Ising model generation device, Ising model generation method, and program

Abstract

The present invention provides an Ising model generation device including: a transform parameter estimation unit that calculates the maximum value of a first coefficient upper-limit parameter and the maximum value of a second coefficient upper-limit parameter for which a transformed expression satisfies a coefficient constraint of an Ising machine and calculates the number of spins in a transformed polynomial and a quadratic expression according to the maximum value of the first coefficient upper-limit parameter and the maximum value of the second coefficient upper-limit parameter; and a parameterized Ising model transform unit that, when the number of spins satisfies a number-of-spins constraint of the Ising machine, transforms a combinatorial optimization problem to an Ising model by replacing an integer variable using binary encoding in a case where a coefficient of a linear expression of a binary variable is not greater than the maximum value of the coefficient upper-limit parameter and otherwise by using a linear expression of a binary variable calculated so that the coefficient of the linear expression of the binary variable is equal to the maximum value of the coefficient upper-limit parameter.

Description

[Technical Field]

[0001] The present disclosure relates to an Ising model generation device, an Ising model generation method, and a program that generate an Ising model used for combinatorial optimization. [Background technology]

[0002] A combinatorial optimization problem is a problem whose solution involves selecting the best option for a specific evaluation index from a large number of combinations of options that satisfy specific conditions. Combinatorial optimization problems appear in a wide range of fields, from business decision-making such as personnel planning and route planning to academic research such as drug discovery. However, it is difficult for conventional computers to solve combinatorial optimization problems efficiently (in polynomial time relative to the problem size). For this reason, technology for performing high-speed combinatorial optimization using computer hardware that utilizes physical phenomena to search for optimal solutions is considered important. Combinatorial optimization problems can be expressed using the Ising model. Hardware that performs optimization of the Ising model is called an Ising machine. Furthermore, combinatorial optimization problems are often formulated as integer-coefficient polynomial optimization problems for integer variables. Therefore, unless otherwise specified, combinatorial optimization below refers to integer-coefficient polynomial optimization for integer variables.

[0003] This section explains the flow of combinatorial optimization using an Ising machine. The user expresses the combinatorial optimization problem they wish to solve as an Ising model. Mathematically, an Ising model is a quadratic polynomial related to multiple variables (spins) that take on the binary values ​​+1 or -1. The method for converting a combinatorial optimization problem into an Ising model will be explained later. The value obtained by substituting the value of each spin (+1 or -1) into the Ising model is called the energy of the Ising model. For the input Ising model, the Ising machine searches for a combination of spin values ​​(+1, -1) that minimizes the energy (called the solution of the Ising model), and returns the solution it finds to the user. The user interprets the resulting combination of spin values ​​as the solution to the original combinatorial optimization problem.

[0004] Ising machines can be realized using quantum processors (computational processors that use the principles of quantum mechanics), degenerate optical parametric oscillators, or dedicated integrated circuits using conventional semiconductor technology. Each of these machines imposes two major constraints on the Ising model to be optimized. One is that the number of spins in the Ising model that can be handled is limited to several tens of thousands. For example, the Ising machine (Advantage Quantum System) based on a quantum processor developed by D-Wave optimizes an Ising model with at most 5,000 spin variables. The other is that the range of coefficients of each term in the Ising model (as a polynomial) must be relatively small. For example, in an Ising machine using a degenerate optical parametric oscillator, the coefficients of the Ising model are limited to a finite number of values ​​due to hardware constraints. Furthermore, in an Ising machine based on a quantum processor, terms with relatively small coefficients are subject to quantum mechanical noise and are essentially ignored during optimization, resulting in a decrease in optimization accuracy (Non-Patent Document 1). As will be discussed later, it is difficult to efficiently find a transformation that satisfies these constraints when converting practical combinatorial optimization problems into an Ising model. Therefore, the hardware limitations of these Ising machines pose a challenge to using them for practical combinatorial optimization.

[0005] This section explains how to convert a combinatorial optimization problem (integer variable integer coefficient polynomial optimization problem) into an Ising model. An integer variable integer coefficient polynomial optimization problem is an optimization problem with inequality constraints using an integer variable integer coefficient polynomial, with an integer variable integer coefficient polynomial as the objective function. Expressed mathematically, it is as follows:

number

number

[0006] It should be noted that Z should be written in blackboard bold, but due to limitations in the functionality of word processing software, it cannot be written correctly, so it is written as a regular Z. The same applies below.

[0007] Additionally, letters such as n, m, z, f, g, u, and b should all be written in italics, but due to limitations in the functionality of word processing software, they are written in bold.

[0008] Also, for letters like f~(z) and g~(z), it is correct to place a tilde directly above the first letter, but due to limitations in word processing software, this cannot be done correctly, so a tilde is placed after the letter in question.

