Calculation methods, calculation systems, and programs

Abstract

To make it possible to implement a multi-spin flip in an Ising machine.SOLUTION: In a calculation method for solving an Ising model, a classical computer computes the Hamiltonian Horg of an Ising model having a plurality of prescribed spin variables from a combinatorial optimization problem with constraints, randomly selects a first spin variable from the plurality of spin variables, sets, from the plurality of spin variables, a prescribed proportion of spin variables other than the first spin variable as a second spin variable, and merges the second spin variable with the first spin variable on the basis of a provisional solution of the Ising model to delete the second spin variable from the Hamiltonian Horg, thereby computing a transformed Hamiltonian H'. The Ising machine determines a solution with respect to the transformed Hamiltonian H', and the classical computer computes the value of the second spin variable from the solution obtained by the Ising machine, to thereby determine the solution to the Ising model.SELECTED DRAWING: Figure 8

Description

[Technical Field]

[0001] This invention relates to a calculation method, a calculation system, and a program for solving the Ising model. [Background technology]

[0002] Ising machines are attracting attention as hardware that has the potential to efficiently solve combinatorial optimization problems. A combinatorial optimization problem is the problem of finding the combination of variables that minimize or maximize an objective function under multiple constraints. When using an Ising machine, a combinatorial optimization problem is transformed into an Ising problem. An Ising problem uses a format called an Ising model, which is given by multiple spin variables with values ​​of +1 or -1. An Ising problem is the problem of finding the spin configuration that gives the ground state energy of the Hamiltonian (energy function) of the Ising model. Various methods for transforming combinatorial optimization problems into Ising problems have been proposed to date (see, for example, Patent Document 1).

[0003] When a constrained combinatorial optimization problem is formulated as an Ising problem, the Hamiltonian H is given by H = H obj +H c It is given by H obj and H c These are the objective function and the constraint term, respectively. When the constraint is satisfied, H c = 0, and if not satisfied, H c The result is >0. A solution that satisfies the constraints is called a feasible solution, and a solution that gives the ground energy is called a ground solution. The hardware of many Ising machines uses single-spin-flip simulated annealing (SA) as its operating principle when searching for a ground solution. Here, a spin flip is the conversion of the spin variable value between +1 and -1.

[0004] For example, consider an Ising model consisting of two spin variables (σ1, σ2) whose Hamiltonian H is given by equation (1).

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[0005] As shown in FIG. 1, the executable solution has lower energy than the non-executable solution, and it becomes an executable solution when (σ1, σ2) is (+1, -1) or (-1, +1), and becomes a non-executable solution when (-1, -1) or (+1, +1). Among the executable solutions, (-1, +1) that gives the ground energy is the ground state solution, and (+1, -1) is a local solution. A local solution is a solution where the energy increases no matter which spin variable is flipped. If either one of the spin variables of the local solution (+1, -1) is flipped, it becomes a non-executable solution. If σ2 of the non-executable solution (-1, -1) is flipped or σ1 of (+1, +1) is flipped, it becomes the ground state solution.

[0006] Here, in SA, a state change that lowers energy (transition from a non-executable solution to an executable solution) is always accepted, while a state change that increases energy (transition from a local solution to a non-executable solution) is probabilistically accepted. This acceptance probability depends on the temperature, being high at high temperatures and low at low temperatures. Therefore, when in the state of a local solution, at low temperatures, a transition to a solution with high energy is less likely to occur, so it is trapped in the local solution and the ground state solution cannot be explored. This leads to deterioration of the performance of the Ising machine.

[0007] In order to escape from the local solution and transition between executable solutions, as shown in FIG. 2, there is a multi-spin flip method of flipping a plurality of spin variables simultaneously. Regarding this, in Non-Patent Document 1, the effectiveness of multi-spin flip that transitions to the ground state solution without passing through a non-executable solution has been discussed, and it has been reported that the performance of SA is improved.

