solution into an executable solution (step 106), and calculates the new probability distribution p(σ) after the post-processing (step 108). The post-processing is defined by the mapping P: Pσ′→σ of the solution σ′ obtained when measuring the quantum state |ψ(T)〉 generated in step 102 in the {σ i z} basis. Specific examples of the post-processing will be described later. The new probability distribution p(σ) is given as in Equation (9).

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[0029] Next, the classical computer calculates the energy expectation value E based on the new probability distribution p(σ) obtained in step 108 (step 110). The energy expectation value E is represented as the weighted sum of the cost function C(σ) (Equation (3)) and the probability distribution p(σ) (Equation (9)), and is defined as in Equation (10).

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[0030] Next, the classical computer updates the current annealing path s (m) (t) to a new annealing path s (m+1) (t) so that the energy expectation value E calculated in step 110 becomes smaller (step 112). The update of the annealing path is performed using a known optimization algorithm such as the gradient descent method or the Powell method.

[0031] Steps 102 to 112 are repeated until the energy expectation value E converges (until the solution stops being updated). As described above, steps 102 and 104 are executed by the quantum device, and steps 106, 108, 110, and 112 are executed by the classical computer.

[0032] As an application example of pVQA, consider a COP that minimizes -x1 under the constraint x1 + x2 = 1 for two spin variables x1, x2 ∈ {0, 1}. The Hamiltonian (energy function) in the QUBO model of this COP is formulated, for example, as in Equation (11). Here, H obj = -x1, H cst = 5(x1 + x2 - 1) 2 is as follows.

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[0033] Figure 2A shows the energy values (values of H) of the solutions of the COP represented by Equation (11). In Figure 2A, (x1, x2) = (1, 0), (0, 1) are feasible solutions, (x1, x2) = (0, 0), (1, 1) are infeasible solutions, and (x1, x2) = (1, 0) is the optimal solution.

[0034] Post - processing can be performed based on a local optimization method. The post - processing based on the local optimization method uses the property that all local minimum solutions are feasible solutions. A local minimum solution is defined as a solution whose energy value cannot be lowered by flipping the spin variable x i . Flipping means converting the value of the spin variable from 0 to 1 or from 1 to 0. The local optimization method is a method that lowers the energy value by flipping the spin variable. Since an infeasible solution is not a local minimum solution, by repeatedly applying the local optimization method to lower the energy value, a feasible solution that is a local minimum solution can be obtained.

[0035] An example of the local optimization method is the greedy method. An example of post - processing using the greedy method for the constrained COP shown in Figure 2A will be described.

[0036] The classical computer calculates the energy value E0 of the tentative solution (x1, x2), the energy value E1 of the solution obtained by flipping x1 of (x1, x2), and the energy value E2 of the solution obtained by flipping x2 of (x1, x2). When E1 is the minimum among E0, E1, and E2, the classical computer performs a post-processing operation of flipping x1 of (x1, x2). When E2 is the minimum among E0, E1, and E2, the classical computer performs a post-processing operation of flipping x2 of (x1, x2). When E0 is the minimum among E0, E1, and E2, the tentative solution (x1, x2) is maintained.

[0037] In the example of FIG. 2A, by the post-processing based on the greedy method, both of the non-executable solutions (0, 0) and (1, 1) are converted into the executable solution (1, 0). On the other hand, the executable solutions (1, 0) and (0, 1) are not converted. Therefore, the probability of the executable solution is increased by the post-processing.

[0038] As shown in FIG. 2B, while the original probability distribution p′ spreads also to the non-executable solutions (0, 0) and (1, 1), the new probability distribution p after the post-processing is distributed only to the executable solutions (0, 0) and (1, 1).

[0039] <Annealing schedule> Next, with reference to FIGS. 3 and 4, the scheduling of the annealing path s(t), which is a parameter of pVQA, will be described. As shown in FIG. 3, the annealing path s(t) has three types of schedules: a continuous schedule, a linear schedule, and a discrete schedule.

[0040] The continuous schedule is generally formulated by a continuous function for the annealing path s(t). Although it shows the best performance among the three types of schedules, its realization on existing quantum devices is difficult. On the other hand, the linear schedule is executable on a quantum annealer, and the discrete schedule is easily executable on a gate-type quantum computer.

