Calculation model and calculation program

Abstract

The calculation model of the present invention is a calculation model that can be applied to an Isin model or QUBO, assigns a plurality of options in a combinatorial optimization problem to any one of values that can be taken by one or more binary variables, and fixes any one of the binary variables on the basis of constraints provided to the combinatorial optimization problem.

Description

Technical Field

[0001] The invention relates to a calculation model and a calculation program. Background Art

[0002] Attempts are being made to use quantum annealing to find the optimal solution to a combinatorial optimization problem. A combinatorial optimization problem can be treated as a minimization problem of finding a combination that minimizes an arbitrary objective function.

[0003] Optimization problems are sometimes subject to various constraints. For example, as described in Patent Document 1, constraints are sometimes added to the objective function so that the objective function does not become minimum when the constraints are violated.

[0004] Prior art literature

[0005] Patent Literature

[0006] Patent Document 1: International Publication No. 2022 / 024329 Summary of the invention

[0007] Problems to be solved by the invention

[0008] When the number of constraints increases, the proportion of the search range that satisfies the constraints decreases. Even when a constraint term is added as a constraint to the objective function, a search is performed in a range that does not satisfy the constraints, so the search efficiency is poor.

[0009] The present invention has been made in view of the above circumstances, and an object of the present invention is to provide a calculation model and a calculation program that are excellent in search efficiency even when constraints are set.

[0010] Technical solutions to solve problems

[0011] In order to solve the above problems, the present invention provides the following technical solutions.

[0012] The computational model of the first aspect is a computational model that can be applied to the Ising model or QUBO (quadratic unconstrained binary optimization). In this computational model, multiple options in a combinatorial optimization problem are assigned to any one of the possible values ​​of one or more binary variables, and any one of the binary variables is fixed based on the constraints given to the combinatorial optimization problem.

[0013] In the calculation model of the above aspect, the variable that can be fixed based on the constraint may be removed from the first calculation model represented by the following equation (1).

[0014] [Mathematical formula 1]

[0015]

[0016] In formula (1), x i 、x j is the 2-valued variable, J ij is the interaction parameter, h i is a parameter imposed on each variable according to external factors, and α is a constant.

[0017] In the calculation model of the above aspect, it is also possible that, when applied to the QUBO, in the case where the variable x that can be fixed is a When the variable x is fixed to 1, it is expressed by the following formula (2). a When it is fixed to 0, it is expressed by the following formula (3).

[0018] [Mathematical formula 2]

[0019]

[0020] [Mathematical formula 3]

[0021]

[0022] In the calculation model of the above aspect, it may also be that, when applied to the Ising model, in the case where the variable x that can be fixed is a When it is fixed to +1, it is expressed by the following formula (4), in which the variable x that can be fixed is a When it is fixed to -1, it is expressed by the following formula (5).

[0023] [Formula 4]

[0024]

[0025] [Formula 5]

[0026]

[0027] In the calculation model of the above aspect, it may also be that the constraint is a constraint on selecting a specific option from among the multiple options.

[0028] In the calculation model of the above aspect, the constraint may be a constraint of not selecting a specific option among the multiple options.

[0029] In the calculation model of the above aspect, the multiple options may also be represented by one-hot representation respectively.

[0030] In the calculation model of the above aspect, it is also possible that the multiple options are represented by binary representation respectively.

[0031] In the calculation model of the above aspect, the plurality of options may be represented by domain wall representations, respectively.

[0032] In the calculation model of the above aspect, the multiple options may also be represented by unary expressions respectively.

[0033] The computing program of the second aspect is a computer program applicable to the Ising model or QUBO. The computing program performs the following processing: a first processing of assigning multiple options in a combinatorial optimization problem to any one of the values ​​that one or more binary variables can take; and a second processing of fixing any one of the binary variables based on the constraints given to the combinatorial optimization problem.

[0034] In the calculation program of the above aspect, the second processing may be performed as follows: finding a variable that can be fixed among the plurality of binary variables based on the constraint; and removing the variable that can be fixed from the first calculation model represented by the above formula (1).

[0035] In the calculation program of the above aspect, when applied to the QUBO, the variable x that can be fixed may be a When the variable x is fixed to 1, the above formula (1) is transformed into the above formula (2). a When is fixed to 0, the above equation (1) is transformed into the above equation (3).