[0009] Here, we describe it as a minimization problem, but in the case of a maximization problem, we can reduce it to a minimization problem by inverting the sign of f~(z). Also, the equality constraints are u i = 0, any inequality (including unbounded ones) can be expressed as g~ i (z) constant term and u i Note that this can be expressed without loss of generality by appropriately defining . Therefore, the above formulation includes polynomial optimization with more general inequality and equality constraints. Transforming this problem into an Ising model involves the following four steps:

[0010] Step 1: Convert to an unconstrained integer polynomial optimization problem Step 2: Convert to an unconstrained binary (0-1) variable polynomial optimization problem Step 3: Conversion to an unconstrained binary quadratic optimization problem (QUBO) Step 4: Conversion to an Ising model Each step will be explained. In Step 1, each constraint 0 ≤ g~ i (z)≦u i The constrained optimization problem is transformed into an equivalent unconstrained optimization problem by introducing a penalty term that expresses the constraint 0 ≤ g~ i (z)≦u i The penalty term expressing i ∈Z is an integer variable polynomial g i p (z,ξ i ) such that:

number

number

number

number

number

[0011] In step 3, the binary polynomial optimization problem is converted into an equivalent binary quadratic optimization problem (QUBO). Several methods are known for this, but in practice, the conversion is often performed monomial by monomial. For details, see Non-Patent Documents 3 and 4.

[0012] In step 4, the 0-1 variable x i Let us consider a QUBO with ±1 variable σ i This is converted to an Ising model with x i =(σ i+1 ) / 2.

[0013] By the above procedure, integer variable integer coefficient polynomial optimization can be converted into an Ising model. [Prior art documents] [Non-patent literature]

[0014] [Non-Patent Document 1] Oku, Daisuke & Tawada, Masashi & Tanaka, Shu & Togawa, Nozomu. (2020). How to Reduce the Bit-width of an Ising Model by Adding Auxiliary Spins. IEEE Transactions on Computers. PP. 1-1. 10.1109 / TC.2020.3045112. [Non-patent document 2] K. Tamura, T. Shirai, H. Katsura, S. Tanaka, N. Togawa, “Performance Comparison of Typical Binary-Integer Encodings in an Ising Machine,” in IEEE Access, vol. 9, pp. 81032-81039, 2021, doi: 10.1109 / ACCESS.2021.3081685. [Non-patent document 3] H. Ishikawa, "Higher-order clique reduction in binary graph cut," 2009 IEEE Conference on Computer Vision and Pattern Recognition, 2009, pp. 2993-3000, doi: 10.1109 / CVPR.2009.5206689. [Non-patent document 4] Boros, E., Crama, Y. & Rodriguez-Heck, E. Compact quadratizations for pseudo-Boolean functions. J Comb Optim 39, 687-707 (2020). https: / / doi.org / 10.1007 / s10878-019-00511-0 Summary of the Invention [Problem to be solved by the invention]

[0015] However, the number of spins in the converted Ising model is generally larger than the number of variables, n, in the original problem. Furthermore, the coefficients of each term in the converted Ising model are also generally larger than the coefficients of each term in the polynomial of the original problem. This can result in the converted Ising model not satisfying the aforementioned hardware constraints of the Ising machine, making it impossible to apply the Ising machine. There is actually some freedom in selecting the specific processing methods for Step 2 and Step 3, and by combining these choices, it may be possible to find a conversion to an Ising model that satisfies the constraints of the Ising machine. However, the search for such a conversion method poses the following challenges A) and B).

[0016] A) Generally, there is a trade-off between the number of spins and the magnitude of the coefficients in the converted Ising model, and conventionally known candidate processing methods for each step have the property of making it easier to satisfy only one of two constraints, such as "small number of spins but large coefficients" or "small coefficients but large number of spins." Conventional conversion methods do not allow for flexible adjustment of this trade-off, and therefore there are many cases where no combination of conventional conversion methods can produce a conversion that satisfies both constraints.

[0017] B) In order to find a conversion method that satisfies both constraints, it is necessary to perform numerous trials to determine whether the converted Ising model satisfies the constraints for each combination of the processing methods selected in Step 2 and Step 3 above. These trials require calculations to actually convert to an Ising model and calculate the number of spins and coefficients. Generally, the number of terms (as polynomials) in the converted Ising model is significantly greater than the number of terms in the polynomials that appear in the original problem, so performing this conversion process each time results in a large computational overhead (computation time and memory). Furthermore, if a conversion method to an Ising model that satisfies both constraints does not exist in the first place, determining whether such a method exists also poses a computational cost issue.

[0018] These problems pose major practical challenges when applying Ising machines to combinatorial optimization.

[0019] Therefore, the present disclosure provides an Ising model generation device that can flexibly adjust the trade-off between the number of spins and coefficients of the converted Ising model and can reduce the number of attempts to convert to an Ising model. [Means for solving the problem]

[0020] The Ising model generation device of the present disclosure includes a transformation parameter estimation unit and a parameterized Ising model transformation unit.

