Coupling matrix compression method, coupling matrix compression device, and program

Abstract

Provided is a new coupling matrix compression device for compressing an input coupling matrix corresponding to an input Ising Hamiltonian in an optimal solution search device that compresses the input coupling matrix, solves the input coupling matrix with an Ising machine using a coupling matrix obtained through compression, and restores a solution corresponding to the input coupling matrix from an output solution of the Ising machine. A coupling matrix compression device according to the disclosed technology comprises a compression determination unit. The compression determination unit reduces the comparison between the value of an Ising Hamiltonian in cases where the spins of the spin pair of interest have the same sign and the value of the Ising Hamiltonian in cases where the spins of the spin pair of interest have different signs to a number division problem and carries out evaluation, thereby determining the possibility of compression of the spin pair of interest, where, among spins coupled by an input coupling matrix or by a combined matrix after compression, the spin pair of interest is a spin pair to be subjected to compression determination, and peripheral spins are spins excluding the spin pair of interest.

Description

Method for compressing a connection matrix, device for compressing a connection matrix, and program

[0001] The disclosed technology relates to a technology for reducing the size of an input problem when solving a combinatorial optimization problem using an Ising machine.

[0002] A combinatorial optimization problem is a problem of selecting the best possible combination for a specific evaluation index from a large number of combinations that satisfy specific conditions. Combinatorial optimization problems appear in a wide range of fields, from business decision-making such as personnel planning, route planning, and drug discovery to academic research. A computer called an Ising machine has been proposed as a computer that is expected to efficiently solve combinatorial optimization problems. An Ising machine takes as input a function called the Ising Hamiltonian, which represents the target combinatorial optimization problem, and finds a solution to the combinatorial optimization problem (Figure 1(a)).

[0003] Specifically, the Ising machine is based on the Ising Hamiltonian H(s → ) (where s → =(s1, s2, …, s N ), s i (i=1,2,…,N) is a variable that takes the value 1 or -1, called spin. → is found as a solution to the combinatorial optimization problem. Here, the Ising Hamiltonian H(s → ) is a function expressed by the following equation: However, J ij =J ji Let's say. J ij is called the coupling coefficient.

[0004] Here, J ij (i,j=1,2,…,N, i≠j) is the (i,j)th element, and J ii The N×N real symmetric matrix J with the (i,i)th element being =0 (i=1, 2, …, N) is called the coupling matrix. Also, the Ising Hamiltonian H(s → ) is the minimum value of s → , that is, s that satisfies Eq. (1) → is called the solution or ground state of the Ising Hamiltonian. → =(s1, s2, …, sN ) is also sometimes called spin.

[0005] Currently proposed Ising machines have an upper limit on the size of the Ising Hamiltonian that can be input due to constraints imposed by implementation methods, etc., and they are unable to solve optimization problems corresponding to Ising Hamiltonians of a larger size. To address this issue, a method has been proposed in which the size of the coupling matrix is ​​compressed before solving the problem and then the obtained solution is restored (Figure 1(b)). For example, Non-Patent Document 1 proposes a compression method that focuses on the coefficient scale difference between the components of the coupling matrix. Furthermore, Non-Patent Document 2 proposes a compression method that focuses on parallelism, a geometric feature of the coupling matrix.

[0006] P. Thai et al., "FastHare: Fast Hamiltonian Reduction for Large-scale Quantum Annealing", https: / / arxiv.org / abs / 2205.05004, 2022. D. Rehfeldt et al., "Faster exact solution of sparse MaxCut and QUBO problems", https: / / arxiv.org / abs / 2202.02305, 2022.

[0007] However, there are some combinatorial optimization problems that cannot be solved because the size of the coupling matrix cannot be sufficiently reduced even when using the methods described in Non-Patent Document 1 and Non-Patent Document 2. Therefore, the disclosed technology aims to provide a new technology for reducing the size of the coupling matrix that represents a combinatorial optimization problem.