[0036] In the calculation program of the above aspect, when applied to the Ising model, the variable x that can be fixed may be a When the variable x is fixed to +1, the above equation (1) is transformed into the above equation (4). a When is fixed to -1, the above equation (1) is transformed into the above equation (5).

[0037] In the calculation program of the above aspect, a constraint of selecting a specific option from among the plurality of options may be provided as the constraint.

[0038] In the calculation program of the above aspect, a constraint of not selecting a specific option among the plurality of options may be provided as the constraint.

[0039] In the calculation program of the above aspect, the plurality of options may be represented by one-hot representation respectively.

[0040] In the calculation program of the above aspect, the plurality of options may be represented by binary representations respectively.

[0041] In the calculation program of the above aspect, the plurality of options may be represented by domain wall representations, respectively.

[0042] In the calculation program of the above aspect, the plurality of options may be represented by unary expressions respectively.

[0043] Effects of the Invention

[0044] The calculation model of the present invention has excellent search efficiency. BRIEF DESCRIPTION OF THE DRAWINGS

[0045] Figure 1 It is a schematic diagram of the Ising model and QUBO.

[0046] Figure 2 This is an example of expressing the possible values ​​of a binary variable by one-hot representation and assigning a plurality of options to the possible values.

[0047] Figure 3 This is an example of expressing the possible values ​​of a binary variable by binary representation, and assigning a plurality of options to the possible values.

[0048] Figure 4 This is an example of expressing possible values ​​of a binary variable by domain wall representation and assigning a plurality of options to the possible values.

[0049] Figure 5 This is an example of expressing the possible values ​​of a binary variable by unary expression and assigning a plurality of options to the possible values.

[0050] Figure 6 An example of a case where the first constraint is set when allocating a plurality of options by one-hot representation is shown.

[0051] Figure 7 An example of a case where the first constraint and the second constraint are set when a plurality of options are assigned by one-hot expression is shown.

[0052] Figure 8 This is an example of a conceptual diagram in which a constraint for selecting a specific option is provided when a plurality of options are assigned using a one-hot representation.

[0053] Fig. 9 This is an example of a conceptual diagram in which a constraint of not selecting a specific option is provided when a plurality of options are assigned using a one-hot representation.

[0054] Fig.10 This is an example of a conceptual diagram in which a constraint of not selecting a specific option is provided when a plurality of options are assigned using a one-hot representation.

[0055] Fig.11 This is an example of a conceptual diagram in which a constraint of not selecting a specific option is provided when a plurality of options are assigned using a one-hot representation.

[0056] Fig.12 This is an example of a conceptual diagram of a case where a constraint of not selecting a specific option is provided when a plurality of options are assigned by binary representation.

[0057] Fig.13 This is an example of a conceptual diagram of a case where a constraint of not selecting a specific option is provided when a plurality of options are allocated by domain wall representation.

[0058] Fig.14 This is an example of a conceptual diagram in which a constraint of not selecting a specific option is provided when a plurality of options are represented by domain wall representation.

[0059] Fig.15 This is an example of a conceptual diagram in which a constraint of not selecting a specific option is provided when a plurality of options are assigned by unary expression.

[0060] Fig.16 This is an example of a conceptual diagram in which a constraint of not selecting a specific option is provided when a plurality of options are represented by a unary expression. DETAILED DESCRIPTION

[0061] Hereinafter, the present embodiment will be described in detail with reference to the accompanying drawings as appropriate. In the accompanying drawings used in the following description, in order to make the features of the present embodiment easy to understand, for convenience, the features are sometimes enlarged and the dimensional ratios of the components are sometimes different from the actual ones. The materials, dimensions, etc. illustrated in the following description are examples, and the present invention is not limited to these, and can be implemented by appropriate changes within the scope of not changing the main purpose.

[0062] "First Implementation Method"

[0063] The calculation model of the first embodiment is a calculation model that can be applied to the Ising model or QUBO for quantum annealing. Quantum annealing is an algorithm for finding a state (ground state) with minimum energy based on the calculation model.

[0064] The Ising model is a model that predicts the overall stable state when multiple factors interact with each other and a strong force is applied to each factor.

[0065] Figure 1 This is a schematic diagram of the Ising model. The Ising model has multiple bits b that interact with each other through a strong force F. Each bit b is composed of a spin s. The spin s represents either an up or down state. The bits b are represented by variables representing binary states. By setting the strong force F, the state in which adjacent spins s are balanced becomes a stable state, or the state in which they are antiparallel becomes a stable state. The strong force F is called an interaction parameter.

