Model generation device, model generation method, permutation generation system, and program

Abstract

To provide a model generation device capable of generating a model having fewer quadratic terms than conventional ones, for generating permutations or partial permutations.SOLUTION: A model generation device includes a problem conversion unit that performs, when a problem to be solved is input, a conversion into a model for finding a permutation in which m numbers are selected and arranged from n numbers without duplication, for two natural numbers m and n satisfying m≤n. The problem conversion unit includes in the model a constraint for finding the permutation, which is that a first column A obtained by selecting m numbers from n numbers and a second column B obtained by selecting n numbers from m numbers satisfy a predetermined condition. The predetermined condition is that the first column A is a permutation and the second column B is an inverse permutation of the first column A.SELECTED DRAWING: Figure 1

Description

[Technical field]

[0001] The present invention relates to a model generation device, a model generation method, a permutation generation system, and a program. [Background technology]

[0002] <Quantum annealer and quasi-quantum annealer> A quantum annealer is a dedicated computer that uses quantum effects to solve combinatorial optimization problems in a short time. However, quantum annealers have some issues, such as a small number of quantum bits and the possibility that correct calculation results may not be obtained due to noise. A pseudo-quantum annealer is a device or software that uses semiconductors such as regular CPUs, GPUs, FPGAs, and ASICs to solve the Ising model or QUBO model (described below) without using quantum effects.

[0003] <IsingモデルとQUBOモデル> Quantum annealers directly handle the Ising model, which is a quadratic expression with variables being quantum bits that take on values ​​of -1 or +1. On the other hand, pseudo quantum annealers often handle the QUBO (Quadratic Unconstrained Binary Optimization) model. The QUBO model is a modification of the Ising model in which each quantum bit takes on a value of 0 or 1 to make it easier to design algorithms, and the QUBO model and the Ising model can be converted to each other equivalently without changing the topology of the model (the shape of the interaction expressed in quadratic terms).

[0004] The Ising model is expressed by the following formula (1). In formula (1), a set V is a set of n nodes, a set E is a set of edges connecting the nodes, and a weight J i,j is the weight of the interaction on the edge ((i, j) ∈ E) connecting node i and node j, and the weight h i is the bias weight of node i, and variable s irepresents the spin of node i (i∈V), which is a quantum bit that can take the value -1 or +1. H(S) is the i is the Hamiltonian of the n-bit spin vector S with elements i = 0 to n-1. Note that the constant C is an offset and may not be written. The first sum term in equation (1) is called the quadratic term, and the second sum term is called the linear term.

[0005]

number

[0006] For example, the Ising model for the node configuration shown in FIG. 11 is expressed by the following formula (2). In FIG. 11, each circle represents a node, the number inside the circle represents the node number i, and the number attached to the circle represents the bias weight h i The number attached to the edge connecting the circles is the weight J of the interaction between the nodes connected by the line. i,j In equation (2), the first through eighth terms, which correspond to the first sum term in equation (1), are quadratic terms, and the ninth through thirteenth terms, which correspond to the second sum term in equation (1), are linear terms. When equation (2) is input, a quantum annealer or quasi-quantum annealer calculates vector S (optimal solution) that minimizes H(S). In the case of equation (2), vector S=[+1, -1, -1, +1, +1, -1], which gives H(S)=-32, is calculated as the optimal solution.

[0007]

number

[0008] The QUBO model is expressed by the following formula (3). In formula (3), the set V is a set of n nodes, and the set E is a set of edges connecting the nodes. The weight W i,j is the weight of the interaction on the edge ((i, j) ∈ E) connecting node i and node j, and the weight W i,i is the bias weight of node i. Note that the weight W i,ican be considered as the weight of the self-loop edge (i, i). i is a bit that can take the value 0 or 1. E(X), i.e., equation (3), is i This is the energy of n-bit vector X, whose elements are i from 0 to n-1. Note that the constant C is an offset and may not be written. In equation (3), the first sum term is also a quadratic term, and the second sum term is a linear term.

[0009]

number

[0010] For example, the QUBO model with the node configuration shown in FIG. 12 is expressed by the following formula (4). In FIG. 12, each circle represents a node, the number inside the circle represents the node number i, and the number attached to the circle represents the bias weight W i,i The number attached to the edge connecting the circles is the weight W of the interaction between the nodes connected by the line. i,j In equation (4), the first to eighth terms, which correspond to the first sum term in equation (3), are quadratic terms, and the ninth to twelfth terms, which correspond to the second sum term in equation (3), are linear terms. When equation (4) is input, the quasi-quantum annealer calculates vector X (optimal solution) that minimizes E(X). In the case of equation (4), vector X = [1, 0, 0, 1, 1, 0], which gives E(X) = -7, is calculated as the optimal solution.

[0011]

number

[0012] The Ising model and the QUBO model can be converted to each other without changing the topology. i The values ​​of -1 and +1 are x i When the values ​​of 0 and 1 are used, x i =(s iFor example, the Ising model shown in FIG. 11 and formula (2) and the QUBO model shown in FIG. 12 and formula (4) can be converted into equivalent solutions, and the optimal solution [+1, -1, -1, +1, +1, -1] where H(S) = -32 corresponds to the optimal solution [1, 0, 0, 1, 1, 0] where E(X) = -7.

[0013] <Graph of quantum annealer> A quantum annealer or a pseudo quantum annealer can find an optimal solution or an approximate solution to a problem that requires the assignment of values ​​to variables that minimizes the value of a quadratic expression in an Ising model or a QUBO model. Some quantum annealers or pseudo quantum annealers have interactions between only some nodes. When solving an Ising model or a QUBO model using a quantum annealer or a pseudo quantum annealer that has interactions between only some nodes, it is necessary to embed the problem to be solved in a graph that represents the shape of the interactions that the quantum annealer or the pseudo quantum annealer has. A graph that represents such a shape of interactions is called a topology, and such embedding is called a graph-minor embedding. FIG. 13 is a diagram showing an example of a graph-minor embedding of an Ising model. Here, an explanation is given by taking as an example a lattice-shaped graph in which each node corresponds to a quantum bit and is connected only to adjacent nodes vertically and horizontally. The Ising model M1 in FIG. 13 is the Ising model shown in FIG. 11. In order to embed the Ising model M1 into a lattice graph, we obtain the Ising model M1Graph-minor embedding and convert it to the Ising model M2. For example, D-Wave's quantum annealer, D-Wave Advantage, has a topology called a Pegasus graph with 5760 nodes.

[0014] In the embedded Ising model M2, node 0 of the Ising model M1 is transformed into a pair of node 0 and node 0'. The qubits s0 and s 0'are constrained to have the same value by adding a penalty term to their interaction weights, the negative value -P. 0' s0=s 0' When s0≠s, the value is negative -P. 0' When , the value P is positive, so s0 = s 0' Similarly, in the embedded Ising model M2, node 5 in Ising model M1 is converted to a pair of node 5 and node 5'. In this way, an Ising model with Graph-minor embedding has a larger number of nodes than the original Ising model. For example, a complete graph that can be Graph-minor embedded in a Pegasus graph with 5760 nodes has 177 nodes. Note that a complete graph is a graph that can correspond to any Ising model, and has weights J between all nodes. i,j is a graph on which

[0015] In addition, in the quantum annealer, the coefficient J i,j and h i has a range of possible values ​​and its resolution (number of effective bits) is also limited. i,j and h i is scaled to fit within these ranges. For example, D-Wave's quantum annealer, D-Wave Advantage, is i,j and h i The possible ranges of values ​​for are i,j ∈[-1.0,+1.0], h i ∈[-4.0,+4.0], and the resolution is about 5 bits. Therefore, J i,j and h i If there are any coefficients with large absolute values, those with small absolute values ​​may be ignored by scaling, so it is preferable that the maximum absolute value of these coefficients is small.