[0008] To solve the above problems, the disclosed technology provides a coupling matrix compression device for an optimal solution search device that compresses an input coupling matrix corresponding to an input Ising Hamiltonian, uses the coupling matrix obtained by compression to find a solution with an Ising machine, and restores a solution corresponding to the input coupling matrix from an output solution of the Ising machine. The coupling matrix compression device according to the disclosed technology includes a compression determination unit. Among the spins coupled by the input coupling matrix or the compressed coupling matrix, a spin pair to be subjected to compression determination is defined as a spin pair of interest, and spins other than the spin pair of interest are defined as peripheral spins. The compression determination unit determines the compressibility of the spin pair of interest by evaluating a comparison between the value of the Ising Hamiltonian when the spin pair of interest has the same sign and the value of the Ising Hamiltonian when the spin pair of interest has opposite signs, by reducing the comparison to a number partitioning problem.

[0009] According to the disclosed technology, the conditions for determining compressibility are generalized, making it possible to find compressible spin pairs in more situations. In addition, by adjusting (including shortening) the calculation time required to solve the number partitioning problem, it is possible to adjust the determination so that it can be completed within a certain time.

[0010] 1 is a flowchart illustrating an optimal solution search method according to the prior art. A diagram illustrating compressibility determination. A diagram illustrating the relationship between a compression criterion vector, compressibility determination conditions, and a number partitioning problem. A functional block diagram showing an example of the configuration of an optimal solution search device according to a first embodiment. A flowchart illustrating an example of the operation of an optimal solution search device. A flowchart illustrating vertex pair list creation processing and compressibility determination processing. A flowchart illustrating connection matrix compression processing. A diagram showing a specific example of matrix compression processing. A flowchart illustrating primitive solution restoration processing. A diagram showing a specific example of primitive solution restoration processing. A diagram showing an example of the functional configuration of a computer.

[0011] Hereinafter, embodiments of the disclosed technology will be described in detail. Note that components having the same functions are assigned the same numbers, and duplicated descriptions will be omitted.

[0012] [Key Points of the Disclosed Technology] Prior to describing the embodiments, the determination of compressibility, which is the key point of the disclosed technology, will be described using an example. First, the determination of compressibility in Non-Patent Document 2 will be described (FIG. 2(a)). In Non-Patent Document 2, if the column vectors that make up a coupling matrix are parallel, the coupling matrix is ​​determined to be compressible.

[0013] The coupling matrix J is a 4×4 real symmetric matrix expressed by the following equation: And then, [J 1,3 ,J 2,3 ] T =k[J 1,4 ,J 2,4 ] T (where k is a real number other than 0). In other words, the vector J3 consisting of the elements of the third column of the connection matrix J → and a vector J4 consisting of the elements of the fourth column of the coupling matrix J → are assumed to be parallel.

[0014] For simplicity, let k>0, and transform the Ising Hamiltonian H(s) by focusing on the terms related to spins s3 and s4, which are candidates for compression, as follows: H(s)=(terms unrelated to spins s3 and s4)+J 1,3 s1s3+J 2,3 s2s3+J 1,4 s1s4+J 2,4 s2s4 = (terms unrelated to spins s3 and s4) + kJ 1,4 s1s3+kJ 2,4 s2s3+J 1,4 s1s4+J 2,4 s2s4 = (terms unrelated to spins s3 and s4) + J 1,4 s1(ks3+s4)+ J 2,4 s2(ks3+s4) = (terms unrelated to spins s3 and s4) + (J 1,4 s1+J 2,4 s2)(ks3+s4) where ks3+s4 can take any value of {-k-1, -k+1, k-1, k+1} depending on the configuration of spins s3 and s4 (i.e., the values ​​taken by spins s3 and s4), but spins s3 and s4 are 1,4 s1+J 2,4Since ks3+s4 does not appear anywhere other than (s2)(ks3+s4), the value of ks3+s4 in the ground state is limited to {-k-1, k+1}. Therefore, by setting ks3+s4 to (k+1)s', the spins s3 and s4 can be compressed into a single spin s'.

[0015] Now, J 3,4 ≠0, and Even in this case, for any s1 and s2, If this holds, s3 and s4 have the same sign in the ground state. In other words, when searching for the ground state, s3 and s4 can be replaced with a single spin (Fig. 2(b)).

[0016] An example of a sufficient condition for equation (5) is as follows (the derivation will be described later): However, v(s1, s2) is given by the following formula.