[0066] The Ising model is represented by the following energy function (cost function).

[0067] [Mathematical formula 6]

[0068]

[0069] Hereinafter, the calculation model represented by equation (1) is referred to as the first calculation model. i 、x j is the input variable. i 、x j A binary variable of +1 or -1. i 、x j Corresponds to Figure 1 The state of the spin s in J ij is the interaction parameter. ij Corresponds to Figure 1 The strong force F in h i is a parameter applied to each bit b by an external factor, such as a magnetic field parameter. The magnetic field parameter can be regarded as a weight of each bit b. α is a constant.

[0070] QUBO (Quadratic Unconstrained Binary Optimization) is a computational model that can be equivalently transformed into the Ising model. In the Ising model, each bit b is represented by a binary variable of +1 or -1, whereas in QUBO, each bit b is represented by a binary variable of 0 or 1. QUBO is represented by the first computational model in the same way as the Ising model. In QUBO, x i 、x j A 2-valued variable that is either 0 or 1.

[0071] The Ising model and QUBO can be applied to combinatorial optimization problems, such as the traveling salesman problem, the knapsack problem, the shift optimization problem, and the delivery planning problem.

[0072] In the Ising model and QUBO, each of the multiple options in the combinatorial optimization problem is represented by a binary variable x i 、x j Furthermore, by finding the variable x that minimizes the energy function i 、x j , can use the Ising model or QUBO to solve combinatorial optimization problems.

[0073] In the calculation model of this embodiment, multiple options in the combinatorial optimization problem are assigned to any of the values ​​that one or more variables can take. The variable is the above-mentioned binary variable.

[0074] The multiple options in a combinatorial optimization problem vary depending on the problem being optimized. For example, if it is a traveling salesman problem, there are options for which cities to visit in which order. For example, if it is a shift optimization problem, there are options for who goes to work and when.

[0075] Assigning this option to more than one variable can take on a value. Figure 2 This is an example of assigning multiple options to any of the possible values ​​of a variable. Figure 2 The left side of is a diagram that assigns a portion of the multiple options to any of the possible values ​​of the three variables. Figure 2 The right side of is a diagram that assigns a portion of the multiple options to any of the values ​​that two variables can take.

[0076] Here, the value that a variable can take refers to the selectable value among the values ​​generated by combining two-value variables. Figure 2 As shown on the left side of , if there are three variables x1, x2, and x3, then the three variables x1, x2, and x3 can choose "1" or "0" respectively. Therefore, there are 8 possible values ​​for the three variables x1, x2, and x3: (0, 0, 0) (0, 0, 1), (0, 1, 0), (1, 0, 0), (0, 1, 1), (1, 1, 0), (1, 0, 1), (1, 1, 1). These 8 values ​​are not always all possible values. Depending on the method of expressing the options described later, the selectable values ​​are different, and the possible values ​​of the three variables x1, x2, and x3 are further limited.

[0077] For example, when one of the three options A, B, and C is selected as the first option, option A is assigned to value a1, option B is assigned to value a2, and option C is assigned to value a3. Values ​​a1, a2, and a3 are values ​​that can be taken by the combination of variables x1, x2, and x3, respectively. Also, for example, when one of the two options D and E is selected as the second option, these options D and E are assigned to values ​​that can be taken by two variables y1 and y2. Option D is assigned to value b1, and option E is assigned to value b2. Values ​​b1 and b2 are values ​​that can be taken by the combination of variables y1 and y2, respectively. In this case, the calculation model can select any one of the combinations of options (a1, b1), (a1, b2), (a2, b1), (a2, b2), (a3, b1), and (a3, b2) as the optimal solution. In addition, it is arbitrary to assign which option to which of the values ​​that the variables can take.

[0078] The possible values ​​of a variable are represented by a combination of multiple binary variables. For example, the values ​​a1, a2, and a3 are represented by a combination of three binary variables x1, x2, and x3, respectively, and the values ​​b1 and b2 are represented by a combination of two binary variables y1 and y2, respectively. The variables x1, x2, x3, y1, and y2 correspond to the variables x in equation (1).i 、x j , corresponding to Figure 1 The spin s of the bit b in Figure 2 ] shows the case of QUBO where the binary variables are 1 and 0. In the case of the Ising model, the binary variables may also be +1 and -1.