[0016] <Combinatorial optimization problems> When solving various combinatorial optimization problems using a quantum annealer or a quantum-inspired annealer, an instance (input) of the combinatorial optimization problem to be solved is converted into an Ising model or a QUBO model. For example, when using a quantum annealer to solve an Ising model, a graph-minor embedding is obtained to embed the topology of the Ising model M1 to be solved into the topology of the quantum annealer to be used, written as an embedded Ising model M2, and annealing is repeated 100 to 10,000 times to obtain the result. Then, the best solution among the obtained approximate solutions is adopted as the solution.

[0017] In the case of a combinatorial optimization problem of the type that finds an optimal permutation (a permutation with the minimum value of the objective function) from all permutations (permutation-type combinatorial optimization problem), in order to represent an instance by an Ising model or a QUBO model, one-hot representation (one-hot encoding) of an n×n binary matrix (n is the number of elements of the permutation) is widely used. Using this method, the number of quadratic terms in both the Ising model and the QUBO model is n 3 -n 2 pieces.

[0018] <one-hot representation> As shown in FIG. 14, the one-hot representation is a representation in which only one place in the variable column becomes 1, and an integer is represented by the position of 1. In the case of the one-hot representation of an Ising model by k qubits s i k - 1 qubits s i become -1, and 1 qubit s i becomes +1, so it is represented by Equation (5). Note that in Equation (5), when expanding the equation, s i 2 = 1 is substituted. This is because s i is -1 or +1 and always becomes 1 when squared. The first summation term on the right side of Equation (5) is a quadratic term, and the second summation term is a linear term. Therefore, in the one-hot representation of an Ising model by k qubits s i the number of quadratic terms is (k 2It becomes (-k) / 2, and the coefficient of the linear term becomes k - 2. In the case of the permutation-type combinatorial optimization problem, each row and each column of the n×n matrix are in one-hot representation. That is, since there are a total of 2n of n rows and n columns for the one-hot representation of n qubits, the number of quadratic terms is n 3 -n 2 and it becomes. The coefficient of the linear term becomes 2(2n - 4).

[0019]

Number

[0020] For the one-hot representation of the QUBO model by k qubits x i among the k qubits, k - 1 qubits x i become 0, and 1 qubit x i becomes 1, so it is represented by Equation (6). Note that in the expansion of Equation (6), x i 2 = x i is substituted. This is because x i is 0 or 1 and its value does not change even when squared. Also in Equation (6), the first summation term on the right side is the quadratic term and the second summation term is the linear term. Therefore, in the one-hot representation of the QUBO model, the number of quadratic terms is (k 2 -k) / 2, and the coefficient of the first order is 1. In the case of the permutation-type combinatorial optimization problem, the reason why the number of quadratic terms is n 3 -n 2 is the same as in the case of the Ising model.

[0021]

Number

[0022] <domain-wall representation> In Non-Patent Document 1, a method of representing the permutation-type combinatorial optimization problem by a QUBO model using the domain-wall representation is disclosed. For k qubits x iThe domain-wall representation of the QUBO model according to [description] is a representation in which 1s continue from the beginning and the rest are all 0s, and an integer is represented by the number of 1s. Figure 14 is a table showing the correspondence between the one-hot representation and the domain-wall representation. When dealing with the domain-wall representation in the QUBO model, guard bits of 1 at the beginning and 0 at the end are provided, and a constraint is given that there is only one place where the value changes between adjacent qubits. The descriptions outside the parentheses in the domain-wall representation of Figure 14 are the guard bits. Note that in the domain-wall representation of the Ising model, the values of the qubits are +1s continuing from the beginning and the rest are all -1s, and guard bits of +1 at the beginning and -1 at the end are provided.

[0023] In Non-Patent Document 1, when the domain-wall representations of each row show different values, the following formula (7) is used as a constraint by utilizing the fact that the sum of column j is n - 1 - j. The first summation term in formula (7) is a constraint for each row of the n×n matrix to be in the domain-wall representation, and the second summation term is a constraint for the sum of column j to be n - 1 - j. By substituting the values of the guard bits, x i,j 2 =x i,j and other substitutions, when formula (7) is expanded, the number of quadratic terms is (n 3 - 3n) / 2, and the maximum absolute value of the coefficient of the linear term is 2n - 3.

[0024]

Number

[0025] Also, in Non-Patent Document 2, a method of representing a partial permutation type combinatorial optimization problem of selecting different m (m < n) from different n as columns using the domain-wall representation in a QUBO model is disclosed.

Prior Art Documents

Non-Patent Documents

[0026]

Non-Patent Document 1

[0027] In the widely used one-hot representation model, the model in Non-Patent Document 1, and the model in Non-Patent Document 2, the number of quadratic terms is n 3 -n 2 pieces, (n 3 -3n) / 2 pieces, (1 / 2)×(mn 2 +mn 2 )-mn, and as the values ​​of m and n become large, the number becomes enormous. As mentioned above, when using the Ising model or QUBO model with a quantum annealer, quadratic terms are associated with the interactions between the quantum annealer's quantum bits. However, in a quantum annealer, there is interaction only between some of the quantum bits. For example, if a quantum annealer with a 5760-node Pegasus graph topology is used to solve the Traveling Salesman Problem, a permutation-type combinatorial optimization problem, using a one-hot representation, the large number of quadratic terms requires combining multiple nodes to correspond to the quantum bits of the original model, so only a maximum of 13 cities can be handled.

[0028] As mentioned above, when using a quantum annealer, the coefficient J i,j and h i If the maximum absolute value of is large, some terms are ignored due to scaling, which can lead to poor accuracy of the solution. i Since θ takes a large value in the above-mentioned one-hot representation model, the model in Non-Patent Document 1, and the model in Non-Patent Document 2, θ is easily affected by scaling, making it difficult to find an optimal solution.

[0029] The present invention has been made in view of the above circumstances, and provides a model generation device, a model generation method, a permutation generation system, and a program for generating a model with fewer quadratic terms than conventional models for generating permutations or partial permutations. Specifically, in conventional models, the formula expressing the number of required quadratic terms is a cubic formula of n or a cubic formula of m and n, but by using the present invention, the formula expressing the number of quadratic terms can be significantly reduced to a quadratic formula of n or a quadratic formula of m and n. [Means for solving the problem]

[0030] The present invention has been made to solve the above-mentioned problems, and one aspect of the present invention is a model generation device that includes a problem conversion unit that, when a problem to be solved is input, converts the problem into a model for finding one permutation in which m numbers are selected from n numbers without duplicates for two natural numbers m, n satisfying m≦n, and the problem conversion unit includes in the model a constraint condition for finding the one permutation that a first string A in which m numbers are selected from the n numbers and a second string B in which n numbers are selected from the m numbers satisfy a predetermined condition, and the predetermined condition is that the first string A is a permutation and the second string B is the inverse permutation of the first string A.