[0017] To determine whether or not the spin is compressible, it is necessary to find the minimum value of equation (7) determined from the spins other than s3 and s4 (in this case, s1 and s2).

[0018] Then, the task of finding the minimum value of equation (7) is performed on the set S={J i3 +J i4 +δ i This is equivalent to solving the number partitioning problem for |i=1,2}. The number partitioning problem can be solved in pseudo-polynomial time using dynamic programming, but it is known that the solution time increases according to the length of the bit string of the input value. However, from the above formula, if the compressed reference vector is δ i =-J i3 -J i4 +ε i and a shorter (or longer) bit string, ε i By selecting, you can adjust the time it takes to solve the number division problem.

[0019] δ i The specific form of can be specified in various ways. For example, J i3 , J i4 The value is calculated by aligning the sizes of the two and taking the difference. Alternatively, for example, δ i =-J i4δ i If all of the conditional expressions (6-1), (6-2), and (6-3) obtained by specifying δ are satisfied, compression is possible. In other words, if δ satisfies the conditional expressions (6-1), (6-2), and (6-3), i If there is at least one form of

[0020] The above can be summarized as shown in Figure 3. i If you specify specifically, the conditions for compressibility are determined according to equations (6-1), (6-2), and (6-3). The optimization problem required for condition determination is reduced to a number division problem. The solution time for the number division problem is δ i It should be noted that equations (6-1), (6-2), and (6-3) are examples of sufficient conditions for equation (5) to hold, and the number of conditional equations does not necessarily have to be three. This concludes the explanation of the [Key Points of the Disclosed Technology].

[0021] 4 is a functional block diagram showing an example of the configuration of an optimal solution search device according to Embodiment 1. The optimal solution search device 4 includes a pair list unit 401, a compressed reference vector unit 402, a compression determination unit 403, a matrix compression unit 404, an Ising machine 405, and a primitive solution restoration unit 406.

[0022] Fig. 5 is a flowchart illustrating an example of the operation of the optimal solution search device 4. Fig. 6(a) is a detailed flowchart of step S502 in Fig. 5. Fig. 6(b) is a detailed flowchart of step S503 in Fig. 5. Fig. 7 is a detailed flowchart of step S505 in Fig. 5. Fig. 9 is a detailed flowchart of step S509 in Fig. 5.

[0023] [Overall Flow] First, the overall flow will be explained using Figure 5. As mentioned above, compression is the process of combining two spins into one and reorganizing the coupling matrix accordingly. The optimal solution search device 4 lists spin pairs (vertex pairs, described later) that are candidates for compression (step S502). Next, the compressibility of the candidate spin pairs for compression is determined (step S503). The number of vertex pairs that are determined to be compressible is defined as M'.

[0024] The optimum solution search device 4 compresses the connection matrix using the vertex pairs determined to be compressible (step S505). If necessary, further compression processing (steps S502, S503, S505) is performed on the compressed connection matrix. The above processing is repeated until the number M' of vertex pairs determined to be compressible in step S503 becomes 0 or the number of compression processing loops l reaches λ. After the compression processing is completed, the optimum solution search device 4 solves the compressed connection matrix using an Ising machine (step S508). Finally, the output solution of the Ising machine is used to restore a solution (primitive solution) corresponding to the connection matrix before compression.

[0025] This concludes the explanation of the overall flow. Below, each step will be explained in detail.

[0026] The optimum solution search device 4 initializes a compression counter l to 0 (step S501).

[0027] [Creating a vertex pair list] <Explanation of vertex pairs> Let G=(V, E) be a weighted undirected graph with a connection matrix J as the adjacency matrix (where V={s1, s2, …, s N}, E={e=(s i , s j )|Edge e=(s i , s j ) for J i,j ≠0}), that is, V has spin s i is a set whose elements are J i,j Spin s that satisfies ≠0 i , s j Pair of (s i , s j ) is a set of elements. Here, since the graph G is an undirected graph, the pair (s i , s j Note that J is not an ordered pair. i,j The edge e=(s i , s j ) is called the weight of the graph. The graph consisting of s1, s2, s3, and s4 shown in Figure 2 is a weighted undirected graph with N=4.