[0079] Each of the options can be represented by one-hot representation, binary representation, domain wall representation, or unary representation.

[0080] One-hot representation is a method of representing N types of information using N variables. In the case of one-hot representation, only one of the N variables is "1", and the other variables are all "0" in the case of QUBO, and all "-1" in the case of the Ising model. Figure 2 The options shown are represented by one-hot representation. One-hot representation requires the same number of variables as the options, but even if one of the variables is overwritten by noise, it does not represent other states and is strong in terms of noise.

[0081] Binary representation is a method of representing N types of information in binary. In the case of binary representation, each variable representing an option is allowed to be "1" at the same time. Figure 3 This is an example of assigning multiple options to the values ​​that a variable represented by binary representation can take. Binary representation can represent more states with fewer variables. For example, Figure 3 As shown, even two variables x1 and x2 can take four values ​​a1 to a4. For example, when the first options are three options A, B, and C, options A to C are assigned to any of the values ​​a1 to a3, and value a4 is not assigned.

[0082] Domain wall representation is a method of representing N types of information at the boundary position where adjacent values ​​are different. Figure 4 This is an example of assigning multiple options to the possible values ​​of a variable represented by domain wall representation. In the case of domain wall representation, the possible values ​​a1, a2, and a3 of the variable are represented by two fixed values ​​z1 and z2 and multiple variables x1 and x2, respectively. One fixed value z2 is fixed to "1", and the other fixed value z1 is fixed to "0" in the case of QUBO and fixed to "-1" in the case of Ising model.

[0083] Unary representation is a method of representing N types of information using the sum of variables. Figure 5 This is an example of the possible values ​​of a variable represented by a unary representation. For example, when the first option is option A, option B, and option C, as for the sum of the variables, option A is assigned 0 (the possible value of the variable is a1), option B is assigned 1 (the possible values ​​of the variable are a2 and a3), and option C is assigned 2 (the possible value of the variable is a4).

[0084] The computational model of this embodiment includes constraints. The computer calculates combinations that would be excluded if it were a human as combinations equivalent to other combinations. When constraints are imposed on the computational model, it is possible to prevent impossible variable combinations that output meaningless solutions from being output as optimal solutions. Constraints include, for example, constraints caused by the representation method of options (hereinafter referred to as first constraints), constraints added to options (hereinafter referred to as second constraints), and the like.

[0085] The first constraint varies depending on the representation method chosen.

[0086] For example, when options are represented by one-hot representation, a constraint in which any one variable is "1" becomes the first constraint.

[0087] For example, when options are represented by binary representation, the constraint of not selecting values ​​that are not assigned to the options among the possible values ​​of the variable becomes the first constraint. In binary representation, the number of possible values ​​of the variable does not match the number of options, and sometimes values ​​that are not assigned to the options are generated among the possible values ​​of the variable.

[0088] For example, when the options are represented by domain wall representation, the constraint that there is only one boundary between adjacent values ​​that are different becomes the first constraint.

[0089] For example, in the case where an option is expressed by a unary expression, the first constraint is not particularly set.

[0090] Figure 6 This shows an example of a case where the first constraint C1 is set for a plurality of states that can be expressed by two variables. Figure 6 In the method of expressing the first constraint C1 as an option, the one-hot expression is used. In this case, for example, the first constraint C1 is set to values ​​a4 to a8, b3, and b4 among the possible values ​​of the variable. In the case of the combination of these variables, the first constraint C1 does not minimize the energy of the objective function.

[0091] Values ​​a4 to a8, b3, and b4 do not satisfy the condition of a one-hot representation where only one of the N variables is "1", and options cannot be assigned. Therefore, in the case of values ​​a4 to a8, b3, and b4, if the energy of the objective function is minimized, a meaningless solution is output, and therefore, a first constraint C1 is set for these values ​​a4 to a8, b3, and b4.

[0092] The second constraint is the condition provided for the combinatorial optimization problem. For example, if it is a traveling salesman problem, "must go to city M for the Nth time" and "must not go to city M for the Nth time" become the second constraints. In addition, if it is a shift optimization problem, "do not let the same person work for more than a few hours", "specify the number of employees per day", and "reflect each employee's vacation wishes" become the second constraints.