Prior Art Documents

Patent Documents

[0008]

Patent Document 1

Non-Patent Documents

[0009] [Non-Patent Document 1] Mohammad Bagherbeik, Parastoo Ashtari, Seyed Farzad Mousavi, Kouichi Kanda, Hirotaka Tamura, Ali Sheikholeslami, "A Permutational Boltzmann Machine with Parallel Tempering for Solving Combinatorial Optimization Problems", in International Conference on Parallel Problem Solving from Nature, 2020, pp. 317-331. [Overview of the project] [Problems that the invention aims to solve]

[0010] However, since existing Ising machines operate on the principle of single-spin flips, implementing multi-spin flips in an Ising machine would require modifying its operating principle, which is extremely difficult.

[0011] This invention has been made in view of the above problems, and aims to provide a calculation method, a calculation system, and a program that enable the implementation of multi-spin flips in an Ising machine. [Means for solving the problem]

[0012] The calculation method according to the present invention is a calculation method for solving the Ising model, wherein a classical computer calculates the Hamiltonian of an Ising model having a predetermined number of spin variables from a constrained combinatorial optimization problem, (b) randomly selects a first spin variable from the number of spin variables, sets a predetermined proportion of spin variables other than the first spin variable as a second spin variable, and calculates a deformed Hamiltonian by merging the second spin variable with the first spin variable based on a provisional solution of the Ising model and eliminating the second spin variable from the Hamiltonian, (c) obtains a solution for the deformed Hamiltonian using an Ising machine, and (d) obtains a solution to the Ising model by calculating the value of the second spin variable from the solution obtained by the Ising machine using a classical computer.

[0013] The computational system according to the present invention is a computational system for solving the Ising model, comprising: a classical computer having a processor that calculates the Hamiltonian of an Ising model having a predetermined number of spin variables from a constrained combinatorial optimization problem, randomly selects a first spin variable from the number of spin variables, sets a predetermined proportion of spin variables other than the first spin variable from the number of spin variables as second spin variables, and calculates a deformed Hamiltonian by merging the second spin variables with the first spin variables based on a provisional solution of the Ising model and eliminating the second spin variables from the Hamiltonian; and an Ising machine that finds a solution for the deformed Hamiltonian. The processor finds a solution to the Ising model by calculating the value of the second spin variable from the solution obtained by the Ising machine.

[0014] The present invention provides a program for causing a classical computer to execute a computational method for solving an Ising model in conjunction with an Ising machine. The program calculates the Hamiltonian of an Ising model having a predetermined number of spin variables from a constrained combinatorial optimization problem, randomly selects a first spin variable from the multiple spin variables, sets a predetermined proportion of the spin variables other than the first spin variable as the second spin variable, calculates a deformed Hamiltonian by merging the second spin variable with the first spin variable based on a provisional solution of the Ising model and eliminating the second spin variable from the Hamiltonian, and then calculates the value of the second spin variable from the solution obtained by the Ising machine for the deformed Hamiltonian to find the solution of the Ising model. [Effects of the Invention]

[0015] According to the present invention, by deforming the Hamiltonian through the merging of spin variables in the Ising model, a process equivalent to a multi-spin flip can be realized with a single-spin flip. [Brief explanation of the drawing]

[0016] [Figure 1] This is a schematic diagram illustrating the simulated annealing (SA) of a single spin flip, which is the operating principle of an Ising machine. [Figure 2] This is a schematic diagram to explain multispin flips. [Figure 3] This block diagram shows the configuration of the calculation system according to this embodiment. [Figure 4] Figure 3 is a block diagram showing the hardware configuration of a classical computer. [Figure 5] This is a conceptual diagram illustrating the deformation of energy terrain using the merge method. [Figure 6A] This is a schematic diagram illustrating an example of the energy landscape of a two-spin Ising model. [Figure 6B]Figure 6A is a schematic diagram representing the deformed energy landscape obtained by merging the spin parameters of the Ising model. [Figure 7A] This is a schematic diagram illustrating an example of the energy landscape of a 4-spin Ising model. [Figure 7B] Figure 7A is a schematic diagram representing the deformed energy landscape obtained by merging the spin parameters of the Ising model. [Figure 8] This flowchart shows the calculation method according to this embodiment. [Modes for carrying out the invention]

[0017] Embodiments of the present invention will be described below with reference to the drawings.