[0021] The first coefficient upper limit parameter is a parameter that determines an upper limit for the coefficients of a linear expression of a binary variable when replacing an integer variable with a linear expression of a binary variable during conversion to an unconstrained binary variable polynomial, and the second coefficient upper limit parameter is a parameter that determines an upper limit for the coefficients of a linear expression of a binary variable when replacing an integer variable with a linear expression of a binary variable during conversion to an unconstrained binary variable quadratic expression.

[0022] The transformation parameter estimation unit calculates the maximum value of the first coefficient upper limit parameter and the maximum value of the second coefficient upper limit parameter at which the transformed expressions satisfy the coefficient constraints of the Ising machine, and calculates the number of spins of the transformed polynomial and quadratic expressions using the maximum value of the first coefficient upper limit parameter and the maximum value of the second coefficient upper limit parameter.

[0023] The parameterized Ising model conversion unit converts the combinatorial optimization problem into an Ising model by using binary encoding if the number of spins satisfies the spin number constraint of the Ising machine, and by replacing the integer variables with a linear expression of binary variables calculated so that the coefficient of the linear expression of binary variables is equal to the maximum value of the coefficient upper limit parameter in all other cases. [Effects of the Invention]

[0024] According to the Ising model generation device of the present disclosure, it is possible to flexibly adjust the trade-off between the number of spins and the coefficients of the converted Ising model, and it is possible to reduce the number of attempts to convert to an Ising model. [Brief explanation of the drawings]

[0025] [Figure 1] FIG. 1 is a block diagram showing the functional configuration of an Ising model generation device according to a first embodiment. [Figure 2] FIG. 2 is a block diagram showing the functional configuration of a conversion parameter estimation unit according to the first embodiment. [Figure 3] FIG. 2 is a block diagram showing the functional configuration of a parameterized Ising model conversion unit according to the first embodiment. [Figure 4] 3 is a flowchart showing the operation of the Ising model generation device according to the first embodiment. [Figure 5] FIG. 10 is a block diagram showing the functional configuration of an Ising model generation device according to a second embodiment. [Figure 6] FIG. 10 is a block diagram showing the functional configuration of a conversion parameter estimation unit according to a second embodiment. [Figure 7] FIG. 10 is a block diagram showing the functional configuration of a parameterized Ising model conversion unit according to a second embodiment. [Figure 8] 10 is a flowchart showing the operation of the Ising model generation device according to the second embodiment. [Figure 9] FIG. 10 is a block diagram showing the functional configuration of an Ising model generation device according to a third embodiment. [Figure 10] 10 is a flowchart showing the operation of the Ising model generation device according to the third embodiment. [Figure 11] FIG. 2 is a diagram showing an example of the functional configuration of a computer. DETAILED DESCRIPTION OF THE INVENTION

[0026] Hereinafter, embodiments of the present disclosure will be described in detail. Note that components having the same functions are assigned the same numbers, and redundant explanations will be omitted. [Example]

[0027] The Ising model generation device of this embodiment is a device that converts a combinatorial optimization problem into an Ising model that satisfies given spin number constraints and coefficient constraints with low computational cost (if no conversion that satisfies the constraints exists, it determines one).

[0028] The first idea realized by the Ising model generation device of this embodiment is (i) to characterize the degree of freedom of the processing of the above-mentioned Steps 1 to 4 regarding conversion to an Ising model by the maximum value of the coefficient upper limit parameter. The maximum value of the coefficient upper limit parameter has the function of adjusting the trade-off between the number of spins and the coefficients of the Ising model. By introducing this degree of freedom, it becomes possible to realize more conversion methods than conventional conversion methods, thereby reducing cases in which constraints cannot be met no matter which conversion method is used.

[0029] The second idea realized by the Ising model generation device of this embodiment is (ii) to directly calculate the maximum value of the coefficient upper limit parameter appropriate for satisfying the coefficient constraint of the Ising model from the polynomial of the original problem without actually converting to an Ising model. The number of spins of the Ising model when conversion is performed using the maximum value of the coefficient upper limit parameter is also calculated. This makes it possible to search for the maximum value of the coefficient upper limit parameter that simultaneously satisfies two constraints (coefficient constraint and spin number constraint) with low computational cost.

[0030] The functional configuration of the Ising model generation device of the first embodiment will be described below with reference to Fig. 1. As shown in the figure, the Ising model generation device 1 of the present embodiment includes a problem / constraint acquisition unit 11, a conversion parameter estimation unit 12, and a parameterized Ising model conversion unit 13.

[0031] The problem / constraint acquisition unit 11 acquires a combinatorial optimization problem and Ising machine constraints (coefficient constraints, spin number constraints) (S11).

[0032] The first coefficient upper limit parameter is a parameter that determines an upper limit for the coefficients of a linear expression of a binary variable when replacing an integer variable with a linear expression of a binary variable during conversion to an unconstrained binary variable polynomial, and the second coefficient upper limit parameter is a parameter that determines an upper limit for the coefficients of a linear expression of a binary variable when replacing an integer variable with a linear expression of a binary variable during conversion to an unconstrained binary variable quadratic expression.