[0031] Another aspect of the present invention is the model generation device described above, wherein the integers constituting each of the first column A and the second column B are represented using binary vectors, and the number of quadratic terms in the expression representing the constraint conditions for representing using the binary vectors is represented by a linear expression of the number of elements of the binary vectors.

[0032] Another aspect of the present invention is the model generation device described above, wherein the integers constituting each of the first column A and the second column B are represented using domain-wall representation.

[0033] Another aspect of the present invention is the model generation device described above, wherein the model is an objective function that satisfies the constraint conditions when the value is minimized, and takes as arguments a binary vector representing the integers constituting each of the first column A and the second column B.

[0034] Another aspect of the present invention is the model generation device described above, where m < n, and the predetermined condition is that the permutation represented by the first column A coincides with the permutation represented by the auxiliary variable matrix, and at least a part defined by the permutation represented by the first column A among the reverse permutations represented by the second column B coincides with the reverse permutation represented by the auxiliary variable matrix.

[0035] Another aspect of the present invention is the model generation device described above, where m = n, and the predetermined condition is that the permutation represented by the first column A coincides with the permutation represented by the auxiliary variable matrix, and the reverse permutation represented by the second column B coincides with the reverse permutation represented by the auxiliary variable matrix.

[0036] Another aspect of the present invention is a model generation method including a step of converting a problem to be solved into a model for finding one permutation in which m numbers are selected from n numbers without duplicates for two natural numbers m, n satisfying m≦n, in which in the step, a constraint that a first string A in which m numbers are selected from the n numbers and a second string B in which n numbers are selected from the m numbers are included in the model as a constraint for finding the one permutation, in which the specified condition is that the first string A is a permutation and the second string B is an inverse permutation of the first string A.

[0037] Another aspect of the present invention includes a model generation device and a model solving device, wherein the model generation device includes a problem conversion unit that, when a problem to be solved is input, converts the problem into a model for finding one permutation in which m numbers are selected from n numbers without duplicates for two natural numbers m and n satisfying m≦n, and the problem conversion unit includes in the model a constraint condition for finding the one permutation that a first string A in which m numbers are selected from the n numbers and a second string B in which n numbers are selected from the m numbers satisfy a predetermined condition, and the predetermined condition is that the first string A is a permutation and the second string B is an inverse permutation of the first string A, and the model solving device is a permutation generation system that calculates an optimal solution that minimizes the value indicated by the model, or an approximation of the optimal solution.

[0038] Another aspect of the present invention is a program for causing a computer to function as a problem conversion unit that, when a problem to be solved is input, converts the computer into a model for finding one permutation in which m numbers are selected from n numbers without duplicates for two natural numbers m, n satisfying m≦n, wherein the problem conversion unit includes in the model a constraint condition for finding the one permutation that a first column A in which m numbers are selected from the n numbers and a second column B in which n numbers are selected from the m numbers satisfy a predetermined condition, and the predetermined condition is that the first column A is a permutation and the second column B is an inverse permutation of the first column A. Effect of the Invention

[0039] According to the present invention, the model generation method, model generation device, permutation generation system, and program can generate a model for generating permutations or partial permutations with fewer quadratic terms than in the past. [Brief description of the drawings]

[0040] [Figure 1] FIG. 1 is a schematic diagram showing a configuration of a permutation generation system 10 according to an embodiment of the present invention. [Diagram 2] 2 is a schematic block diagram showing the configuration of a model generating device 11 in the embodiment. FIG. [Diagram 3] 4 is a flowchart illustrating an example of the operation of the model generating device 11 in the embodiment. [Figure 4] FIG. 1 is a schematic diagram showing an example of a traveling salesman problem. [Diagram 5] FIG. 1 is a schematic diagram illustrating an example of a quantum bit in a conventional model. [Figure 6] FIG. 1 is a diagram for explaining the correspondence between a conventional matrix X and a permutation π. [Figure 7] FIG. 2 is a schematic diagram illustrating a first model of the above-mentioned embodiment. [Figure 8] FIG. 11 is a schematic diagram illustrating a third model of the embodiment. [Figure 9] FIG. 13 is a schematic diagram illustrating a fifth model according to the embodiment. [Figure 10] 11 is a table showing an example of an expression of integers using binary vectors in the embodiment. [Figure 11] FIG. 1 is a schematic diagram illustrating an example of a node configuration of a conventional Ising model. [Figure 12] FIG. 1 is a schematic diagram showing an example of a node configuration in a conventional QUBO model. [Figure 13] FIG. 1 is a diagram for explaining embedding of a conventional Ising model into a graph. [Figure 14] 1 is a table showing a conventional correspondence between one-hot representation and domain-wall representation. DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS

[0041] Hereinafter, an embodiment of the present invention will be described with reference to the drawings. Fig. 1 is a schematic diagram showing a configuration of a permutation generation system 10 according to an embodiment of the present invention. The permutation generation system 10 includes a model generation device 11 and a quantum annealer 12. The permutation generation system 10 calculates an optimal solution or an approximate solution to a permutation type combinatorial optimization problem or a partial permutation type combinatorial optimization problem, for example, a traveling salesman problem.

[0042] When a permutation type combinatorial optimization problem or a partial permutation type combinatorial optimization problem is input, the model generation device 11 converts the optimization problem into an Ising model or a QUBO model, and has the quantum annealer 12 calculate a solution to the model. The model generation device 11 also converts the solution calculated by the quantum annealer 12 into a solution to the input optimization problem and outputs it. For example, if the input optimization problem is a traveling salesman problem, the model generation device 11 outputs the order of the cities to be visited when a list of cities to be visited is input. The model generation device 11 may be realized by one or more computers reading and executing a program.

[0043] The quantum annealer 12 (model solving device) is a quantum annealing quantum computer (quantum annealer) or a pseudo quantum annealer, and calculates an optimal solution that minimizes the value indicated by the set model, or an approximate solution to the optimal solution. Note that the pseudo quantum annealer may be any one that solves the Ising model or the QUBO model, and includes not only those that simulate quantum annealing, but also others.

[0044] 2 is a schematic block diagram showing the configuration of the model generation device 11 in this embodiment. The model generation device 11 includes a problem conversion unit 111, a model setting unit 112, a solution conversion unit 113, and a solution acquisition unit 114. When a permutation type combinatorial optimization problem or a partial permutation type combinatorial optimization problem (a problem to be solved) is input, the problem conversion unit 111 converts the problem to be solved into an Ising model or a QUBO model. The problem to be solved may be input by an operator of the model generation device 11 using an input device such as a keyboard, a mouse, or a touch panel, or may be received from another device.