[0041] The linear schedule has two variational parameters s1 and s2, representing the values of s(t) at t = 0 and t = T respectively. The annealing path s(t) of the linear schedule is given by Equation (12).

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[0042] Note that in previous research, s1 = s(0) = 0 and s2 = s(T) = 1 were fixed, but it should be noted that in this embodiment, s1 and s2 are adopted as variational parameters.

[0043] The discrete schedule is applied to the quantum approximate optimization algorithm (QAOA). The discrete schedule has p layers, and each layer is composed of two variational parameters. Therefore, there are 2p variational parameters s i (i ∈ {1,..., 2p}). The annealing path s(t) of each layer l (l ∈ {1,..., p}) takes a value of 0 or 1 as represented by Equation (13).

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[0044] Here, s0 = 0, and s i-1 ≤ s i (i ∈ {1,..., 2p}), s 2p ≤ T. During t ∈ [s 2p , T], the quantum state does not evolve in time. From Equations (6) and (13), in each layer l, after the quantum state evolves in time by the Ising Hamiltonian H Ising , it then evolves in time by the transverse magnetic field Hamiltonian H q .

[0045] Figure 4 shows the optimal annealing paths for continuous schedules for different annealing times (T = 0.1, T = 1, T = 10). The multiple lines in each graph of Figure 4 correspond to the annealing paths for 10 different graph partitioning problems with 8 nodes.

[0046] At T = 0.1, quantum annealing starts with the Ising Hamiltonian H Ising and suddenly transitions to the transverse magnetic field Hamiltonian H q at the midpoint of the annealing time T. At T = 1, quantum annealing starts with the Ising Hamiltonian H Ising and continues with a continuous function and ends with the transverse magnetic field Hamiltonian H q At T = 10, quantum annealing starts with the transverse magnetic field Hamiltonian H q and ends with the Ising Hamiltonian H Ising which is similar to the standard annealing path. As the annealing time increases, the success probability of obtaining the optimal solution to the above graph partitioning problem increases. Here, the success probability indicates the probability of obtaining the optimal solution when calculating the quantum annealing along the optimal annealing path s(t) shown in FIG. 4 and then performing the above post-processing.

[0047] In FIG. 2A, an example of a COP with one constraint is shown, but the pVQA of this embodiment is also applicable to a COP with multiple constraints. Examples of a COP with one constraint include the graph partitioning problem and the knapsack problem. An example of a COP with multiple independent constraints includes the graph coloring problem. Examples of a COP with multiple non-independent constraints include the traveling salesman problem and the quadratic assignment problem.

[0048] FIG. 5 shows an example where multiple constraints are independent of each other. For the spin variables x i arranged in 3 rows and 3 columns, assume there are 3 constraints that the sum of the spin variables x i in each row (horizontal direction) is 1. In this case, since the constraints are defined independently for each row, these 3 constraints are independent of each other. In the example of FIG. 5, the first row and the second row satisfy the constraints, but the third row has a sum of spin variables x i equal to 2 and does not satisfy the constraints.

[0049] FIG. 6 shows an example where multiple constraints are not independent of each other. For the spin variables x i arranged in 3 rows and 3 columns, for the spin variables x iThe sum is 1, and the spin variable x in each column (vertical direction) i There are six constraints that the sum is 1. As is clear from Fig. 6, since two constraints, one for the row and one for the column, are imposed on one spin variable x i , these six constraints are not independent of each other. In the example of Fig. 6, the first row, the second row, the first column, and the second column satisfy the constraints, but the third row and the third column do not satisfy the constraints.

[0050] <pVQA's Scope of Application> Next, regarding the COP, the scope of application of pVQA will be examined. The Hamiltonian Q of the QUBO model is expressed as in Equation (14-1). Here, Q obj is the objective function, Q cst is the constraint term, and A > 0 is the constraint coefficient. Furthermore, the Hamiltonian Q′ for post-processing is expressed as in Equation (14-2). Here, Q′ cst c is the constraint term, and A′ > 0 is the constraint coefficient. [Number]

[0051] x i ∈ {0, 1} (i ∈ V), consider the COP that minimizes or maximizes the objective function Q obj under the linear constraints shown in Equation (15). Here, M is the number of constraints. [Number]

[0052] Regarding this COP, the following theorem can be derived. Theorem: When the value of the constraint coefficient A′ is sufficiently large (A′ > ΔQ obj ), and the following first condition, second condition, and third condition are satisfied, there exists a local optimization method that converts all non-feasible solutions into feasible solutions. Here, ΔQ obj is the maximum value of the energy change of Q i when the spin variable x obj is flipped. (i) First condition: satisfying formula (16).