[0028] vertex s of graph G i , s j(i≠j) pair is a vertex pair (s i , s j ) below. i , s j ) is simply represented as a vertex pair (i, j). Also, the i and j in the vertex pair (i, j) are called vertices. The vertex pair (i, j) is not an ordered pair, just a pair, like the edges of the undirected graph G. In other words, the vertex pair (i, j) and the vertex pair (j, i) represent the same vertex pair. Therefore, there are N(N-1) / 2 vertex pairs for the graph G.

[0029] <Creating a List> The pair list unit 401 receives the input connection matrix or the connection matrix output by the matrix compression unit 404 as input, and outputs a vertex pair list. This will be explained using FIG. 6(a). The vertex pairs can be selected in any way, but if they can be selected in an order that increases the compressibility with respect to the definition of the compression criterion vector described below, the list is created in that order. Below, an example is shown of the procedure for selecting vertex pairs based on a set of column vectors with high parallelism among the column vectors that make up the connection matrix.

[0030] It is assumed that an N×N connection matrix is ​​input to the pair list section. For each vertex pair (i, j) (i, j=1, 2, ..., N, i≠j), P(i, j) is calculated using the following equation (step S601). J i → , J j → Let P(i, j) be a vector consisting of the elements of the i-th column of the coupling matrix J and a vector consisting of the elements of the j-th column of the coupling matrix J, respectively. i → and vector J j → The dot product of vector J i → and the magnitude of the vector J j → It can be said that the value is obtained by dividing by the product of the magnitudes of

[0031] In addition, That is, the vector J i →is a zero vector, the matrix obtained by excluding the i-th row and i-th column from the coupling matrix J is set as the coupling matrix J again, and processing is performed. i There are two solutions: 1 and -1.

[0032] Using P(i, j) calculated in step S601, M vertex pairs (i, j) with the largest absolute value |P(i, j)| of P(i, j) are selected in order, and a list L={p1=(u 1,1 ,u 1,2 ), p2=(u 2,1 ,u 2,2 ),…, p M =(u M,1 ,u M,2 )} is generated (step S602). That is, for a, b=1, 2, ..., M, if a≦b, then |P(u a,1 , u a,2 )|≧|P(u b,1 , u b,2 Generate a list L so that || is satisfied.

[0033] In the following, the vertex pair list generated from the input connection matrix is ​​L 0 =(p 0 1,p 0 2,…,p 0 M ), the vertex pair list generated from the connection matrix after l compression times is L l =(p l 1,p l 2,…,p l M This concludes the explanation of [Creating a vertex pair list].

[0034] [Determination of Compressibility] The compression determination unit 403 determines the compression candidate vertex pair list L from the pair list unit 401. l Receive the vertex pair list Lt l ={p l 1=(v 1,1 ,v 1,2 ), p l 2=(v 2,1 ,v 2,2 ),…, p l M' =(v M',1 ,vM',2 )} is output. This will be explained using FIG. 6(b).

[0035] The compression determination unit 403 determines whether the vertex pair p l m For (m=1, 2, ..., M), do the following (Loop 1).

[0036] <Acquisition of Compression Criterion Vector> The compression determination unit 403 acquires p l m =(v m,1 ,v m,2 ) corresponding to the compressed reference vector (δ + ) → =(δ + 1(p l m ), δ + 2(p l m )),(δ - ) → =(δ - 1(p l m ), δ - 2(p l m )) is obtained (step S611).

[0037] <Condition judgment> Assume that the joint matrix to be compressed is an N × N matrix. m,1 and u m,2 The spins corresponding to s a , s b The compression determination unit 403 calculates the following expressions (11), (12), and (13), and if expression (11) is greater than 0 and expressions (12) and (13) are equal to or greater than 0, s a =s b If the value is not equal to the value of the original data, it is determined that equal sign compression is possible. Otherwise, it is determined that equal sign compression is not possible. however,

[0038] The compression determination unit 403 further calculates the following equations (15), (16), and (17), and if equation (15) is greater than 0 and equations (16) and (17) are equal to or greater than 0, s a =-s bIf the value is 0, it is determined that compression (decoding compression) is possible, and if not, it is determined that decoding compression is not possible. however,

[0039] If equal sign compression or inverse sign compression is possible (Yes in step S612), the compression determination unit 403 determines whether p l m After the loop 1 is completed, the compression determination unit 403 adds the vertex pair list Lt l and Lt. l The number of elements M' and the compression type for each element (equal sign compression / inverse sign compression) are output.