[0045] Through this conversion, the problem conversion unit 111 generates a model for determining one permutation in which m numbers are selected from n numbers without duplication for two natural numbers m and n satisfying m≦n. Details of the model will be described later, but the problem conversion unit 111 includes in the model, as a constraint for determining the one permutation, that a first string A in which m numbers are selected from n numbers and a second string B in which n numbers are selected from m numbers satisfy a predetermined condition. The predetermined condition is that the first string A is a permutation and the second string B is an inverse permutation of the first string A. This means that if the first string A is [A(0),A(1),...A(m-1)] in which m integers are selected from n integers between 0 and n-1, and the second string B is [B(0),B(1),...B(n-1)] in which n integers are selected from m integers between 0 and m-1, then there are no duplicates in A(0),A(1),...A(m-1), and for all i (0≦i≦m-1), B(A(i))=i is satisfied. This predetermined condition is also called B(i) being the inverse function of A(i), or A(i) and B(i) being dual.

[0046] The model setting unit 112 sets the model generated by the problem conversion unit 111 in the quantum annealer 12, and causes the quantum annealer 12 to calculate an optimal solution or an approximate solution of the model. When setting the model in the quantum annealer 12, the model setting unit 112 may collectively associate multiple nodes of the quantum annealer 12 with quantum bits of the model in order to generate interactions between the nodes of the quantum annealer 12 that correspond to the quadratic terms of the model.

[0047] The solution acquisition unit 114 acquires the solution (optimum solution or approximate solution) calculated by the quantum annealer 12. The solution conversion unit 113 converts the solution acquired by the solution acquisition unit 114 into a solution to the problem input to the problem conversion unit 111 and outputs it. The output by the solution conversion unit 113 may be displayed on a display or the like, or may be the transmission of information indicating the solution to another device.

[0048] FIG. 3 is a flowchart for explaining an example of the operation of the model generating device 11 in this embodiment. First, the problem conversion unit 111 accepts an input of a problem to be solved and converts the problem to be solved into a QUBO model (step Sa1). Next, the model setting unit 112 sets the QUBO model converted in step Sa1 in the quantum annealer 12 and causes the quantum annealer 12 to calculate a solution of the model (step Sa2). The solution acquisition unit 114 acquires the calculated solution from the quantum annealer 12 (step Sa3). The solution conversion unit 113 converts the solution acquired in step Sa3 into the solution of the problem to be solved input in step Sa1 and outputs it (step Sa4). Note that, although FIG. 3 shows an example in which the QUBO model is used, the Ising model may be used instead of the QUBO model.

[0049] Before explaining the details of the model generated by the problem conversion unit 111, a conventional model using one-hot representation will be explained. FIG. 4 is a schematic diagram showing an example of the traveling salesman problem. The traveling salesman problem in FIG. 4 is to find the order in which to visit four cities numbered 0 to 3 using the shortest route. It is assumed that the solution is the route [1, 3, 2, 0], that is, the order of city number 1, city number 3, city number 2, and city number 0. Note that d 1,3 is the distance from city number 1 to city number 3, and d 3,2 is the distance from city number 3 to city number 2, and d 2,0 is the distance from city number 2 to city number 0, and d 0,1 is the distance from city number 0 to city number 1. In addition, the distance from city number 1 to city number 2 and the distance from city number 0 to city number 3 are also given as traveling salesman problems.

[0050] FIG. 5 is a schematic diagram for explaining an example of a quantum bit in a conventional model. In the case of the traveling salesman problem in FIG. 3, in the conventional model using the one-hot representation, x 0,0 , x 0,1 , …, x 3,3 It uses 16 quantum bits, where x i,j is a quantum bit in the QUBO model that can take on either the value 0 or 1. When the value is 1, it indicates that the i-th city j will be visited. The quantum bit that indicates the tour [1, 3, 2, 0] is x, as shown in Figure 5. 0,1 , x 1,3 , x 2,2 , x 3,0 is 1, otherwise it is 0.

[0051] To obtain such a solution, the one-hot representation includes in the model a constraint that only one quantum bit is 1 in each row of the matrix in Figure 5, and only one quantum bit is 1 in each column. The following equation (8) is an example of the objective function E(X) of a conventional model.

[0052]

number

[0053] The first sum term in equation (8) is a constraint term that restricts only one quantum bit to be 1 in each row of matrix X. This first sum term takes its minimum value, 0, when only one quantum bit is 1 and the rest are 0 in each row of matrix X. The second sum term is a constraint term that restricts only one quantum bit to be 1 in each column of matrix X. This second sum term takes its minimum value, 0, when only one quantum bit is 1 and the rest are 0 in each column of matrix X. The third sum term is a term that represents the total distance due to cycling. d in the third sum term j,kis the distance from city numbered j to city numbered k. P multiplied by the first and second sum terms is a weight to prioritize the constraints over the calculation of the travel route distance. The matrix X that minimizes this objective function E(X) is the optimal solution. The quadratic expression of the elements of matrix X expanded into this objective function is the QUBO model. The expressions for the first and second sum terms are expressions for obtaining a one-hot representation, and the number of quadratic terms contained in the quadratic expression of the elements of column X obtained by expanding this part is n. 3 -n 2 It becomes.

[0054] FIG. 6 is a diagram for explaining the correspondence between the conventional matrix X and the permutation π. Here, we focus on the quantum bits that are 1 for each row of the matrix X that is the conventional solution. The quantum bit that is 1 in the 0th row is x, which corresponds to the city number 1 that is visited in the 0th place. 0,1 The quantum bit with a value of 1 in the first row is x, which corresponds to the first city to be visited, number 3. 1,3 The quantum bit with a value of 1 in the second row is x, which corresponds to the second city to be visited. 2,2 The quantum bit with a value of 1 in the third row is x, which corresponds to the third city to be visited, number 0. 3,0 That is, these rows are injective from row numbers (the order of visits) to city numbers, and represent the permutation π:[π(0), π(1), π(2), π(3)]=[1, 3, 2, 0].

[0055] On the other hand, if we look at the quantum bits that are 1 in each column of matrix X, the quantum bit that is 1 in the 0th column corresponds to the third quantum bit, which is the order of visiting city number 0. 3,0 The quantum bit with a value of 1 in the first column corresponds to the 0th place in the order to visit the city with the number 1, x 0,1 The quantum bit with a value of 1 in the second column corresponds to the second order of visiting city number 2, x 2,2 The quantum bit with a value of 1 in the third column corresponds to the first place in the order to visit city number 3, x 1,3That is, these sequences are injective from the sequence number (city number) to the order of visiting the cities, and the permutation π -1 :[π -1 (0), π -1 (1), π -1 (2), π -1 (3)]=[3, 0, 2, 1].

[0056] In this way, in the conventional model using one-hot representation for rows and columns, permutation π and inverse permutation π -1 In the one-hot representation, where only one place is 1 and the place is an integer, the constraint is expressed as Σx i,j Since the equation includes a term that squares x, the QUBO model obtained by expanding the equation has a large number of quadratic terms, as explained using equation (6). On the other hand, the constraint equation includes x, which is included in the term that squares x. i,j There is a domain-wall representation as an expression with a small number of . In the case of the QUBO model, the domain-wall representation starts with 1 and the rest are all 0, and the number of 1s represents an integer. The domain-wall representation that can represent k values ​​(for example, integers from 0 to k-1) is expressed by formula (9). In formula (9), x -1 , x k are guard bits, and x -1 =1, x k =0. In the case of the Ising model, the domain-wall representation starts with +1 and the rest are all -1, and the number of +1s represents an integer. The domain-wall representation that can represent k values ​​(for example, integers from 0 to k-1) is expressed by equation (10). Note that in equation (9), s -1 , s k is a guard bit, and s -1 =1, s k = 0. In both cases, the number of quadratic terms is k-1, and the maximum absolute value of the coefficients of the linear terms is 1. However, it is not possible to use domain-wall representation for both rows and columns of a matrix.