[0093] The second constraint is set within the range of conditions that satisfy the first constraint C1. Figure 7 This is an example of a case where the first constraint C1 and the second constraint C2 are set for a plurality of states that can be expressed by two variables. The second constraint C2 is set for example to any one of the values ​​a1 to a3, b1, and b2 that can be selected even within the condition where the first constraint C1 is set. Figure 7 As shown, a second constraint C2 is set for the value a2. For example, the second constraint C2 may be a constraint for selecting the value a2 or a constraint for not selecting the value a2. For example, in the case where the constraint is set in a manner of selecting a2, when a2 is selected, the energy function represented by equation (1) is minimum. For example, in the case where the constraint is set in a manner of not selecting the value a2, when the value a2 is selected, the energy function represented by equation (1) is not minimum.

[0094] As a method of applying constraints to a computational model, there is a penalty method. The penalty method is a method of providing a constraint term so that the energy is not minimized relative to the possible values ​​of the variable that produces an unselectable solution (meaningless solution). For example, the following formula is an example of an energy function with a restriction term added to the objective term by the penalty method.

[0095] [Formula 7]

[0096] H(x)=H cost (x)+λH penalty (x)…(6)

[0097] In the above formula (6), H cost (x) is the target item, H penalty (x) is the constraint term, and λ is the coefficient. For example, by penalty (x) is set to a large value when both or either of the first constraint C1 and the second constraint C2 are not satisfied. When both or either of the first constraint C1 and the second constraint C2 are not satisfied, the energy function H(x) is not minimized.

[0098] The penalty method does not minimize the energy function when the variables that do not satisfy the constraints can take values ​​a4 to a8, b3, and b4.

[0099] That is, even if the combination of options that the calculation model can select is any one of (a1, b1), (a1, b2), (a2, b1), (a2, b2), (a3, b1), (a3, b2), the search range of the calculation model is all combinations of (a1, b1), (a1, b2), (a2, b1), (a2, b2), (a3, b1), (a3, b2), (a4, b1), (a4, b2), (a5, b1), (a5, b2), (a6, b1), (a6, b2), (a7, b1), (a7, b2), (a8, b1), (a8, b2).

[0100] That is, (a4, b1), (a4, b2), (a5, b1), (a5, b2), (a6, b1), (a6, b2), (a7, b1), (a7, b2), (a8, b1), and (a8, b2), which do not need to be searched, will be included in the search range. If all constraints are set only by the penalty method, the search efficiency will be poor.

[0101] In contrast, the calculation model of the present embodiment fixes any one of a plurality of binary variables based on the constraints provided to the combinatorial optimization problem. If a part of the variables is fixed, the search range based on the calculation model search becomes narrower, and the operation efficiency is improved. The variables that can be fixed in the calculation model vary depending on the constraints provided to the calculation model.

[0102] First, variables that can be fixed are obtained based on the constraints provided to the calculation model. Hereinafter, a method of obtaining variables that can be fixed in the case of the first mode and the case of the second mode under the constraints provided to the calculation model will be described.

[0103] The first mode is the case where a constraint is set on the computational model to select a specific option from multiple options. For example, if it is a traveling salesman problem, it is determined as "the Nth one goes to M cities", providing a constraint of fixed selection for the computational model.

[0104] In the first mode, since a specific option is selected, any one of the values ​​a1 to a3 that the variable can take is selected. The values ​​a4 to a8 that the variable can take cannot be selected based on the first constraint C1.

[0105] For example, Figure 8 As shown, a constraint is provided to select value a2 from values ​​a1 to a8 (select option B). In this case, all variables x1 to x3 can be fixed. For example, each variable x1 to x3 can be fixed to x1=0, x2=1, and x3=0.

[0106] The constraint that fixes the variables x1 to x3 is equivalent to the second constraint C2, but includes the first constraint C1. Therefore, when selecting an option based on the constraint, the first constraint C1 is not required, and the constraints can be reduced. This is also the case when the option is represented by binary representation or domain wall representation. When the option is represented by unary, the first constraint C1 is not provided initially.

[0107] In the first mode, when a constraint is provided for selecting the value a2, x1=0, x2=1, and x3=0 are fixed. In this case, x2 is first removed from the first calculation model. Formula (1) is transformed into the following formula (2). In this case, a=2. In formula (2), S / {a} means removing a from the set S of subscripts of all variables. a corresponds to the subscript of the variable to be fixed.