[0040] This concludes the explanation of "Determining compressibility".

[0041] 5, if M'>0 (there is a vertex pair to be compressed, Yes in step S504), the process proceeds to step S505. If M'=0 (there is no vertex pair to be compressed, No in step S504), the process proceeds to step S508.

[0042] [Compression of the connection matrix] The matrix compression unit 404 receives the compression target vertex pair list Lt l Received Lt l The connection matrix before compression is compressed based on the above equation, and a small size connection matrix is ​​output. This will be explained using FIG. 7. The connection matrix before compression is J 0 (n 0 ×n 0 matrix), the combined matrix after l compressions is J l (n l ×n l matrix).

[0043] Compression is p l 1=(v 1,1 ,v 1,2 ), p l 2=(v 2,1 ,v 2,2 ), p l 3=(v 3,1 ,v 3,2), ... shall be performed in this order. During compression, rows and columns are added to the combined matrix. For example, when adding a row / column to a 5x5 matrix, the added row / column will be numbered "6" after "5" (Fig. 8(b)). Also, during compression, rows and columns are deleted from the combined matrix, but the numbers of the remaining rows / columns will not be changed. For example, if the third and fourth rows / columns are deleted from the 6x6 matrix in Fig. 8(b) (Fig. 8(c)), the row / column numbers of the compressed 4x4 matrix will be "1, 2, 5, 6" (Fig. 8(d)). Therefore, n l ×n l The matrix obtained by compressing a-1 times (compressing a-1 vertex pairs) is (n l -(a-1))×(n l -(a-1)) matrix, and the largest row / column number is n l It is +(a-1).

[0044] The matrix compression unit 404 initializes a vertex pair counter a to 1 (step S701), and l From p l a =(v a,1 ,v a,2 ) and the target connection matrix J l (Step S701). l is n l ×n l It is a matrix. <Compression process 1> p l a =(v a,1 ,v a,2 If the compression type of (n) is equal sign compression (Yes in step S703), the matrix compression unit 404 performs the following process (step S704). l -(a-1))×(n l -(a-1)) and expand the connection matrix to l +a line is the following formula: nth l +Let column a be the following formula: Then, the v a,1 row, v. a,1 column, v a,2 row, v. a,2 Remove the column and (n l -a)×(n l-a) Generate a matrix. Figure 8 shows an example of merging the third and fourth rows / columns in a 5x5 matrix.

[0045] <Compression process 2> p l a =(v a,1 ,v a,2 If the compression type of (n) is inverse compression (No in step S703), the matrix compression unit 404 performs the following process (step S705). l -(a-1))×(n l -(a-1)) and expand the connection matrix to l +a line nth l +a column And then, v a,1 row, v. a,1 column, v a,2 row, v. a,2 Remove the column and (n l -a)×(n l -a) Generate a matrix.

[0046] The matrix compression unit 404 increments the vertex pair counter a, and the number of compression processes is M'(list Lt l If it has not reached M' (No in step S707), the next vertex pair p l a When M' is reached (Yes in step S707), the connection matrix J after l+1 compressions is obtained and the next compression process is performed. (l+1) Output.

[0047] This concludes the explanation of [Compression of Connection Matrix].

[0048] Returning to the flow of Fig. 5, the optimal solution search device 4 increments the compression counter l (step S506) and determines whether the number of searches for vertex pairs to be compressed and the number of times the connection matrix has been compressed has reached λ. If λ has not been reached (No in step S507), the process returns to step S502 and searches for the next vertex pair to be compressed. If λ has been reached (Yes in step S507), the process proceeds to step S508. The final value of the compression counter (total number of compressions) is set to I (maximum λ).

[0049] [Solution by Ising machine] Coupling matrix J I(n I ×n I matrix) into the Ising machine, and n I Solution of s-dimensional vector I→ is obtained (step S508).