[0057]

number

[0058]

number

[0059] Therefore, in the model of this embodiment, a matrix A of quantum bits whose rows are domain-wall representations and a matrix B of quantum bits whose columns are domain-wall representations are separately prepared, and the constraints include that column A indicated by the rows of matrix A is a permutation, and column B indicated by the columns of matrix B is the inverse permutation of column A.

[0060] 7 is a schematic diagram illustrating a first model of the present embodiment. The first model is a QUBO model for a permutation-type combinatorial optimization problem, and as shown in FIG. 8, the quantum bit a 0,0 , …a 3,2 and the quantum bit b that composes matrix B 0,0 , … b 2,3 That is, when the number of elements of the permutation is n, the quantum bit a that composes the matrix A is 0,0 , …a n-1,n-2 and the quantum bit b that composes matrix B 0,0 , … b n-2,n-1 The matrix A represents column A by making each row a domain-wall representation.

[0061] If each row of matrix A represents the permutation [1, 3, 2, 0], then a 0,0 is 1, and a 0,1 and a 0,2 is 0, and a 1,0 From a 1,2 is 1, and a 2,0 and a 2,1 is 1, and a 2,2 is 0, and a 2,0 From a 2,2 is 0.

[0062] If each column of matrix B represents the permutation [3, 0, 2, 1], then b 0,0 From b2,0 is 1, and b 0,1 From b 2,1 is 0, and b 0,2 and b 1,2 is 1, and b 2,2 is 0, and b 0,3 is 1, and b 1,3 and b 2,3 is 0.

[0063] Here, a of matrix A 0,-1 From a 3,-1 In a, a quantum bit called a guard bit is set, whose value is fixed to 1. 0,3 From a 3,3 Let be a guard bit with value 0. Similarly, b of matrix B -1,0 From b -1,3 Let be a guard bit with value 1, and b 3,0 From b 3,3 is a guard bit with a value of 0. Then, r i,j =a i,j-1 -a i,j The matrix R obtained from matrix A by i,j =b i-1,j -b i,j Matrix C obtained from matrix B by the above corresponds to the one-hot representation of each column of matrix B. Therefore, if matrix R and matrix C are the same, column A is a permutation and column B is the inverse permutation of column A.

[0064] The following formula (11) is a formula showing the constraint conditions included in the first model in this embodiment. The first sum term in formula (11) is a term of the constraint condition that each row of matrix A is a domain-wall representation, that is, the row of matrix A is a column in which n numbers are selected and arranged. The second sum term is a term of the constraint condition that each column of matrix B is a domain-wall representation, that is, the column of matrix B is a column in which n numbers are selected and arranged. The third sum term is a term of the constraint condition that matrix R and matrix C in FIG. 7 match, that is, the column A indicated by the row of matrix A is a permutation, and the column B indicated by the column of matrix B is the inverse permutation of column A. Therefore, the problem conversion unit 111 includes the constraint condition shown in formula (11) in the objective function, and performs the substitution of the values ​​of the guard bits, a i,j 2 =a i,j , b i,j 2 =b i,j By expanding the model, such as by substituting the above, it can be made into a QUBO model for permutation-type combinatorial optimization problems.

[0065]

number

[0066] In the polynomial obtained by expanding equation (11), the number of quadratic terms is 6n 2 -12n+4. Therefore, the number of quadratic terms is n 3 -n 2 By using the one-hot representation, the number of quadratic terms can be reduced for n≧6.

[0067] In the first model, the model for the permutation-type combinatorial optimization problem was described as a QUBO model in which qubits take values of 0 or 1, but it may also be an Ising model in which qubits take values of -1 or +1. Equation (12) is an equation showing the constraint conditions included in the second model in the present embodiment. The second model is an Ising model for the permutation-type combinatorial optimization problem, but the equation of the constraint conditions is the same as Equation (11) which was the constraint condition of the QUBO model. However, at the time of expansion, the value of the guard bit to be substituted is -1 or +1, and a i,j 2 = 1, b i,j 2 = 1 are substituted, which is different from the first model. In this way, the problem conversion unit 111 can be an Ising model for the permutation-type combinatorial optimization problem by expanding the constraint conditions shown in Equation (12) and including them in the objective function.

[0068]

Number

[0069] In the polynomial obtained by expanding the constraint condition equation (12), the number of quadratic terms is 6n 2 - 12n + 4. Therefore, the number of quadratic terms can be reduced when n ≥ 6 compared to the one-hot representation where the number of quadratic terms is n 3 - n 2 . Furthermore, the maximum absolute value of the coefficient of the linear term is 2. Therefore, the maximum absolute value of the coefficient of the linear term can be reduced when n > 3 compared to the one-hot representation where the coefficient of the linear term is 2n - 4.

[0070] FIG. 8 is a schematic diagram for explaining the third model of the present embodiment. The third model is a QUBO model for the partial permutation-type combinatorial optimization problem. A partial permutation is a permutation formed by arranging m numbers without repetition from n numbers for two natural numbers m and n that satisfy m < n. The example in FIG. 8 is an example of obtaining a partial permutation of arranging 3 numbers selected from 5 numbers, and the qubits a 0,0 , … a 2,3and the quantum bit b that composes matrix B 0,0 , … b 1,4 That is, the quantum bit a that constitutes the matrix A is used. 0,0 , …a m-1,n-2 and the quantum bit b that composes matrix B 0,0 , … b m-2,n-1 is used.

[0071] In the third model, each row of matrix A is a domain-wall representation, and each column of matrix B is a domain-wall representation. However, unlike the first model, the columns of matrix B whose inverse permutations are undefined can take the value of either 0 or 1. i,j =a i,j-1 -a i,j A matrix R with elements c i,j =b i-1,j -b i,j The first model is also similar to the first model in that if the matrix C whose elements are:

[0072] If the rows of matrix A represent the permutation [3, 0, 1], then the first row represents 3, so a 0,0 From a 0,2 is 1, and the remaining a 0,3 is 0. The second line represents 0, so a 1,0 From a 1,3 is 0. The third line represents 1, so a 2,0 is 1, and the remaining a 2,1 From a 2,3 is 0. If each column of matrix B represents the column [1, 2, undefined, 0, undefined], the first column represents 1, so b 0,0 is 1, and b 1,0 is 0. The second column represents 2, so b 0,1 and b 1,1 is 1. The third column is undefined, so b 0,2 and b 1,2 is either 0 or 1. The fourth column represents 0, so b 0,3 and b 1,3 is 0. The fifth column is undefined, so b 0,4 and b 1,4 is either 0 or 1.