[0108] [Mathematical formula 8]

[0109]

[0110] Next, x1 is removed from the first calculation model. Formula (2) is transformed into the following formula (3). In this case, a=1. In formula (3), S / {a} means removing a from the set S obtained by removing 2 from the set of subscripts of all variables.

[0111] [Mathematical formula 9]

[0112]

[0113] Next, x3 is removed from the first calculation model. Formula (3) is transformed again into a formula with a=3. That is, in formula (3), S / {a} means removing a from the set S of subscripts of all variables with 2 and 1 removed. In the above example, the variables are removed in the order of x2, x1, and x3, but the order is not limited to this. In this case, the search range of the calculation model is limited to (a2, b1) and (a2, b2). That is, the calculation model of this embodiment can reduce the search range from 16 to 2 compared to the penalty method.

[0114] Here, as an example, a case where a constraint is provided for selecting the value a2 (selecting option B) is shown, but the same applies to the case where a value a1 is selected (selecting option A) and the case where a value a3 is selected (selecting option C). When the value a1 is selected, the search range of the calculation model is limited to (a1, b1) and (a1, b2). When the value a3 is selected, the search range of the calculation model is limited to (a3, b1) and (a3, b2).

[0115] The second mode is to set a constraint on the computational model not to select a specific option among multiple options. For example, if it is a traveling salesman problem, a constraint of not selecting the option of "going to M city for the Nth time" (i.e. "not going to M city for the Nth time") is provided.

[0116] For example, Fig. 9 As shown, a constraint is provided that does not select value a2 (does not select option B) from the possible values ​​a1 to a8 of the variable. For example, when the options are represented by a one-hot representation, this constraint can be replaced by a constraint that selects the possible value a1 of the variable or the possible value a3 of the variable by combining with the first constraint C1.

[0117] The value a1 that the variable can take and the value a3 that the variable can take are in common that the variable x2 is 0. That is, in this case, the variable that can be fixed is x2.

[0118] Next, the variables that can be fixed are fixed, and the variables that can be fixed are removed from the first calculation model represented by the above-mentioned equation (1).

[0119] In the second mode, when a constraint is provided that the value a2 is not selected, x2 is fixed to 0. In this case, equation (1) is transformed into equation (3). In this case, a=2. In equation (3), S / {a} means removing a from the set S of subscripts of all variables. a corresponds to the subscript of the variable to be fixed.

[0120] In this case, the search range of the calculation model is limited to (a1, b1), (a1, b2), (a3, b1), (a3, b2), (a4, b1), (a4, b2), (a7, b1), (a7, b2). (a4, b1), (a4, b2), (a7, b1), (a7, b2) are not selected by adding the first constraint C1 as a constraint term to the energy function. That is, the calculation model of this embodiment can reduce the search range from 16 to 8 compared to the penalty method.

[0121] Here, as an example, the case where the constraint of not selecting value a2 (not selecting option B) is shown, but in the case where the constraint of not selecting value a1 (not selecting option A) and the case where the constraint of not selecting value a3 (not selecting option C) are not selected, the search range of the calculation model can also be limited in the same order.

[0122] For example, Fig.10 This is an example of a case where a constraint is provided in which the value a1 is not selected (option A is not selected). In this case, x1 can be fixed to 0. In this case, equation (1) is also transformed into equation (3) (where a=1). In this case, the search range of the calculation model is limited to (a2, b1), (a2, b2), (a3, b1), (a3, b2), (a6, b1), (a6, b2), (a7, b1), (a7, b2). (a6, b1), (a6, b2), (a7, b1), (a7, b2) are not selected by adding the first constraint C1 as a constraint term to the energy function.

[0123] For example, Fig.11 This is an example of a case where a constraint is provided in which the value a3 is not selected (option C is not selected). In this case, x3 can be fixed to 0. In this case, equation (1) is also transformed into equation (3) (where a=3). In this case, the search range of the calculation model is limited to (a1, b1), (a1, b2), (a2, b1), (a2, b2), (a5, b1), (a5, b2), (a7, b1), (a7, b2). (a5, b1), (a5, b2), (a7, b1), (a7, b2) are not selected by adding the first constraint C1 as a constraint term to the energy function.

[0124] As described above, if the computational model fixes any one of the multiple binary variables based on the constraints provided to the combinatorial optimization problem, the computational model can narrow the search range of the computational model search. If the search range of the computational model is narrowed, the search efficiency of the computational model is improved and the computational load is also reduced.