[0050] [Restoration of primitive solution] The primitive solution restoration part is s I→ is used as input, and the input coupling matrix J 0 The solution s corresponding to 0→ This will be explained with reference to FIG. 9. First, the connection matrix J I Solution s for I→ From the coupling matrix J I-1 Solution s for I-1→ We explain the procedure to restore s. By definition, I→ is n I dimensional vector, s I-1→ oha n I-1 is an n-dimensional vector I <n I-1 The coupling matrix J I The row / column number of I , the coupling matrix J I-1 The row / column number of I-1 As, i I ≦n I-1 is restored as follows:

[0051] i I >n (I-1) For this, we distinguish between equal sign compression and inverse sign compression. If it is compressible (equal sign compression), and restored, If it is possible to compress it (reverse coding compression), and restore.

[0052] Figure 10 shows how the solution (σ1,σ2,σ3) is calculated using a 3x3 matrix obtained by compressing the 3rd and 4th rows / columns of a 5x5 matrix and the 1st and 5th matrices, and how the solution corresponding to the original 5x5 matrix is ​​restored. The 3rd and 4th rows / columns are compressed using equal sign processing, and the 1st and 5th rows / columns are compressed using inverse sign processing.

[0053] Based on the above, a detailed flow of the primitive solution restoration (step S509) will be described with reference to FIG. I→ (Step S901). Next, the processing matrix counter L is set to I (Step S902). L=I means that the connection matrix after I-times compression is restored. Next, s is obtained using the L-th compression information. L→ From s L-1→ (step S903). This process can be carried out by replacing I with L in equations (23) to (27). Next, the processing matrix counter is decremented (step S904). Next, the primitive solution restoration unit 406 determines whether the solution corresponding to the input coupling matrix has been restored. If the solution corresponding to the input coupling matrix has not been restored (No in step S905), the process returns to step S903 and the next restoration process is carried out. If the solution corresponding to the input coupling matrix has been restored (Yes in step S905), s0 → is output as a primitive solution (step S906).

[0054] This concludes the explanation of [restoring the primitive solution].

[0055] [Computation time can be reduced] Consider the Ising model consisting of four spins as shown in Figure 2, and determine whether spins 3 and 4 can be compressed to s3 = s4. To make this determination, it is necessary to find the following equation (28). In the following, δ i It is shown that by appropriately defining , the calculation time required to obtain equation (28) using dynamic programming can be reduced.

[0056] J 13 +J 14 =123456789, J 23 +J 24 = 12345678. When δ1 = 10 and δ2 = 20 are selected, the execution time of the dynamic programming is 2.130 (s). On the other hand, when δ1 = -(J13 + J14) + 1 and δ2 = -(J23 + J24) + 1 are selected, the execution time of the dynamic programming is 2.335 (ns). In this way, when the compressed reference vector is δ i =-J i3 -J i4 +εi and a shorter bit string, ε i By selecting this, it is possible to shorten the time required to solve the number division problem. This makes it possible to deal with situations where a compression decision must be made within a certain time.

[0057] [Supplementary Information] In the above [Determining Compressibility], the compression reference vector can be defined adaptively for each vertex pair, but it may also be defined independently of each vertex before creating the list, rather than adaptively. Also, instead of defining two for each pair, only one may be defined and used.

[0058] In the above-mentioned [Compression of Connection Matrix], if there is a pair of vertices that includes the same vertex among the vertex pairs that are determined to be compressible, for example, p1=(v 1,1 ,v 1,2 ), p2=(v 2,1 ,v 2,2 ), (v 1,1 =v 2,1 , v 1,2 ≠v 2,2 ), or (v 1,1 ≠v 2,1 , v 1,2 =v 2,2 ) In this case, there is a pair p a , p b (a <b)のうち、p b In this case, the vertex pair list Lt l The number of elements M' of v should be adjusted appropriately. 1,1 =v 2,1 In the case of v 1,1 (=v 2,1 ),v 1,2 ,v 2,2 It may be considered that the three spins can be compressed into one spin, and the corresponding three rows / columns of the coupling matrix may be compressed.

[0059] [Derivation of equations (6-1), (6-2), and (6-3), etc.] <Preparation> <What we want to show> The following equation (30) holds for any fixed s1 and s2.