[0033] The transformation parameter estimation unit 12 calculates the maximum value of the first coefficient upper limit parameter and the maximum value of the second coefficient upper limit parameter at which the transformed expressions satisfy the coefficient constraints of the Ising machine, and calculates the number of spins of the transformed polynomial and quadratic expressions using the maximum value of the first coefficient upper limit parameter and the maximum value of the second coefficient upper limit parameter (S12).

[0034] If the number of spins satisfies the spin number constraint of the Ising machine, the parameterized Ising model conversion unit 13 converts the combinatorial optimization problem into an Ising model by binary encoding if the coefficient of the linear expression of the binary variables does not become larger than the maximum value of the coefficient upper limit parameter, and by replacing the integer variables with a linear expression of the binary variables calculated so that the coefficient of the linear expression of the binary variables becomes equal to the maximum value of the coefficient upper limit parameter in other cases (S13).

[0035] As shown in Fig. 2, the conversion parameter estimation unit 12 includes a preprocessing unit 121, a parameter calculation unit 122, and a spin number calculation unit 123. As shown in Fig. 3, the parameterized Ising model conversion unit 13 includes a first conversion processing unit 131, a second conversion processing unit 132, a third conversion processing unit 133, and an Ising model conversion processing unit 134. The operation of each component will be described below with reference to Fig. 4.

[0036] Preprocessing unit 121 executes preprocessing for calculating the maximum values ​​of the first and second coefficient upper limit parameters described below (S121, see <Details of Step S121>).

[0037] The parameter calculation unit 122 calculates the maximum value of the first coefficient upper limit parameter and the maximum value of the second coefficient upper limit parameter that satisfy the coefficient constraint of the Ising machine (S122, see <Details of Step S122>).

[0038] The spin number calculation unit 123 calculates the number of spins of the polynomial and quadratic expression after transformation using the maximum value of the first coefficient upper limit parameter and the maximum value of the second coefficient upper limit parameter (S123).

[0039] The Ising model generation device 1 determines whether the spin number satisfies the spin number constraint of the Ising machine. If the spin number does not satisfy the spin number constraint of the Ising machine, the Ising model generation device 1 outputs a result (conversion not possible) (S1A). If the spin number satisfies the spin number constraint of the Ising machine, the parameterized Ising model conversion unit 13 executes the following process.

[0040] The first conversion processing unit 131 converts the combinatorial optimization problem into an unconstrained integer variable polynomial optimization problem (S131, similar to Step 1 above).

[0041] The second conversion processing unit 132 converts the unconstrained integer variable polynomial optimization problem into the unconstrained binary variable polynomial optimization problem by binary encoding if the coefficient of the linear expression of the binary variables is not greater than the maximum value of the first coefficient upper limit parameter, and by replacing the integer variables with a linear expression of the binary variables calculated so that the coefficient of the linear expression of the binary variables is equal to the maximum value of the first coefficient upper limit parameter in other cases (S132, see <Details of Step S132>).

[0042] The third conversion processing unit 133 converts the unconstrained binary polynomial optimization problem into an unconstrained binary quadratic optimization problem by using binary encoding if the coefficient of the linear expression of the binary variables is not greater than the maximum value of the second coefficient upper limit parameter, and by replacing the integer variables with a linear expression of the binary variables calculated so that the coefficient of the linear expression of the binary variables is equal to the maximum value of the second coefficient upper limit parameter in other cases (S133, see <Details of Step S133>).

[0043] The Ising model conversion processing unit 134 converts the unconstrained binary variable quadratic optimization problem into an Ising model (S134, similar to the above step 4). The Ising model generation device 1 outputs the Ising model (S1B).

[0044] <Details of step S121> For the sake of explanation, the coefficient constraints of the Ising machine are M - (<0) or more M +(>0). To be precise, the coefficient constraint should be given as the range of values ​​that the ratio of the maximum and minimum absolute values ​​of the coefficients of the Ising model can take, rather than the range of the coefficient values ​​themselves. However, in this specification, since only integer coefficient Ising models need to be considered in the construction of the conversion, we will assume that the minimum absolute value of the coefficient is given as 1, and will express the coefficient constraint in the above notation. The calculation method is the penalty term g i p (z,ξ i ), so in the following we will consider the constraint 0≦g~ i (z)≦u i As the penalty term corresponding to

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[0045] Step 1: Convert the objective function f~(z) of the constrained polynomial optimization into the objective function f~(z) of the unconstrained optimization

number

[0046] Step 2 is a process of replacing each integer variable of F~(z,ξ) with a linear expression of binary variables, but this replacement is not performed when calculating the number of spins and coefficients of the converted Ising model. Instead, the preprocessing unit 121 calculates an appropriate z=(z j ) j ,ξ=(ξ i ) i C z =(C z,j ) j ,C ξ =(C ξ,i ) i There are several possible ways to achieve this, but the following method is an example. z,j The preprocessing unit 121 calculates each ξ in the polynomial F~(z,ξ).i The upper limit of the range ξ i =u i-1 The polynomial substituted with