[0073] The following formula (13) is a formula showing the constraint conditions included in the third model in this embodiment. The first sum term in formula (13) is a term of the constraint condition that each row of matrix A is a domain-wall representation, that is, the row of matrix A is a column of m numbers selected from n numbers, where n, m are natural numbers such that n>m. The second sum term is a term of the constraint condition that each column of matrix B is a domain-wall representation, that is, the column of matrix B is a column of n numbers selected from m numbers, where n, m are natural numbers such that n>m. The third sum term is a term of the constraint condition that matrix R and matrix C in FIG. 8 are the same, that is, the column A indicated by the row of matrix A is a permutation, and the column B indicated by the column of matrix B is the inverse permutation of column A. Therefore, the problem conversion unit 111 can expand the objective function by including the constraint condition shown in equation (13) to create a QUBO model for a partial permutation type combinatorial optimization problem in which m numbers are selected and arranged without duplicates from n numbers.

[0074]

number

[0075] In the polynomial obtained by expanding the constraint equation (13), the number of quadratic terms is 6mn-6m-6n+4. Therefore, the number of quadratic terms is (1 / 2)×(m 2 n+mn 2 )-mn, the number of quadratic terms can be reduced when m+n≧14.

[0076] In the third model, the model for partial permutation type combinatorial optimization problems has been described as a QUBO model in which the quantum bit takes a value of 0 or 1, but it may be an Ising model in which the quantum bit takes a value of -1 or +1. Equation (14) is an equation showing the constraint conditions included in the fourth model in this embodiment. The fourth model is an Ising model for partial permutation type combinatorial optimization problems, but the equation for the constraint conditions is the same as equation (13), which was the constraint condition of the QUBO model. However, when expanding, the value of the guard bit to be substituted is -1 or +1, and a i,j 2 =1, b i,j 2 = 1. In this way, the problem conversion unit 111 can convert the constraint condition shown in the formula (14) into an Ising model of the partial permutation type combinatorial optimization problem by expanding the objective function.

[0077]

number

[0078] In the polynomial obtained by expanding the constraint equation (14), the number of quadratic terms is 6mn-6m-6n+4. Therefore, the number of quadratic terms is (1 / 2)×(m 2 n+mn 2 The number of quadratic terms can be reduced for n ≥ 6 compared to the m+n-3 one-hot representation. Furthermore, the maximum absolute value of the coefficient of the linear term is 2, and the maximum absolute value of the coefficient of the linear term can be reduced for m+n > 5 compared to the m+n-3 one-hot representation.

[0079] FIG. 9 is a schematic diagram for explaining the fifth model of the present embodiment. Like the third model, the fifth model is also a QUBO model for partial permutation type combinatorial optimization problems. However, the quantum bit x 0,0 x m-1,n-1The fifth model differs from the third model in that it uses an auxiliary variable matrix X with elements of . In the fifth model, the condition that column B is the inverse permutation of column A is that matrix X and matrix R are the same and matrix X and matrix C correspond to each other.

[0080] Equation (15) is an equation showing the constraint conditions included in the fifth model in this embodiment. The first sum term in equation (15) is a term of the constraint condition that each row of matrix A is a domain-wall representation. The second sum term is a term of the constraint condition that each column of matrix B is a domain-wall representation. The third sum term is a term of the condition that matrix X and matrix R in FIG. 9 match. In other words, the permutation represented by matrix A matches the permutation represented by auxiliary variable matrix X. The fourth sum term is a term of the condition that matrix X and matrix C in FIG. 9 correspond to each other. Here, "correspondence" means that x i,j When is 1, b i-1,j -b i,j Also, the x that is 1 in column j is i,j When there is no, the fourth sum term is b in column j. i,j The value of is not restricted. That is, at least a part of the inverse permutation represented by matrix B that is defined by the permutation represented by matrix A matches the inverse permutation represented by auxiliary variable matrix X. The third and fourth sum terms are constraints that column A indicated by the rows of matrix A is a permutation, and column B indicated by the columns of matrix B is the inverse permutation of column A. Therefore, the problem conversion unit 111 can develop the constraints shown in formula (15) in the objective function to create a QUBO model of a partial permutation type combinatorial optimization problem of a partial permutation in which m numbers are selected and arranged without duplication from n numbers.

[0081]

number

[0082] In the polynomial obtained by expanding the constraint equation (15), the number of quadratic terms is 6mn-4m-4n, and the number of quadratic terms is (1 / 2) × (m 2 n+mn 2)-mn, the number of quadratic terms can be reduced when m+n≧14. Also, the objective function for the travel distance in the traveling salesman problem can be expressed as x i,j By using i,j In some cases, it may be possible to reduce the number of quadratic terms in the objective function compared to writing it using

[0083] Equation (16) is an equation showing the constraint condition included in the sixth model in this embodiment. The sixth model is similar to the fifth model in that it uses a quantum bit s 0,0 From s m-1,n-1 This is an Ising model for partial permutation combinatorial optimization problems using an auxiliary variable matrix S with elements s. The first summation term in equation (15) is a constraint term that makes each row of matrix A a domain-wall representation. The second summation term is a constraint term that makes each column of matrix B a domain-wall representation. The third summation term is a condition term that matrices S and R match. Here, matching means that s i,j When is +1, a i,j-1 -a i,j is 2, and s i,j When is -1, a i,j-1 -a i,j is 0. The fourth summation term is the condition that matrix S and matrix C correspond to each other. Here, correspondence means that s i,j When is 1, b i-1,j -b i,j Also, s that is 1 in column j is i,j When there is no, the fourth sum term is b in column j. i,j The value of is not restricted. The third and fourth sum terms are constraints that column A indicated by the rows of matrix A is a permutation, and column B indicated by the columns of matrix B is the inverse permutation of column A. Therefore, by including the constraints shown in formula (16) in the objective function and expanding it, problem conversion unit 111 can create an Ising model of a partial permutation type combinatorial optimization problem of a partial permutation in which m numbers are selected and arranged from n numbers without duplication.

[0084]

number

[0085] In the polynomial obtained by expanding the constraint equation (16), the number of quadratic terms is 6mn-4m-4n, and the number of quadratic terms is (1 / 2) × (m 2 n+mn 2 )-mn, the number of quadratic terms can be reduced when m+n≧14. Furthermore, the maximum absolute value of the coefficient of the linear term is 3. Therefore, the maximum absolute value of the coefficient of the linear term can be reduced when m+n>6 compared to the one-hot representation in which the coefficient of the linear term is m+n-3. Also, when the objective function, such as the distance traveled in the traveling salesman problem, is expressed as x i,j By using the fourth model, i,j In some cases, it may be possible to reduce the number of quadratic terms in the objective function compared to writing it using

[0086] Equation (17) is an equation showing the constraint conditions included in the seventh model in this embodiment. The seventh model is a QUBO model for permutation-type combinatorial optimization problems, but uses the auxiliary variable matrix X as in the fifth model. The first sum term in equation (17) is a term of the constraint condition in which each row of matrix A is a domain-wall representation. The second sum term is a term of the constraint condition in which each column of matrix B is a domain-wall representation. The third sum term is a term of the condition that matrix X and matrix R match. That is, the permutation represented by matrix A matches the permutation represented by auxiliary variable matrix X. The fourth sum term is a term of the condition that matrix X and matrix C match. That is, the inverse permutation represented by matrix B matches the inverse permutation represented by auxiliary variable matrix X. The third sum term and the fourth sum term are constraint conditions that column A indicated by the rows of matrix A is a permutation, and column B indicated by the columns of matrix B is the inverse permutation of column A. Therefore, the problem conversion unit 111 can generate a QUBO model for the permutation type combinatorial optimization problem by expanding the objective function while including the constraint condition shown in formula (17).