[0018] <Calculation System> First, the configuration of the computing system in this embodiment will be described. As shown in Figure 3, the computing system 10 in this embodiment comprises a classical computer 20 and an Ising machine 30 connected to the classical computer 20. The classical computer 20 is a von Neumann type computer such as a personal computer.

[0019] The Ising machine 30 is a well-known Ising machine, such as the D-Wave 2000Q machine, CMOS annealing machine, or Digital Annealer (registered trademark).

[0020] As shown in Figure 4, the classical computer 20 comprises a processor 202, memory 204, storage device 206, input unit 208, display 210, and I / F 212, and these devices are connected via a bus.

[0021] The processor 202 includes a Central Processing Unit (CPU) and other components, and executes various processes according to programs stored in memory 204 and storage device 206. Details of the calculation methods performed by the processor 202 will be described later (see Figures 5 to 8).

[0022] Furthermore, instead of a general-purpose computer such as a CPU, a dedicated computer such as an Application Specific Integrated Circuits (ASIC) or Field Programmable Gate Array (FPGA) for executing the calculation method of this embodiment may be used as the processor 202.

[0023] Memory 204 has Read Only Memory (ROM) and Random Access Memory (RAM). ROM stores boot programs such as the BIOS. When the processor 202 reads a program stored in ROM or a program stored in storage device 206, these programs are loaded into RAM.

[0024] The storage device 206 has a non-temporary computer-readable storage medium and stores a program for executing the calculation method of this embodiment, data necessary for executing the program, and the like. Examples of computer-readable storage media include hard disk drives (HDDs), solid state drives (SSDs), optical discs, and the like.

[0025] Alternatively, the program executed by processor 202 may be stored on another computer connected via a network, and processor 202 may read the program from the other computer via I / F212.

[0026] The input unit 208 has input devices such as a mouse and a keyboard. The display 210 is a display such as a Liquid Crystal Display (LCD) and displays the results of processing performed by the processor 202.

[0027] I / F212 is an interface for connecting the classic computer 20 to a network such as a Local Area Network (LAN), Wide Area Network (WAN), and / or the Internet.

[0028] <Calculation method> Next, the calculation method of this embodiment will be described with reference to Figures 5 to 8. In this embodiment, in order to realize multi-spin flips in the Ising machine 30, a merging method is used to deform the energy landscape of the Ising model by merging at least some of the spin variables of the Ising model, as shown in Figure 5. This makes the single-spin flip in the deformed Hamiltonian equivalent to the multi-spin flip in the original Hamiltonian.

[0029] The merging method will be explained below using examples of two-spin and four-spin Ising models.

[0030] As an Ising model for a two-spin system, we consider an Ising model (hereinafter referred to as Ising model I) that gives the same Hamiltonian H1(σ1, σ2) as equation (1) above. Ising model I has a one-hot constraint that satisfies the constraint when one of σ1 and σ2 is +1 and the other is -1. Figure 6A shows the energy landscape of H1(σ1, σ2). Ising model I has a local solution (+1, -1) with an energy value of -2 and a base solution (-1, +1) with an energy value of -4. The remaining non-feasible solutions are (-1, -1) and (+1, +1), which give energy values ​​of 20 and 14, respectively. In Ising model I, a double spin flip is required to transition between feasible solutions.

[0031] Now, assume we are in a local optimum of H1(σ1, σ2). Since σ1σ2=-1 in the local optimum, we can use σ2=-σ1 to merge σ2 with σ1 and eliminate σ2 from H1(σ1, σ2), which gives us the transformed Hamiltonian H1′(σ1) as shown in equation (2).