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[0053] To prove this theorem, it is necessary to show that all non - executable solutions are not minimal solutions under the above - mentioned conditions. x i Assuming that x ∈ {0, 1} (i ∈ V) does not satisfy the c - th constraint, formula (18) or formula (19) is satisfied.

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[0054] When formula (18) is satisfied, formula (20) is derived from the first condition.

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[0055] Assume that a non - executable solution that does not satisfy the c - th constraint is a minimal solution, and a proof by contradiction is carried out. Formula (20) and the third condition (that is, the constraints are independent of each other) suggest that when the value of Σa i in formula (15) increases due to the flip of the spin variable x i c x i and A′ > ΔQ obj , the energy value (the value of Q′) becomes lower. However, if this non - executable solution is a minimal solution, the energy value should not become lower by flipping. Therefore, by flipping, Σa i c x iThe value of i should not increase when x i c = 1, a i should be ≥ 0, and when x i c = 0, a i should be ≤ 0, which means that equation (21) holds for any x

Equation

[0056] However, equation (21) contradicts the second condition that there exists an executable solution. Therefore, the non-executable solution is not a minimum solution. Note that for non-executable solutions that satisfy equation (19), it can also be proven by a similar argument that the non-executable solution is not a minimum solution.

[0057] When multiple constraints are not independent of each other, the above theorem does not apply, and post-processing based on local optimization methods cannot be used. However, if an ad-hoc post-processing method is used, pVQA is applicable. As shown in Figure 6, taking six non-independent constraints where the sum of the spin variables x i in each row is 1 and the sum of the spin variables x i in each column is 1 as an example, an ad-hoc post-processing method will be explained.

[0058] Assume that the spin variable x i is in the initial state 702 of Figure 7. In the initial state 702, the second column, the third column, and the first to third rows do not satisfy the constraints. First, for the second and third columns where the sum of x i is 2 or more, flip x i = 1 to x i = 0 so that the sum of x i becomes 1 and the energy value becomes as small as possible. Assume that this results in a transition from state 702 to state 704.

[0059] In state 704, all columns satisfy the constraints, but the second and third rows do not satisfy the constraints. Next, for the second row where the sum of x i is 2 or more, x isuch that the sum of x is 1 and the energy value is minimized as much as possible i set x i = 1 to x

[0060] = 0. Suppose that the state has transitioned from state 704 to state 706 i In state 706, the second column, third column, first row, and second row satisfy the constraints, but in the first column and third row, the sum of x i is 0 and does not satisfy the constraints. Next, for the third row where the sum of x i is 0, set x i = 0 to x i = 1. By doing so, if the state transitions from state 706 to state 708, all constraints are satisfied. Such an ad-hoc post-processing method is described in the following non-patent literature S. Kanamaru, et al., “Mapping Constrained Slot-Placement Problems to Ising Models and its Evaluations by an Ising Machine,”2019 IEEE 9th International Conference on Consumer Electronics (ICCE-Berlin), Berlin, Germany, 2019.

[0061] <Example of Constrained COP> Next, as an example of a COP with one constraint, the graph partitioning problem (GPP) and the quadratic knapsack problem (QKP) will be described. Both are NP-hard COPs

[0062] The GPP is a graph G g with a set V g of an even number of vertices (nodes) and an edge set E g = (V g , E g ). Given V gWhen dividing it into two subsets of equal size, it divides in such a way that the number of edges (cut number) connecting the two subsets is minimized. For each vertex i ∈ V of the graph g a spin variable x i is defined, and when vertex i belongs to the first subset, x i = 1, and when vertex i belongs to the second subset, x i = 0 is set.

[0063] This GPP has linear equality constraints given by Equation (22). Here, |V g | represents the number of vertices.

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[0064] In this GPP, Q obj and Q cst are given as in Equation (23). Here, k i is the degree of node i.