[0087]

number

[0088] In the polynomial obtained by expanding the constraint equation (17), the number of quadratic terms is 6n 2 -8n. Therefore, the number of quadratic terms is n 3 -n 2 The number of quadratic terms can be reduced for n ≥ 6 compared to the one-hot representation. Furthermore, the maximum absolute value of the coefficient of the linear term is 3, which is a small value. Also, if the objective function, such as the distance traveled in the traveling salesman problem, is expressed as x i,j By describing it using a, like the first model, i,j In some cases, it may be possible to reduce the number of quadratic terms in the objective function compared to writing it using

[0089] Equation (18) is an equation showing the constraint condition included in the eighth model in this embodiment. The eighth model is an Ising model for a permutation type combinatorial optimization problem, but uses an auxiliary variable matrix S as in the sixth model. The first sum term in equation (18) is a term of the constraint condition that each row of matrix A is a domain-wall representation. The second sum term is a term of the constraint condition that each column of matrix B is a domain-wall representation. The third sum term is a term of the condition that matrix S and matrix R match. The fourth sum term is a term of the condition that matrix S and matrix C match. The third sum term and the fourth sum term are constraint conditions that column A indicated by the row of matrix A is a permutation, and column B indicated by the column of matrix B is the inverse permutation of column A. Therefore, the problem conversion unit 111 can make an Ising model of a permutation type combinatorial optimization problem by expanding the objective function including the constraint condition shown in equation (18).

[0090]

number

[0091] In the polynomial obtained by expanding the constraint equation (18), the number of quadratic terms is 6n 2 -8n. Therefore, the number of quadratic terms is n 3 -n 2The number of quadratic terms can be reduced for n ≥ 6 by using the one-hot representation. Furthermore, the maximum absolute value of the coefficient of the linear term is 2, which is a small value. Also, if the objective function, such as the distance traveled in the traveling salesman problem, is expressed as x i,j By using i,j In some cases, it may be possible to reduce the number of quadratic terms in the objective function compared to writing it using

[0092] Equation (19) is an example of the second model of the traveling salesman problem in this embodiment. In the example of equation (19), the terms in parentheses multiplied by the weight P are constraints included in the second model, and are consistent with equation (12). The remaining terms are terms that indicate the total distance traveled when traveling. Note that d j,k is the distance from the jth city to the kth city. P is a weight greater than 1 for prioritizing the constraint condition. That is, the problem conversion unit 111 converts the distance between cities d j,k When this is input, equation (19) is expanded and converted to the Ising model.

[0093]

number

[0094] Equation (20) is an example of the sixth model of the subgraph isomorphism problem in this embodiment. In the example of equation (20), the terms in the brackets multiplied by the weight P are constraints included in the sixth model, and are consistent with equation (16). The terms outside the brackets are (u, v) ∈ E G and (u',v')∈E H It is shown that π(u)=u' and π(v)=v' for u, v ...

[0095]

number

[0096] In the above-mentioned first to eighth models, the domain-wall representation is used as an example of expressing each integer constituting each of columns A and B using a binary vector, but other representations using binary vectors may be used. That is, the model generated by the problem conversion unit 111 may be an objective function whose value is minimized when a constraint condition is satisfied that a first column A in which m numbers are selected from n numbers is a permutation, and a second column B in which n numbers are selected from m numbers is an inverse permutation of the first column A, and whose argument is a binary vector representing the integers constituting each of the first column A and the second column B.

[0097] FIG. 10 is a table showing an example of a representation of an integer using a binary vector in this embodiment. Bit inversion is a representation in which the bits of the domain-wall representation are inverted. Alternating bits is a representation in which the same value is used to represent an integer in consecutive positions. As shown in FIG. 10, for example, in the case of 4 bits, 2 is represented as [1, 1, 0, 0] in the domain-wall representation, as [0, 0, 1, 1] in the bit inversion of the domain-wall representation, and as 0[1, 0, 0, 1]0 in the alternating bits. The numbers outside the parentheses in the alternating bits are guard bits.

[0098] In the case of alternating bits in the QUBO model, the constraint is that the sum of the values ​​of adjacent quantum bits must be 1 so that the bit values ​​alternate, but guard bits are provided at both ends so that there are positions where the same value occurs consecutively. As in the example of FIG. 10, when the number of bits is even, guard bits of the same value are provided at both ends. When the number of bits is odd, guard bits of different values ​​are provided at both ends. FIG. 10 shows an example of the QUBO model in which quantum bits take values ​​of 0 or 1, but bit inversion, alternating bits, etc. can also be used in the Ising model. In each of the above models, a domain-wall representation is used, but these bit inversions and alternating bits may also be used.

[0099] In the domain-wall representation, the number of quadratic terms is expressed by a linear expression of k, as shown in equation (9) of the QUBO model and equation (10) of the Ising model. In addition, when the constraint is expressed as a sum of quadratic expressions, the quadratic expression contains only two quantum bits. In the case of the bit inversion of FIG. 10, only the value of the guard bit is different from the domain-wall representation, and the number of quadratic terms is expressed by a linear expression of k, and the constraint expressions are equations (9) and (10). In the case of alternating bits, the condition is that the values ​​of adjacent quantum bits are different, so the constraint can be expressed, for example, in the case of the QUBO model, by equation (21). In this way, in the case of alternating bits, the number of quadratic terms is expressed by a linear expression of k, and when the constraint is expressed as a sum of quadratic expressions, the quadratic expression contains only adjacent quantum bits, i.e., only two. Thus, in the example of FIG. 10, when the constraint condition is expressed as a sum of quadratic expressions, since the quadratic expression contains only two elements of the binary vector, the number of quadratic terms is a linear expression with the number k of quadratic expressions. On the other hand, in one-hot representation, the constraint condition is equations (5) and (6), which are quadratic expressions containing elements of all binary vectors, so the number of quadratic terms is a quadratic expression with the number k of elements of the binary vector. Therefore, by using an expression in which the number of quadratic terms of the expression expressing the constraint condition is a linear expression with the number k of quadratic expressions, as in the example of FIG. 10, the number of quadratic terms can be reduced more than by using one-hot representation.

[0100]

number

[0101] The present invention may be embodied as follows. (1) One embodiment is a model generation device. When a problem to be solved is input, for two natural numbers m and n satisfying m ≤ n, it includes a problem conversion unit that converts the problem into a model for obtaining one permutation formed by selecting m numbers without repetition from n numbers and arranging them. The problem conversion unit includes, as a constraint condition when obtaining the one permutation, that a first column A selected from the n numbers and a second column B selected from the m numbers satisfy a predetermined condition. The predetermined condition is that the first column A is a permutation and the second column B is the reverse permutation of the first column A.

[0102] (2) Another embodiment is the model generation device of (1), wherein the integers constituting each of the first column A and the second column B are represented using binary vectors, and the number of quadratic terms in the expression representing the constraint condition for representing using the binary vectors is represented by a linear expression of the number of elements of the binary vectors.

[0103] (3) Another embodiment is the model generation device of (2), wherein the integers constituting each of the first column A and the second column B are represented using domain - wall representation.