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[0032] H1′(σ1) is independent of the merged spin variable σ2. Figure 6B shows the energy landscape of H1′(σ1). In the original H1(σ1, σ2), there was an energy barrier between the local solution and the ground solution, but in H1′(σ1), the energy barrier has disappeared. When H1′(σ1) is input into the Ising machine 30, the ground solution of H1′(σ1) can be found with a single spin flip. In H1′(σ1), the energy value is -2 when σ1=+1 and the energy value is -4 when σ1=-1, so σ1=-1 is the ground solution. That is, in H1′(σ1), σ1=+1 is no longer a local solution, and a single spin flip transitions to the ground solution σ1=-1.

[0033] Next, we find the value of the eliminated σ2 from σ2 = -σ1. From the basis solution σ1 = -1 of H1'(σ1), we obtain (σ1, σ2) = (-1, +1), and the energy function takes its minimum value of -4. Therefore, (σ1, σ2) = (-1, +1) is the basis solution of the original H1(σ1, σ2). Thus, the single spin flip from σ1 = +1 to σ1 = -1 in H1'(σ1) is equivalent to the double spin flip from (σ1, σ2) = (+1, -1) to (σ1, σ2) = (-1, +1) in the original H1(σ1, σ2).

[0034] Next, we consider the Ising model of a four-spin system that gives the Hamiltonian H2(σ1, σ2, σ3, σ4) in equation (3) (hereinafter referred to as Ising model II).

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[0035] The Ising model II has a one-hot constraint that satisfies the constraint when one spin variable is +1 and the other spin variable is -1 for {σ1, σ2}, {σ3, σ4}, {σ1, σ3}, and {σ2, σ4}. Figure 7A shows the energy landscape of H2(σ1, σ2, σ3, σ4). The Ising model II has a local solution (-1, +1, +1, -1) that gives an energy value of +1 and a base solution (+1, -1, -1, +1) that gives an energy value of -1. The remaining non-feasible solutions are (+1, +1, +1, -1), (+1, -1, +1, -1), and (+1, -1, +1, +1) that give a large energy value of 799. Four spin flips are required to transition between feasible solutions in the Ising model II.

[0036] Let us assume that we are currently in a local optimum of H2(σ1, σ2, σ3, σ4). Of the four spin variables, let σ1 be the spin variable used as the criterion for merging (hereinafter referred to as the first spin variable), and let σ2, σ3, and σ4 be the spin variables merged into σ1 (hereinafter referred to as the second spin variables). In the local optimum, σ1σ2=-1, σ1σ3=-1, and σ1σ4=+1. Therefore, by using σ2=-σ1, σ3=-σ1, and σ4=σ1 to merge σ2, σ3, and σ4 into σ1 and eliminating σ2, σ3, and σ4 from H2(σ1, σ2, σ3, σ4), we obtain the transformed Hamiltonian H2′(σ1) as shown in equation (4).

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[0037] H2′(σ1) is independent of the merged second spin variables σ2, σ3, and σ4. Figure 7B shows the energy landscape of H2′(σ1). In the original H2(σ1, σ2, σ3, σ4), there was an energy barrier between the local solution and the ground solution, but in H2′(σ1), the energy barrier has disappeared. When H2′(σ1) is input into the Ising machine 30, the ground solution of H2′(σ1) can be found with a single spin flip. In H2′(σ1), the energy value is +1 when σ1=-1 and the energy value is -1 when σ1=+1, so σ1=+1 is the ground solution. That is, in H2′(σ1), σ1=-1 is no longer a local solution, and a single spin flip transitions to the ground solution σ1=+1.