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[0047] <Details of Step S122> Any two variables z j ,z j' and for a positive integer C, define the following quantity:

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[0048] Step 3 converts a monomial of degree 3 or higher into a quadratic one, and if the coefficient a of the monomial is positive, it has a parameter for the conversion. The value of this parameter is C 2 ≦M + If a C that satisfies / a is adopted, the Ising model after the transformation satisfies the given coefficient constraint (proof omitted). In reality, the binary variable polynomial that is the input for <Step 3> cannot be obtained because the transformation in <Step 2> is not performed, but it is easy to calculate how many monomials of what degree the polynomial after the transformation in <Step 2> has. Therefore, when the parameter for the transformation of each term to a quadratic expression is set to C=1, the number of variables of the polynomial after the transformation can be calculated efficiently (without performing all the transformation processes). Setting the parameter for the transformation of all monomials to C=1 may be a good idea in some cases. 2 ≦M + This will result in estimating a larger number of variables than when the largest C that satisfies / a is used, but the trade-off between the magnitude of the error in the number of variables and the processing time can be adjusted by, for example, partially calculating only those terms whose coefficients are known to be small after the transformation in Step 2.

[0049] <Details of Step S132> The second conversion processing unit 132 introduces the maximum value of the first coefficient upper limit parameter in the above step 2. In step 2, the integer variable z is set to n z binary variables x z,1 ,…,x z,n_z First-order equation of

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[0050] (α) The coefficient of the binary linear equation after substitution is at most C, and (β) A transformation method with property (α) is always n z Property (α) means that the constraints on the coefficients of the Ising model can be satisfied by adjusting C, and property (β) means that the above method is the best in terms of the number of spins among the transformation methods that have property (α).

[0051] Therefore, the above conversion method is an efficient conversion that takes into maximum consideration the number of spins and coefficient constraints of the converted Ising model, and makes it possible to flexibly adjust the trade-off between the two constraints by using the maximum value C of the first coefficient upper limit parameter.

[0052] <Details of Step S133> The third conversion processing unit 133 introduces the maximum value of the second coefficient upper limit parameter in the above step 3. Step 3 is a process of converting each cubic or higher monomial included in the polynomial of the objective function to be optimized into an equivalent optimization problem by replacing it with an appropriate quadratic expression. It is known that when the coefficient of a cubic or higher monomial is negative, this can be achieved by adding one binary variable (Non-Patent Document 3). On the other hand, when the coefficient is positive, a method has been proposed in which multiple binary variables are added and replaced. These methods do not allow for adjustment of the trade-off between the magnitude of the coefficient of each term in the expression after monomial replacement and the number of added variables, and therefore do not allow for flexible adjustment of the trade-off between the two constraints of the converted Ising model.

[0053] Therefore, the third conversion processing unit 133 introduces the maximum value of the second coefficient upper limit parameter as follows: First, the third conversion processing unit 133 converts a dth degree monomial with a positive coefficient

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[0054] The operation of the Ising machine generator described above is one example of a method for realizing ideas (i) and (ii). Two examples (Example 2 and Example 3) will be used to explain the flow of processing for converting a combinatorial optimization problem into an Ising model using the functions realized by ideas (i) and (ii). The coefficient constraints of an Ising machine include cases where the coefficients are required to be strictly within a certain range (hard constraints) and cases where the coefficients are required to be within as small a range as possible (soft constraints). Example 2 deals with conversion under hard constraints, and Example 3 deals with conversion under soft constraints. [Example]

[0055] Hereinafter, with reference to FIG. 5, a functional configuration of an Ising model generating device 2 according to a second embodiment, which can determine whether the spin number constraint is satisfied more strictly than in the first embodiment, will be described.

[0056] As shown in the figure, the Ising model generation device 2 of this embodiment includes a problem / constraint acquisition unit 11, a conversion parameter estimation unit 22, and a parameterized Ising model conversion unit .

[0057] As shown in Fig. 6, the conversion parameter estimation unit 22 includes a preprocessing unit 121, a parameter calculation unit 222, and a spin number calculation unit 123. As shown in Fig. 7, the parameterized Ising model conversion unit 23 includes a first conversion processing unit 131, a second conversion processing unit 232, a third conversion processing unit 133, and an Ising model conversion processing unit 134. Hereinafter, the operation of each component will be described with reference to Fig. 8, focusing on processing different from that in the first embodiment.

[0058] The preprocessing unit 121 executes S121 in the same manner as in the first embodiment.

[0059] The parameter calculation unit 222 calculates the maximum value of the first coefficient upper limit parameter that satisfies the coefficient constraint of the Ising machine (S222A).

[0060] The second conversion processing unit 232 executes the conversion process of <<Step 2>> based on the maximum value of the first coefficient upper limit parameter (S232).

[0061] The parameter calculation unit 222 calculates the maximum value of the second coefficient upper limit parameter based on the unconstrained binary variable polynomial optimization problem actually obtained by the conversion process in Step 2 (S222B).