[0060] The cases are divided according to the size of H(s1,s2,+1,+1) and H(s1,s2,-1,-1). If H(s1,s2,+1,+1)≧H(s1,s2,-1,-1), it is shown that the following equation (31) holds for any fixed s1 and s2. When H(s1, s2, +1, +1) ≦ H(s1, s2, −1, −1), it is shown that the following equation (32) holds for any fixed s1 and s2.

[0061] <[1]: If H(s1,s2,+1,+1)≧H(s1,s2,-1,-1)> To derive the sufficient condition, set one condition. This is nothing other than equation (6-1). (The proof is omitted.) Now, δ1 (1) ,δ2 (1) ,δ1 (2) ,δ2 (2) Using Equation (33), C2 ≥ 0, and C3 ≥ 0 are sufficient conditions for equation (31) to hold.

[0062] δ that satisfies equation (35) i (j) is not unambiguously determined, By defining this, we obtain equations (6-2) and (6-3).

[0063] <[2]: When H(s1,s2,+1,+1)≦H(s1,s2,-1,-1)> Similarly, to derive the sufficient condition, set one condition. This is nothing other than equation (6-1). (The proof is omitted.) δ1 (1) ,δ2 (1) ,δ1 (2) ,δ2 (2) Using this, the same result as in [1] can be obtained.

[0064] The decision formulas (11)-(14) for the same coding compression can be obtained by the same idea as above. The decision formulas (15)-(18) for the inverse coding compression can be obtained by replacing the formula (30) with the following: is obtained as a sufficient condition to satisfy

[0065] δ that satisfies equation (35) or equation (42) i (j) Here is another example. Using k,k',k''>0,

[0066] The above is the explanation of [the derivation of equations (6-1), (6-2), (6-3), etc.].

[0067] [Processor, Program, Recording Medium] The functions performed by the components described herein may be implemented in circuitry or processing circuitry, including general-purpose processors, application-specific processors, integrated circuits, ASICs (Application Specific Integrated Circuits), a CPU (a Central Processing Unit), conventional circuits, and / or combinations thereof, programmed to perform the described functions. A processor includes transistors and other circuits and is considered to be circuitry or processing circuitry. A processor may also be a programmed processor that executes a program stored in a memory.

[0068] In this specification, a circuitry, unit, or means is hardware that is programmed to realize or performs the described functions, which may be any hardware disclosed herein or any hardware known to be programmed to realize or perform the described functions.

[0069] If the hardware is a processor considered to be a type of circuitry, the circuitry, means, or unit is a combination of the hardware and software used to configure the hardware and / or processor.

[0070] The various processes described above can be implemented by loading a program that executes each step of the above method into the recording unit 2020 of the computer 2000 shown in Figure 11 and operating the control unit 2010, input unit 2030, output unit 2040, display unit 2050, etc.

[0071] The program describing the processing contents can be recorded on a computer-readable recording medium, which may be, for example, a magnetic recording device, an optical disk, a magneto-optical recording medium, a semiconductor memory, or any other suitable recording medium.

[0072] The program may be distributed by, for example, selling, transferring, lending, etc. portable recording media such as DVDs and CD-ROMs on which the program is recorded. Furthermore, the program may be stored in a storage device of a server computer, and then transferred from the server computer to other computers via a network, thereby distributing the program.

[0073] A computer that executes such a program may first temporarily store the program recorded on a portable recording medium or transferred from a server computer in its own storage device. Then, when executing a process, the computer reads the program stored on its own recording medium and executes the process in accordance with the read program. Alternatively, the computer may read the program directly from a portable recording medium and execute the process in accordance with the program. Furthermore, the computer may execute the process in accordance with the program each time a program is transferred from a server computer to the computer. Alternatively, the server computer may not transfer the program to the computer, but may instead execute the process through a so-called ASP (Application Service Provider) service, which realizes the processing function by issuing an execution instruction and obtaining the results. Furthermore, the server computer may execute the process at the terminal using a so-called SaaS (Software as a Service) service, which allows users to use part of a server computer along with the program. In this embodiment, the program includes information used for processing by an electronic computer that is equivalent to a program (such as data that is not a direct instruction to a computer but has properties that dictate computer processing).