[0038] Next, we use the values ​​of the second spin variable, σ2=-σ1, σ3=-σ1, and σ4=σ1 obtained by merging. From the basis solution σ1=+1 of H2′(σ1), we obtain (σ1, σ2, σ3, σ4)=(+1, -1, -1, +1), and the energy function takes its minimum value of -1. Therefore, (σ1, σ2, σ3, σ4)=(+1, -1, -1, +1) is the basis solution of the original H2(σ1, σ2, σ3, σ4). Thus, a single spin flip from σ1=-1 to σ1=+1 in H2′(σ1) is equivalent to a four-spin flip from (σ1, σ2, σ3, σ4)=(-1, +1, +1, -1) to (σ1, σ2, σ3, σ4)=(+1, -1, -1, +1) in the original H2(σ1, σ2, σ3, σ4).

[0039] In the above examples of Ising models for 2-spin and 4-spin systems, all remaining spin variables were merged with one spin variable. However, in actual Ising models, the number of spin variables is very large, so it is possible to probabilistically determine which second spin variable is merged, such that only a portion (e.g., 30%) of all spin variables are merged. It is also possible for the user to specify the second spin variable. In simulated annealing (SA), even if an energy barrier remains, at high temperatures, the system probabilistically transitions to a solution with higher energy, making it possible to overcome the energy barrier.

[0040] The merging method of the Ising model of the two-spin system and the four-spin system was exemplified, but the merging method can be applied to the Ising model consisting of any N spin variables. Generally, the Hamiltonian H of the Ising model consisting of N spin variables σ i (i ∈ {1, 2,..., N}) is defined as in Equation (5).

Equation

[0041] For the Ising model of Equation (5), the following theorem for the merging method can be derived as follows. <Theorem> For all i ∈ {1, 2,..., N}, as parameters giving the information of the set of spin variables to be multi-spin flipped, m i and s i are defined as in Equation (6).

Equation

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[0042] At this time, for all i, σ i σ mi =s i For a state that satisfies the condition, as shown in equation (11), H M ({s i} i=1 N , {m i} i=1 N H and H have the same energy value.

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[0043] H M ({s i} i=1 N , {m i} i=1 N ;H) represents the deformed Hamiltonian obtained by the merge method. This theorem means that the energy value does not change even when the Hamiltonian is deformed in the merge method. Since the spin variable takes a value of +1 or -1, σ i σ mi =s i From σ i =s i σ mi We obtain this relationship from H in equation (5) to m i i that satisfies ≠i σ i By eliminating H in equation (7) M ({s i} i=1 N , {m i} i=1 N You can obtain H).

[0044] Next, the flow of the calculation method performed by the calculation system 10 will be explained with reference to the flowchart shown in Figure 8. The calculation method of this embodiment is a hybrid algorithm that incorporates the merge method described above into SA.

[0045] First, the classical computer 20 solves the Ising model Hamiltonian H from a constrained combinatorial optimization problem. org The following steps are performed: (Step 802) calculate the initial solution (provisional solution) of the Ising model and set the initial temperature of SA (e.g., 100 degrees) (Step 804).

[0046] Next, the classical computer 20 randomly selects a first spin variable from all the spin variables of the Ising model to serve as the basis for merging, and sets a predetermined percentage of the spin variables other than the first spin variable (for example, 30% of all spin variables) as the second spin variable. Then, based on the current provisional solution, the classical computer 20 merges the second spin variable with the first spin variable to obtain the original Hamiltonian H org The deformed Hamiltonian H' is obtained by eliminating the second spin variable (step 806). The deformed Hamiltonian H' is input into the Ising machine 30.

[0047] The Ising machine 30 finds a solution for the deformed Hamiltonian H′ and updates the solution (step 808). Next, the classical computer 20 finds a new provisional solution for the Ising model by calculating the value of the second spin variable, which was eliminated by the merge in step 806, from the solution obtained by the Ising machine 30 (step 810). Then, the classical computer 20 converts the Hamiltonian back to the original H org Return to the previous state (step 812).