[0062] The spin number calculation unit 123 executes S123 in the same manner as in the first embodiment.

[0063] If the spin number constraint is met, steps S133, S134 and S1B are performed. If the spin number constraint is not met, step S1A is performed.

[0064] While the above process allows for more precise determination of whether the spin number constraint is satisfied, it also requires the actual conversion processing time to output a result that the conversion is not possible if the constraint is not satisfied. Note that even if the calculation of appropriate parameters in <<Step 3>> involves actual conversion processing, the key point remains the same: to calculate appropriate parameters to be used in <<Step 2>> without numerous trials involving actual conversion. [Example]

[0065] Hereinafter, with reference to FIG. 9, a functional configuration of an Ising model generation device 3 according to a third embodiment that can perform optimal operations when the coefficient constraint of the Ising model is a soft constraint will be described.

[0066] As shown in the figure, the Ising model generation device 3 of this embodiment includes a problem / constraint acquisition unit 11, a conversion parameter estimation unit 32, a parameterized Ising model conversion unit 13, and a relaxation coefficient constraint setting / updating unit 33.

[0067] The operation of each component will be described below with reference to FIG. 10, focusing on the processing that differs from the first and second embodiments.

[0068] In the second embodiment, a conversion with as few spins as possible is performed under the given coefficient constraint, and if the number of spins violates the constraint, it is immediately determined that the conversion is "impossible."

[0069] If the coefficient constraint is a soft constraint, and the number of spins after the conversion violates the constraint, the coefficient constraint can be relaxed and the process of Example 2 can be repeated until a conversion method that satisfies the spin number constraint is found, thereby obtaining a conversion that "strictly satisfies the spin number constraint and has as small a coefficient range as possible."

[0070] First, the calculation of parameters to satisfy the coefficient constraints and the number of spins after conversion are performed by the method disclosed in the embodiment 2. That is, the problem / constraint acquisition unit 11 executes step S11, and the conversion parameter estimation unit 32 executes the process of estimating the conversion parameters and the number of spins in the same manner as in the embodiment 2 (S32A).

[0071] If the spin number constraint is satisfied, the parameterized Ising model conversion unit 13 executes the conversion process (S13), and the Ising model generation device 3 outputs the Ising model and ends the process (S1B).

[0072] If the spin number constraint is not satisfied, the relaxation coefficient constraint setting / updating unit 33 sets an initial value for the relaxation coefficient constraint (S33A). This initial value is set so that the range of the coefficient constraint is sufficiently large.

[0073] The transformation parameter estimation unit 32 executes a process of estimating the transformation parameters and the number of spins based on the initial values ​​of the relaxed coefficient constraints (S32B). If the range of the coefficient constraints is relaxed to be sufficiently large, the calculated number of spins will be the smallest among all possible transformation methods. Therefore, if the spin number constraints are not satisfied even after executing step S32B, the Ising model generation device 1 outputs the result (transformation not possible) (S1A).

[0074] If the spin number constraint is satisfied after executing step S32B, there exists a coefficient constraint between the original coefficient constraint and the relaxed coefficient constraint that narrows the coefficient range as much as possible while still satisfying the spin number constraint. Therefore, such a relaxed coefficient constraint is searched for by further repeating the parameter and spin number calculation (for example, by binary search).

[0075] Specifically, if the spin number constraint is satisfied after step S32B is executed, the relaxation coefficient constraint setting / updating unit 33 sets the upper and lower limits of the relaxation coefficient constraint (S33B).

[0076] The transformation parameter estimation unit 32 executes the process of estimating the transformation parameters and the number of spins based on the upper and lower limits of the relaxation coefficient constraints (S32C).

[0077] The relaxation coefficient constraint setting / updating unit 33 executes a determination of the spin number constraint (S33C), and updates the upper and lower limits of the relaxation coefficient constraint (S33D).

[0078] The processes of steps S32C, S33C, and S33D are repeatedly executed until the relaxation coefficient constraints converge. When the relaxation coefficient constraints converge, the parameterized Ising model conversion unit 13 executes the conversion process (S13), and the Ising model generation device 3 outputs the Ising model and ends (S1B).

[0079] The calculation of parameters and spin numbers is repeated in the above process, but the key point is that this can be done with low computational cost because it is calculated directly from the polynomials of the original problem without converting to an actual Ising model.

[0080] The above-mentioned Examples 1 to 3 are merely examples and do not limit the scope of application of the present invention.

[0081] <Effects> The Ising model generating devices 1 to 3 disclosed in the first to third embodiments have the following effects.

[0082] When converting combinatorial optimization problems to Ising models, the trade-off between the number of spins and coefficients of the converted Ising model can be flexibly adjusted. This makes it possible to convert combinatorial optimization problems that could not achieve both spin number and coefficient constraints with conventional methods so that the constraints are compatible in many cases.