[0125] Furthermore, although the case where any one of the variables x1 to x3 is fixed to 0 has been described as an example as the second mode, any one of the variables x1 to x3 may be fixed to 1.

[0126] In this case, equation (1) is transformed into equation (2). a corresponds to the subscript of the variable fixed to 1. In equation (2), S / {a} means removing a from the set S of subscripts of all variables.

[0127] For example, in the case where x2 is fixed to 1, the search range of the calculation model is limited to (a2, b1), (a2, b2), (a5, b1), (a5, b2), (a6, b1), (a6, b2), (a8, b1), (a8, b2). (a5, b1), (a5, b2), (a6, b1), (a6, b2), (a8, b1), (a8, b2) are not selected by adding the first constraint C1 as a constraint term to the objective function.

[0128] So far, the case where the options are represented by one-hot representation has been described as an example. However, when the options are represented by other representations, any one of a plurality of binary variables can be fixed in the same order.

[0129] Fig.12This is an example of a fixed condition for a variable when expressing an option in binary representation. For example, when a constraint is provided that the value a2 is not selected, x1 is fixed to 0. In this case, equation (1) is also transformed into equation (3) (where a=1). In this case, the search range of the calculation model is limited to (a1, b1), (a1, b2), (a3, b1), (a3, b2). Fig.12 As shown, the constraint of not selecting the value a2 takes into account the first constraint C1. That is, the calculation model of this embodiment can reduce the search range from 8 to 4 compared with the penalty method.

[0130] Fig.13 This is an example of a fixed condition for a variable when expressing an option through a domain wall representation. For example, a constraint is provided that the value a1 is not selected. In this case, x2 is fixed to 0. In this case, equation (1) is also transformed into equation (3) (where a=2). In this case, the search range of the calculation model is limited to (a2, b1), (a2, b2), (a3, b1), (a3, b2). Fig.13 As shown, the constraint of not selecting the value a1 takes into account the first constraint C1. That is, the calculation model of this embodiment can reduce the search range from 8 to 4 compared with the penalty method.

[0131] Fig.14 This is an example of a fixed condition for a variable when expressing an option through domain wall representation. For example, if a constraint is provided that the value a3 is not selected, x1 is fixed to 1. In this case, equation (1) is transformed into equation (2) (where a=1). In this case, the search range of the calculation model is limited to (a1, b1), (a1, b2), (a2, b1), (a2, b2). Fig.14 As shown, the constraint of not selecting the value a3 compromises the first constraint C1.

[0132] Fig.15 An example of a fixed condition for a variable when an option is represented by a unary representation. For example, when a constraint is provided that the value a1 is not selected, x1 is fixed to 1. In this case, equation (1) is also transformed into equation (2) (where a=1). In this case, the search range of the calculation model is limited to (a2, b1), (a2, b2), (a3, b1), (a3, b2). That is, the calculation model of this embodiment can reduce the search range from 8 to 4 compared to the penalty method. As another example, x2 can be fixed to 1 instead of fixing x1 to 1.

[0133] Fig.16This is an example of a fixed condition for a variable when expressing an option by a unary expression. For example, when a constraint is provided that the value a3 is not selected, x2 is fixed to 0. In this case, equation (1) is also transformed into equation (3) (where a=2). In this case, the search range of the calculation model is limited to (a1, b1), (a1, b2), (a2, b1), (a2, b2). As another example, x1 can be fixed to 0 instead of x2 being fixed to 0.

[0134] In addition, the case where the binary variable is 1 or 0 when applied to QUBO has been described above. In the case of the Ising model in which the binary variable is represented by +1 or -1, the formula (1) is transformed as follows.

[0135] For example, in will be able to fix the variable x a When it is fixed to +1, equation (1) is transformed into the following equation (4).

[0136] [Formula 10]

[0137]

[0138] For example, in will be able to fix the variable x a When it is fixed to -1, the formula (1) is transformed into the following formula (5).

[0139] [Mathematical formula 11]

[0140]

[0141] This computational model can be applied to an Ising machine that specializes in computing the Ising model or QUBO, for example. The computational model is stored in the Ising machine as a computational program, for example. The computational program provides instructions to the processor so that the processor performs processing according to the above computational model. For example, machines such as quantum annealing machines (D-wave, NEC), coherent Ising machines (NTT), analog bifurcation machines (Toshiba), digital annealers (Fujitsu), and CMOS annealers (Hitachi) are examples of Ising machines.