[0048] Next, the classical computer 20 lowers the temperature in SA by a predetermined temperature interval (step 814) and determines whether the current temperature has reached a preset final temperature (e.g., 1 degree) (step 816). If the current temperature has not reached the final temperature (step 816; NO), the process returns to step 806 and steps 806-814 are repeated. By repeating steps 806-814 until the final temperature is reached, the provisional solution gradually approaches the optimal solution. On the other hand, if the current temperature has reached the final temperature (step 816; YES), the most recently obtained provisional solution is determined as the base solution of the Ising model, and the calculation method of this embodiment ends.

[0049] In this way, by deforming the Hamiltonian of the Ising model using the merge method and obtaining a solution for the deformed Hamiltonian using the Ising machine 30, a process equivalent to a multi-spin flip can be realized with a single-spin flip. Therefore, a multi-spin flip can be implemented without changing the operating principle (single-spin flip) of the Ising machine 30, and the performance of the Ising machine 30 can be improved.

[0050] To evaluate the calculation method using the merge technique of this embodiment, we applied the method to the quadratic knapsack problem (QKP), which is an NP-hard problem. Compared to the calculation method without the merge technique, we succeeded in reducing the residual energy (magnitude of error from the optimal value) by an average of 68%. Similarly, when we applied the calculation method of this embodiment to the quadratic assignment problem (QAP), we were able to reduce the residual energy by an average of 15%.

[0051] It should be noted that the present invention is not limited to the embodiments described above, and various modifications are possible without departing from the spirit of the invention. Other embodiments and modifications made by those skilled in the art are also included in the present invention. [Explanation of Symbols]

[0052] 10 Computing Systems 20 Classical Computers 30 Ising Machine 202 processors 204 memory 206 Storage device 208 Input section 210 displays 212 I / F

Claims

1. A computational method for solving the Ising model, Classical computers, (a) Compute the Hamiltonian of an Ising model with a given number of spin variables from a constrained combinatorial optimization problem, (b) Randomly select a first spin variable from the plurality of spin variables, set a predetermined proportion of spin variables other than the first spin variable from the plurality of spin variables as a second spin variable, and calculate the deformed Hamiltonian by merging the second spin variable with the first spin variable based on the provisional solution of the Ising model and eliminating the second spin variable from the Hamiltonian. Using the Ising machine, (c) Find the solution for the modified Hamiltonian, According to the aforementioned classical computer, (d) A method for obtaining the solution of the Ising model by calculating the value of the second spin variable from the solution obtained by the Ising machine.

2. The aforementioned classical computer sets the initial temperature in simulated annealing, and after processes (b) to (d), lowers the temperature. The calculation method according to claim 1, wherein the basis solution of the Ising model is obtained by repeating the processes (b) to (d) until a predetermined final temperature is reached.

3. A computational system for solving the Ising model, A classical computer having a processor that calculates the Hamiltonian of an Ising model having a predetermined number of spin variables from a constrained combinatorial optimization problem, randomly selects a first spin variable from the plurality of spin variables, sets a predetermined proportion of spin variables other than the first spin variable as second spin variables from the plurality of spin variables, and calculates a deformed Hamiltonian by merging the second spin variables with the first spin variables based on a provisional solution of the Ising model and eliminating the second spin variables from the Hamiltonian, and An Ising machine for finding a solution to the aforementioned deformed Hamiltonian, Equipped with, The aforementioned processor, A computational system for obtaining the solution to the Ising model by calculating the value of the second spin variable from the solution obtained by the Ising machine.

4. This is a program that allows a classical computer to execute a computational method for solving the Ising model in conjunction with an Ising machine. We compute the Hamiltonian of an Ising model with a given number of spin variables from a constrained combinatorial optimization problem. A first spin variable is randomly selected from the aforementioned plurality of spin variables, a predetermined proportion of spin variables other than the first spin variable are set as the second spin variable, and the second spin variable is merged with the first spin variable based on the provisional solution of the Ising model to eliminate the second spin variable from the Hamiltonian, thereby calculating the deformed Hamiltonian. A program for causing a process to obtain a solution to the Ising model by calculating the value of the second spin variable from the solution obtained by the Ising machine for the modified Hamiltonian.

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