[0083] By searching for an appropriate conversion method for a combinatorial optimization problem without repeatedly performing trials involving conversion to an Ising model, it is possible to determine with low computational cost (computation time and memory) whether a conversion to an Ising model that satisfies both the number of spins and coefficient constraints exists. Also, if a conversion to an Ising model that satisfies both the number of spins and coefficient constraints exists, the search for that conversion method can be performed with low computational cost.

[0084] <Additional Notes> The device disclosed herein may, for example, be a single hardware entity having an input unit to which a keyboard or the like can be connected, an output unit to which an LCD display or the like can be connected, a communication unit to which a communication device (e.g., a communication cable) capable of communicating with an external device can be connected, a CPU (which may also include a central processing unit, cache memory, registers, etc.), memories such as RAM and ROM, an external storage device such as a hard disk, and buses connecting these input unit, output unit, communication unit, CPU, RAM, ROM, and external storage device so that data can be exchanged between them. If necessary, the hardware entity may also be provided with a device (drive) capable of reading and writing to a recording medium such as a CD-ROM. A physical entity equipped with such hardware resources includes a general-purpose computer.

[0085] The external storage device of the hardware entity stores the programs required to realize the above-mentioned functions and the data required for processing these programs (the programs may be stored in a ROM, which is a read-only storage device, for example, instead of an external storage device). Data obtained by processing these programs is stored in RAM, the external storage device, etc. as appropriate.

[0086] In a hardware entity, each program stored in an external storage device (or ROM, etc.) and the data required to process each program are loaded into memory as needed, and interpreted, executed, and processed by the CPU as appropriate, resulting in the CPU realizing a predetermined function (each component represented as a unit, means, etc., above).

[0087] The present disclosure is not limited to the above-described embodiments, and modifications can be made as appropriate without departing from the spirit of the present disclosure. Furthermore, the processes described in the above embodiments may not only be executed in chronological order according to the order described, but may also be executed in parallel or individually depending on the processing capabilities of the devices that execute the processes or as needed.

[0088] As described above, when the processing functions of the hardware entities (apparatuses of the present disclosure) described in the above embodiments are realized by a computer, the processing contents of the functions that the hardware entities should have are described by a program. Then, by executing this program on a computer, the processing functions of the hardware entities are realized on the computer.

[0089] The various processes described above can be implemented by loading a program that executes each step of the above method into the recording unit 10020 of the computer 10000 shown in Figure 11 and operating the control unit 10010, input unit 10030, output unit 10040, etc.

[0090] The program describing the processing contents can be recorded on a computer-readable recording medium. Examples of computer-readable recording media include magnetic recording devices, optical disks, magneto-optical recording media, and semiconductor memories. Specifically, examples of magnetic recording devices include hard disk drives, flexible disks, and magnetic tapes; optical disks include DVDs (Digital Versatile Discs), DVD-RAMs (Random Access Memory), CD-ROMs (Compact Disc Read Only Memory), and CD-Rs (Recordable) / RWs (Rewritable); magneto-optical recording media include MOs (Magneto-Optical discs), and semiconductor memories include EEP-ROMs (Electrically Erasable and Programmable-Read Only Memory).

[0091] The program may be distributed, for example, by selling, transferring, lending, etc. a portable recording medium such as a DVD or CD-ROM on which the program is recorded. Furthermore, the program may be stored in a storage device of a server computer, and then transferred from the server computer to another computer via a network, thereby distributing the program.

[0092] A computer that executes such a program may first temporarily store the program recorded on a portable recording medium or transferred from a server computer in its own storage device. Then, when executing a process, the computer reads the program stored on its own recording medium and executes the process in accordance with the read program. Alternatively, the computer may read the program directly from a portable recording medium and execute the process in accordance with the program. Furthermore, the computer may execute the process in accordance with the received program each time a program is transferred from a server computer to the computer. Alternatively, the server computer may not transfer the program to the computer, but may execute the process through a so-called ASP (Application Service Provider) service, which realizes the processing function by issuing an execution instruction and obtaining the results. In this embodiment, the program includes information used for processing by a computer that is equivalent to a program (such as data that is not a direct instruction to the computer but has properties that define computer processing).

[0093] In addition, in this embodiment, a hardware entity is configured by executing a predetermined program on a computer, but at least a part of the processing contents may be realized by hardware.

Claims

1. the first coefficient upper limit parameter is a parameter that determines an upper limit of a coefficient of a linear expression of a binary variable when an integer variable is replaced with the linear expression of a binary variable in the conversion to an unconstrained polynomial of a binary variable; the second coefficient upper limit parameter is a parameter that determines an upper limit of a coefficient of a linear expression of binary variables when an integer variable is replaced with a linear expression of binary variables during conversion to an unconstrained quadratic expression of binary variables; a transformation parameter estimation unit that calculates a maximum value of the first coefficient upper limit parameter and a maximum value of the second coefficient upper limit parameter such that the transformed expressions satisfy the coefficient constraint of the Ising machine, and calculates the number of spins of the transformed polynomial and quadratic expressions using the maximum value of the first coefficient upper limit parameter and the maximum value of the second coefficient upper limit parameter; and a parameterized Ising model conversion unit that converts the combinatorial optimization problem into an Ising model by using binary encoding if the number of spins satisfies the spin number constraint of the Ising machine and if the coefficient of the linear expression of binary variables does not become larger than the maximum value of the coefficient upper limit parameter, and by replacing integer variables with a linear expression of binary variables calculated so that the coefficient of the linear expression of binary variables becomes equal to the maximum value of the coefficient upper limit parameter in other cases. Ising model generator.