[0104] (4) Another embodiment is the model generation device of (2) or (3), wherein the model has a minimum value when satisfying the constraint condition, and is an objective function taking as arguments the binary vectors representing the integers constituting each of the first column A and the second column B.

[0105] (5) Another embodiment is the model generation device according to any one of (1) to (4), wherein m < n, and the predetermined condition is that the permutation represented by the first column A coincides with the permutation represented by the auxiliary variable matrix, and at least a part defined by the permutation represented by the first column A among the reverse permutation represented by the second column B coincides with the reverse permutation represented by the auxiliary variable matrix.

[0106] (6) Also, another embodiment is the model generation device according to any one of (1) to (4), wherein m=n, and the predetermined condition is that the permutation represented by the first column A matches the permutation represented by the auxiliary variable matrix, and the inverse permutation represented by the second column B matches the inverse permutation represented by the auxiliary variable matrix.

[0107] (7) In another embodiment, there is provided a model generation method, comprising a step of converting a problem to be solved into a model for finding one permutation in which m numbers are selected from n numbers without duplicates for two natural numbers m, n satisfying m≦n, in which in the step, a first string A in which m numbers are selected from the n numbers and a second string B in which n numbers are selected from the m numbers satisfy a predetermined condition, the first string A being a permutation and the second string B being an inverse permutation of the first string A.

[0108] (8) In another embodiment, a permutation generation system includes a model generation device and a model solving device. The model generation device includes a problem conversion unit that converts a problem to be solved into a model for finding one permutation in which m numbers are selected from n numbers without duplicates for two natural numbers m and n satisfying m≦n, the problem conversion unit includes in the model a constraint condition for finding the one permutation that a first string A in which m numbers are selected from the n numbers and a second string B in which n numbers are selected from the m numbers satisfy a predetermined condition, the predetermined condition being that the first string A is a permutation and the second string B is an inverse permutation of the first string A, and the model solving device calculates an optimal solution that minimizes the value indicated by the model, or an approximation of the optimal solution.

[0109] (9) Yet another embodiment is a program for causing a computer to function as a problem conversion unit that, when a problem to be solved is input, converts the problem into a model for finding one permutation in which m numbers are selected from n numbers without duplicates for two natural numbers m, n satisfying m≦n, wherein the problem conversion unit includes in the model, as a constraint condition for finding the one permutation, that a first column A in which m numbers are selected from the n numbers and a second column B in which n numbers are selected from the m numbers satisfy a predetermined condition, and the predetermined condition is that the first column A is a permutation and the second column B is an inverse permutation of the first column A.

[0110] 1 may be recorded on a computer-readable recording medium, and the program recorded on the recording medium may be read into a computer system and executed to realize the model generation device 11 and the quantum annealer 12. Note that the term "computer system" here includes hardware such as an OS and peripheral devices.

[0111] In addition, "computer-readable recording medium" refers to portable media such as flexible disks, optical magnetic disks, ROMs, and CD-ROMs, and storage devices such as hard disks built into computer systems. Furthermore, "computer-readable recording medium" also includes those that dynamically hold a program for a short period of time, such as a communication line when transmitting a program via a network such as the Internet or a communication line such as a telephone line, and those that hold a program for a certain period of time, such as volatile memory inside a computer system that serves as a server or client in such cases. Furthermore, the above program may be one that realizes part of the above-mentioned functions, or may be one that can realize the above-mentioned functions in combination with a program already recorded in the computer system.

[0112] Although an embodiment of the present invention has been described in detail above with reference to the drawings, the specific configuration is not limited to this embodiment, and design changes and the like that do not deviate from the gist of the present invention are also included. [Explanation of symbols]

[0113] 11 Model generation device 12 Quantum Annealer 111 Problem Conversion Section 112 Model Setting Section 113 Solution conversion section 114 Solution acquisition part

Claims

1. a problem conversion unit that converts, when a problem to be solved is input, into a model for finding a permutation in which m numbers are selected from n numbers without duplication for two natural numbers m and n satisfying m≦n, the problem conversion unit includes in the model, as a constraint condition for determining the one permutation, a condition that a first column A in which m numbers are selected from the n numbers and a second column B in which n numbers are selected from the m numbers satisfy a predetermined condition; The predetermined condition is that the first sequence A is a permutation, and the second sequence B is an inverse permutation of the first sequence A. Model generation device.

2. The integers constituting each of the first column A and the second column B are represented by a binary vector; 2. The model generating device according to claim 1, wherein the number of quadratic terms in an equation expressing a constraint condition to be expressed using said binary vector is expressed as a linear expression of the number of elements of said binary vector.

3. 3. The model generating device according to claim 2, wherein the integers constituting each of the first column A and the second column B are expressed using a domain-wall representation.

4. 4. The model generating device according to claim 2, wherein the model is an objective function that satisfies the constraint condition when a value is minimized and has as arguments a binary vector representing integers constituting each of the first column A and the second column B.

5. m <nであり、 2. The model generating device according to claim 1, wherein the predetermined condition is that the permutation represented by the first column A matches the permutation represented by an auxiliary variable matrix, and at least a portion of the inverse permutation represented by the second column B that is defined by the permutation represented by the first column A matches the inverse permutation represented by the auxiliary variable matrix.

6. m=n, 2. The model generating device according to claim 1, wherein the predetermined condition is that the permutation represented by the first column A matches the permutation represented by the auxiliary variable matrix, and the inverse permutation represented by the second column B matches the inverse permutation represented by the auxiliary variable matrix.

7. The method includes a step of converting a problem to be solved into a model for finding a permutation of m numbers selected from n numbers without duplicates for two natural numbers m and n satisfying m≦n, In the step, a first sequence A in which m numbers are selected from the n numbers and a second sequence B in which n numbers are selected from the m numbers are included in the model as a constraint condition for determining the one permutation; The predetermined condition is that the first sequence A is a permutation, and the second sequence B is an inverse permutation of the first sequence A. Model generation method.

8. A model generating device and a model solving device are provided, The model generating device comprises: a problem conversion unit that converts, when a problem to be solved is input, into a model for finding a permutation in which m numbers are selected from n numbers without duplication for two natural numbers m and n satisfying m≦n, the problem conversion unit includes in the model, as a constraint condition for determining the one permutation, a condition that a first column A in which m numbers are selected from the n numbers and a second column B in which n numbers are selected from the m numbers satisfy a predetermined condition; the predetermined condition is that the first sequence A is a permutation, and the second sequence B is an inverse permutation of the first sequence A; The model solving device calculates an optimal solution that minimizes the value indicated by the model, or an approximation of the optimal solution. Permutation generation system.

9. Computer, A program for functioning as a problem conversion unit that converts, when a problem to be solved is input, into a model for finding a permutation in which m numbers are selected from n numbers without duplication and arranged for two natural numbers m and n satisfying m≦n, the problem conversion unit includes in the model, as a constraint condition for determining the one permutation, a condition that a first column A in which m numbers are selected from the n numbers and a second column B in which n numbers are selected from the m numbers satisfy a predetermined condition; The predetermined condition is that the first sequence A is a permutation, and the second sequence B is an inverse permutation of the first sequence A. program.

Images