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[0065] QKP is a COP that, when given a knapsack with capacity C and n items, finds the combination of items that maximizes the total value of the items placed in the knapsack without exceeding the capacity C. Let the capacity and value of the i-th item be w i , p ii respectively, and let the additional value when the i-th item and the j-th item are placed in the knapsack simultaneously be p ij . Let x i = 1 when the i-th item is placed in the knapsack, and x i = 0 when it is not placed.

[0066] This QKP has linear inequality constraints given by Equation (24).

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[0067] In this QKP, Q in Equation (14-1) obj and Q cst are given as in Equation (25).

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[0068] <System Configuration> Next, the configuration of the quantum computing system that executes the quantum computing method of this embodiment will be described. As shown in FIG. 8, the quantum computing system 10 of this embodiment includes a classical computer 20 and a quantum device 30 connected to the classical computer 20.

[0069] The quantum device 30 is a quantum device such as a quantum annealer or a gate-type quantum computer. As shown in Steps 102 and 104 of FIG. 1, the quantum device 30 executes quantum annealing along the annealing path to generate a quantum state and generate a probability distribution of the quantum state.

[0070] The classical computer 20 is a von Neumann-type computer such as a personal computer. As shown in FIG. 9, the classical computer 20 includes a processor 202, a memory 204, a storage device 206, an input unit 208, a display 210, and an I / F 212, and these devices are connected via a bus.

[0071] The processor 202 has a Central Processing Unit (CPU) or the like, and executes various processes according to programs stored in the memory 204 and the storage device 206. As shown in Steps 106 to 112 of FIG. 1, the processor 202 performs post-processing, calculation of a new probability distribution, calculation of an energy expectation value, and update of the annealing path.

[0072] Note that, instead of a general-purpose computer such as a CPU, as the processor 202, an application specific integrated circuit (ASIC), a field programmable gate array (FPGA), or other dedicated computer for executing the quantum computing method of the present embodiment may be employed.

[0073] The memory 204 has a read only memory (ROM) and a random access memory (RAM). The ROM stores a boot program such as BIOS. When the processor 202 reads a program stored in the ROM or a program stored in the storage device 206, these programs are loaded into the RAM.

[0074] The storage device 206 has a non-transitory computer-readable storage medium and stores a program for causing the processor 202 to execute the quantum computing method of the present embodiment in combination with the quantum device 30, data necessary for the execution of the program, and the like. Examples of the computer-readable storage medium include a hard disk drive (HDD), a solid state drive (SSD), and an optical disk.

[0075] Note that the program executed by the processor 202 may be stored in another computer connected via a network, and the processor 202 may read the program from the other computer via the I / F 212.

[0076] The input unit 208 has an input device such as a mouse and a keyboard. The display 210 is a display such as a liquid crystal display (LCD) and displays the result of the process executed by the processor 202.

[0077] I / F212 is an interface for connecting the classical computer 20 to a network such as a Local Area Network (LAN), a Wide Area Network (WAN), and / or the Internet.

[0078] Next, Examples 1 and 2 in which pVQA is executed by a quantum annealer and a gate-type quantum computer will be described. In Example 1, GPP is the target, and in Example 2, QKP is the target.

Example

[0079] Table 1 shows the results (residual energy Re and success probability p suc ) obtained by executing pVQA and conventional quantum annealing (QA) on a quantum annealer for 10 problem instances GPP_n_i of GPP. Here, n = |V g |, and i represents the label of the random graph. The residual energy Re represents the difference between the average value and the optimal value of the cost function (Equation (3)). The success probability p suc represents the probability of obtaining the optimal solution. The average value of the cost function and the success probability p suc are calculated based on the top 50 solutions based on the cost function among 100 solutions obtained by repeating quantum annealing. A linear annealing path was used, and the annealing path was updated by the grid search method. In Table 1, the numerical values in parentheses represent the standard deviation. The smaller the value of Re and the larger the value of p suc , the better the performance of the algorithm.

[0080]

Table 1

[0081] From Table 1, it can be seen that for 5 GPPs with |V g | = 32, pVQA reduces the residual energy Re and improves the success probability p suc compared to QA.

[0082] Table 2 shows the success probability p g obtained by executing pVQA and VQA on a gate - type quantum computer for a 4 - node GPP (|V suc | = 4). In Table 2, the numerical values in parentheses represent the standard deviation. Figure 10 shows a schematic diagram of the 4 - node GPP actually used. The dotted line in Figure 10 represents the optimal solution. Using the annealing path of the discrete schedule, the annealing path was updated by the Powell method. The success probability p suc was calculated from 100 solutions obtained by repeating quantum annealing along the annealing path.