2. The Ising model generation device according to claim 1, The transformation parameter estimation unit a parameter calculation unit that calculates a maximum value of the first coefficient upper limit parameter and a maximum value of the second coefficient upper limit parameter that satisfy the coefficient constraint of the Ising machine; a spin number calculation unit that calculates the number of spins of the polynomial and quadratic expression after transformation using the maximum value of the first coefficient upper limit parameter and the maximum value of the second coefficient upper limit parameter; The parameterized Ising model conversion unit a first conversion processing unit that converts the combinatorial optimization problem into an unconstrained integer variable polynomial optimization problem when the number of spins satisfies a spin number constraint of the Ising machine; a second conversion processing unit that converts the unconstrained integer variable polynomial optimization problem into an unconstrained binary variable polynomial optimization problem by using binary encoding when a coefficient of the linear expression of the binary variables is not greater than a maximum value of the first coefficient upper limit parameter, and by replacing integer variables with a linear expression of binary variables calculated so that a coefficient of the linear expression of the binary variables is equal to the maximum value of the first coefficient upper limit parameter in other cases; a third conversion processing unit that converts the unconstrained binary polynomial optimization problem into an unconstrained binary quadratic optimization problem by using binary encoding when a coefficient of the linear expression of the binary variables is not greater than a maximum value of the second coefficient upper limit parameter, and by replacing integer variables with a linear expression of binary variables calculated so that a coefficient of the linear expression of the binary variables is equal to the maximum value of the second coefficient upper limit parameter in other cases; an Ising model conversion processing unit that converts the unconstrained binary variable quadratic optimization problem into an Ising model; Ising model generator.

3. An Ising model generation method executed by an Ising model generation device, the first coefficient upper limit parameter is a parameter that determines an upper limit of a coefficient of a linear expression of a binary variable when an integer variable is replaced with the linear expression of a binary variable in the conversion to an unconstrained polynomial of a binary variable; the second coefficient upper limit parameter is a parameter that determines an upper limit of a coefficient of a linear expression of binary variables when an integer variable is replaced with a linear expression of binary variables during conversion to an unconstrained quadratic expression of binary variables; calculating a maximum value of the first coefficient upper limit parameter and a maximum value of the second coefficient upper limit parameter such that the transformed expressions satisfy the coefficient constraint of the Ising machine, and calculating the number of spins of the transformed polynomial and quadratic expressions according to the maximum value of the first coefficient upper limit parameter and the maximum value of the second coefficient upper limit parameter; When the number of spins satisfies the spin number constraint of the Ising machine, converting the combinatorial optimization problem into an Ising model by binary encoding if the coefficient of the linear expression of binary variables is not greater than the maximum value of the coefficient upper limit parameter, and by replacing integer variables with a linear expression of binary variables calculated so that the coefficient of the linear expression of binary variables is equal to the maximum value of the coefficient upper limit parameter in other cases. Ising model generation method.

4. The Ising model generation method according to claim 3, Calculating a maximum value of the first coefficient upper limit parameter and a maximum value of the second coefficient upper limit parameter that satisfy the coefficient constraint of the Ising machine; calculating the number of spins of the polynomial and quadratic expression after transformation using the maximum value of the first coefficient upper limit parameter and the maximum value of the second coefficient upper limit parameter; converting the combinatorial optimization problem into an unconstrained integer variable polynomial optimization problem if the spin number satisfies the spin number constraint of the Ising machine; converting the unconstrained integer polynomial optimization problem into an unconstrained binary polynomial optimization problem by substituting integer variables with linear expressions of binary variables calculated by binary encoding if the coefficients of the linear expressions of binary variables are not greater than the maximum value of the first coefficient upper limit parameter, and otherwise, such that the coefficients of the linear expressions of binary variables are equal to the maximum value of the first coefficient upper limit parameter; converting the unconstrained binary polynomial optimization problem into an unconstrained binary quadratic optimization problem by replacing integer variables with linear expressions of binary variables calculated by binary encoding if the coefficients of the linear expressions of binary variables are not greater than the maximum value of the second coefficient upper limit parameter, and otherwise such that the coefficients of the linear expressions of binary variables are equal to the maximum value of the second coefficient upper limit parameter; converting the unconstrained binary variable quadratic optimization problem into an Ising model. Ising model generation method.

5. A program that causes a computer to function as the Ising model generation device according to claim 1 or 2.

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