[0074] Furthermore, in this embodiment, the device is configured by executing a predetermined program on a computer, but at least a part of the processing contents may be realized by hardware.

[0075] 4 Optimum solution search device 401 Pair list unit 402 Compressed reference vector unit 403 Compression determination unit 404 Matrix compression unit 405 Ising machine 406 Primitive solution restoration unit 407 Recording unit 2000 Computer 2010 Control unit 2020 Recording unit 2030 Input unit 2040 Output unit 2050 Display unit

Claims

1. A method for searching for an optimal solution in which an input coupling matrix corresponding to an input Ising Hamiltonian is compressed, a solution is obtained with an Ising machine using the coupling matrix obtained by compression, and a solution corresponding to the input coupling matrix is ​​restored from the output solution of the Ising machine, the method compressing the input coupling matrix, wherein, of the spins coupled by the input coupling matrix or the compressed coupling matrix, a spin pair to be subjected to compression judgment is defined as a spin pair of interest, and spins other than the spin pair of interest are defined as peripheral spins, and a compression judgment unit compares the value of the Ising Hamiltonian when the spin pair of interest has the same sign with the value of the Ising Hamiltonian when the spin pair of interest has opposite signs, and evaluates the result by reducing it to a number partitioning problem.

2. A coupling matrix compression method according to claim 1, wherein H() is an Ising Hamiltonian, + / - is the configuration of the spin pair of interest, and σ is the configuration of the peripheral spins, and the compression determination unit performs the following first Hamiltonian comparison for any fixed σ: If it is determined that the spin pair of interest is compressible with the same sign, then for any fixed σ, the following second Hamiltonian comparison If it is determined that the following holds, it is determined that the spin pair of interest can be compressed by inverse sign compression.

3. The coupling matrix compression method according to claim 2, wherein the coupling matrix is ​​an N×N matrix, the target spin pair is s1 and s2, and one of the peripheral spins is s i , s1 and s i The coupling coefficient of J i1 , s2 and s i The coupling coefficient of J i2 , the s i The constant δ is determined for each i As such, the number division problem is and determining whether the first Hamiltonian comparison and the second Hamiltonian comparison are successful or not using the obtained minimum value.

4. The method for compressing a connection matrix according to claim 3, wherein the δ i The method for compressing a connection matrix adjusts the computational complexity of the number division problem by adjusting the above.

5. An optimal solution search device that compresses an input coupling matrix corresponding to an input Ising Hamiltonian, solves with an Ising machine using the coupling matrix obtained by compression, and restores a solution corresponding to the input coupling matrix from the output solution of the Ising machine, the device compressing the input coupling matrix, comprising: a coupling matrix compression device having a compression determination unit that determines the compressibility of the spin pair of interest by comparing the value of the Ising Hamiltonian when the spin pair of interest has the same sign with the value of the Ising Hamiltonian when the spin pair of interest has different signs, by reducing it to a number partitioning problem and evaluating the comparison, with a spin pair that is the target of compression determination among spins coupled by the input coupling matrix or the coupling matrix after compression and spins other than the spin pair of interest being peripheral spins.

6. A coupling matrix compression device according to claim 5, wherein H() is an Ising Hamiltonian, + / - is the configuration of the spin pair of interest, and σ is the configuration of the peripheral spins, and the compression determination unit performs the following first Hamiltonian comparison for any fixed σ: If it is determined that the spin pair of interest is compressible with the same sign, then for any fixed σ, the following second Hamiltonian comparison If it is determined that the above holds, it is determined that the spin pair of interest can be compressed by inverse sign compression.

7. A coupling matrix compression device according to claim 6, wherein the coupling matrix is ​​an N×N matrix, the target spin pair is s1 and s2, and one of the peripheral spins is s i , s1 and s i The coupling coefficient of J i1 , s2 and s i The coupling coefficient of J i2 , the s i The constant δ is determined for each i As such, the number division problem is and determining whether the first Hamiltonian comparison and the second Hamiltonian comparison are successful or not using the obtained minimum value.

8. A program for causing a computer to function as the connection matrix compression device according to any one of claims 5 to 7.

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