[0083]

Table 2

[0084] From Table 2, it can be seen that pVQA has a higher success probability p suc than VQA, and pVQA is significantly superior in performance to VQA.

Example

[0085] Table 3 shows the results (residual energy Re and success probability p suc ) obtained by executing pVQA and QA on a quantum annealer for 7 problem instances QKP_n_i of QKP. n represents the number of items, and i represents the label of the instance that is a benchmark. The calculation methods of the residual energy Re and the success probability p suc are the same as those in Table 1. Also in Table 3, the numerical values in parentheses represent the standard deviation.

[0086]

Table 3

[0087] From Table 3, it can be seen that for pVQA compared to QA, the residual energy Re is significantly reduced and the success probability p suc is improved, and it can be seen that pVQA is significantly superior in performance to QA.

[0088] According to the quantum computing method and system of this embodiment, by using pVQA, which is an algorithm that combines the variational optimization method of the annealing path and the post-processing method, it is possible to efficiently solve the COP on a quantum device with limited coherence time. In addition, it is possible to improve the accuracy of the solution compared to the conventional method. Further, the quantum computing method of this embodiment is applicable regardless of the type of quantum device.

[0089] Note that the present invention is not limited to the above-described embodiment, and various modifications are possible without departing from the spirit of the present invention. Other embodiments and variations made by those skilled in the art are also included in the present invention.

Explanation of Reference Numerals

[0090] 10 Quantum computing system 20 Classical computer 30 Quantum device 202 Processor 204 Memory 206 Storage device 208 Input unit 210 Display 212 I / F

Claims

1. A quantum computing method for solving a combinatorial optimization problem with constraints using an Ising model represented by a plurality of spin variables, comprising: executing quantum annealing according to an annealing path, which is a parameter representing the strength of quantum fluctuations of the Ising model, by a quantum device to generate a quantum state of the plurality of spin variables; generating a probability distribution of the quantum state by the quantum device; executing post-processing by a classical computer to convert an infeasible solution that does not satisfy the constraints into a feasible solution that satisfies the constraints based on a solution representing the quantum state, and calculating a new probability distribution after the post-processing; calculating an energy expectation value of the Ising model by the classical computer based on the new probability distribution; updating the annealing path to a new annealing path by the classical computer so that the energy expectation value becomes smaller.

2. The quantum computing method according to claim 1, wherein the optimal solution of the combinatorial optimization problem is obtained by repeating the processes of generating the quantum state, generating the probability distribution, performing the post-processing, calculating the energy expectation value, and updating the annealing path until the energy expectation value converges.

3. The quantum computing method according to claim 1, wherein the post-processing is executed based on a local optimization method.

4. A quantum computing system for solving a combinatorial optimization problem with constraints using an Ising model represented by a plurality of spin variables, comprising: a quantum device that executes quantum annealing according to an annealing path, which is a parameter representing the strength of quantum fluctuations of the Ising model, to generate a quantum state of the plurality of spin variables and generate a probability distribution of the quantum state; a classical computer that executes post-processing to convert an infeasible solution that does not satisfy the constraints into a feasible solution that satisfies the constraints based on a solution representing the quantum state, calculates a new probability distribution after the post-processing, calculates an energy expectation value of the Ising model based on the new probability distribution, and updates the annealing path to a new annealing path so that the energy expectation value becomes smaller; A quantum computing system comprising the above.

5. A program for causing a classical computer to execute in combination with a quantum device a quantum computing method for solving a combinatorial optimization problem with constraints using an Ising model represented by a plurality of spin variables, when a probability distribution of quantum states of the plurality of spin variables is generated by execution of quantum annealing by the quantum device, based on a solution representing the quantum state, performing post-processing to convert a non-executable solution that does not satisfy the constraint into an executable solution that satisfies the constraint, and calculating a new probability distribution after the post-processing, calculating an energy expectation value of the Ising model based on the new probability distribution, A program having a step of updating an annealing path, which is a parameter of the quantum annealing, to a new annealing path so that the energy expectation value becomes smaller.

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