[0142] The Ising machine may also be a quantum gate computer. For example, if QAOA (Quantum Approximate Optimization Algorithm) is used, the Ising model or QUBO can be calculated using a quantum gate computer.

[0143] Although the embodiments of the present invention have been described in detail with reference to the drawings, each structure and combination thereof in each embodiment are merely examples, and addition, omission, substitution and other changes of structures are possible without departing from the gist of the present invention.

[0144] Description of Reference Numerals

[0145] b…bit, s…spin, F…strong force, a1~a8, b1~b4…value, x1~x3, y1, y2…2-value variable, A, B, C, D, E…options, C1…first constraint, C2…second constraint.

Claims

1. A computational model, wherein: The computational model can be applied to the Ising model or quadratic unconstrained binary optimization, which is denoted as QUBO. Assign multiple options in a combinatorial optimization problem to any of the possible values ​​of one or more 2-valued variables. Either one of the two-valued variables is fixed based on constraints given to the combinatorial optimization problem.

2. The calculation model according to claim 1, wherein: The variables that can be fixed based on the constraints are removed from the first calculation model represented by the following equation, In formula (1), x i 、x j is the 2-valued variable, J ij is the interaction parameter, h i is a parameter imposed on each variable according to external factors, and α is a constant.

3. The calculation model according to claim 2, wherein: In the case of the QUBO, In the variable x that can be fixed a When is fixed to 1, it is expressed by the following formula (2): In the variable x that can be fixed a When is fixed to 0, it is expressed by the following formula (3):

4. The computational model according to claim 2, wherein: In the case of application to the Ising model, In the variable x that can be fixed a When is fixed to +1, it is expressed by the following formula (4): In the variable x that can be fixed a When is fixed to -1, it is expressed by the following formula (5):

5. The computational model according to claim 1, wherein: The constraint is a constraint on selecting a specific option among the plurality of options.

6. The computational model according to claim 1, wherein: The constraint is a constraint of not selecting a specific option among the plurality of options.

7. The computational model according to claim 1, wherein: The multiple options are respectively represented by one-hot representation.

8. The computational model according to claim 1, wherein: The plurality of options are respectively represented by binary representation.

9. The computational model according to claim 1, wherein: The plurality of options are each represented by a domain wall representation.

10. The computational model according to claim 1, wherein: The multiple options are respectively represented by unary expressions.

11. A computing program, wherein: The computational procedure can be applied to the Ising model or quadratic unconstrained binary optimization, denoted as QUBO, The calculation procedure performs the following processing: A first process of assigning a plurality of options in a combinatorial optimization problem to any of the possible values ​​of one or more binary variables; and A second process of fixing any one of the two-valued variables based on a constraint given to the combinatorial optimization problem.

12. The computing program according to claim 11, wherein: The second process is performed as follows: Finding a variable that can be fixed among the plurality of binary variables based on the constraint; and The variables that can be fixed are removed from the first calculation model represented by the following equation, In formula (1), x i 、x j is the 2-valued variable, J ij is the interaction parameter, h i is a parameter imposed on each variable according to external factors, and α is a constant.

13. The computing program according to claim 12, wherein: In the case of the QUBO, In the variable x that can be fixed a When it is fixed to 1, the above formula (1) is transformed into the following formula (2): In the variable x that can be fixed a When it is fixed to 0, the above formula (1) is transformed into the following formula (3):

14. The computing program according to claim 12, wherein: In the case of application to the Ising model, In the variable x that can be fixed a When it is fixed to +1, the above formula (1) is transformed into the following formula (4): In the variable x that can be fixed a When it is fixed to -1, the above formula (1) is transformed into the following formula (5):

15. The computing program according to claim 11, wherein: As the constraint, a constraint of selecting a specific option among the plurality of options is provided.

16. The computing program according to claim 11, wherein: As the constraint, a constraint of not selecting a specific option among the plurality of options is provided.

17. The computing program according to claim 11, wherein: The multiple options are respectively represented by one-hot representation.

18. The computing program according to claim 11, wherein: The plurality of options are respectively represented by binary representation.

19. The computing program according to claim 11, wherein: The plurality of options are each represented by a domain wall representation.

20. The computing program according to claim 11, wherein: The multiple options are respectively represented by unary expressions.

Images