JP7182173B2 - Variable embedding method and processing system - Google Patents

Description

The present invention relates to a method and processing system for embedding variables in hardware graphs.

Conventionally, the applicant has developed a technique for quickly solving an optimization problem for searching for a global optimum value of an evaluation function composed of a combination of multiple variables. Dedicated hardware is generally developed to solve such combinatorial optimization problems at high speed.
This hardware has a specific fixed architecture and can solve combinatorial optimization problems faster than ordinary general-purpose computers. Because this kind of hardware has a specific fixed architecture, it imposes some restrictions on solving combinatorial optimization problems. The first constraint is that there is a limit to the absolute number of variables that can be processed at once. A second constraint is that there is a limit to the number of interactions between the variables. In order to efficiently process optimization problems using dedicated hardware, it is required to execute processing under such constraint conditions.

In order to deal with the first constraint, the technique described in Patent Document 1 divides the variables of the optimization problem into subproblems that can be embedded in the hardware graph, and repeats the optimization process of the subproblems. I'm trying to get a high-precision solution to the original optimization problem. However, no specific method is shown to embed as many variables as possible in the hardware graph. A specific method for embedding the variables of a given optimization problem in a hardware graph is given in Japanese Patent Application Laid-Open No. 2002-200021.

The technique described in Patent Literature 2 refers to a problem graph to be solved and a hardware graph, and embeds variables for each vertex of the hardware graph. The variable embedding process described in Patent Document 2 is divided into a first half and a second half. The first half embeds all the variables of a given problem while allowing redundant assignment of multiple variables to the vertices of the hardware graph. Also in the second half, only one variable is assigned to each vertex on the hardware graph. In addition, Non-Patent Document 1 can also be cited as a reference.

US Publication No. 2017-255629 U.S. Pat. No. 9,501,747

Okuyama et al., "Graph Embedding Algorithm for Ising Computer", The Institute of Electronics, Information and Communication Engineers, vol.116, no.116, IEICE Technical Report, COMP2016-11, pp.97-103, June 2016

The inventors have confirmed that if the technology described in Patent Document 2 described in the Background Art column is adopted, it takes a long processing time, especially in the latter half of the processing. This is because the technique described in Patent Document 2 permits duplication once and embeds it in the first half, so it is necessary to correct it in the second half. Therefore, it takes a lot of time to complete all these processes. In particular, when it is necessary to repeatedly embed a large-scale problem into a hardware graph while dividing it into subproblems, as in the technique described in Patent Document 1, it is necessary to reduce the processing time for each time. important for reducing processing time.
SUMMARY OF THE INVENTION It is an object of the present invention to provide a method and processing system for embedding variables in a hardware graph that can shorten the processing time.

According to the inventions described in claims 1 and 13, when embedding at least a part of all variables to the vertices of the hardware graph, it is necessary to duplicately assign the variables of the optimization problem to the vertices of the hardware graph. Then, the variables that do not require overlapping assignment are selected and embedded in the vertices of the hardware graph without using the variables that require overlapping assignment as the variables of the subproblem. For this reason, it becomes possible to perform the processing without spending time for the processing of resolving duplicate assignments, thereby shortening the processing time.
For example, the process of judging whether duplicate assignment is necessary starts from the vertex to which no variable is assigned on the hardware graph, and the vertex assigned to the embedded variable adjacent to the variable to be additionally embedded. can be constructed on the hardware graph using only unused vertices.
For example, if the above path exists, by appropriately embedding additional variables to be embedded or combined embedded variables for each vertex on the path, while reproducing the interaction between variables in the problem graph , can embed variables in the hardware graph. The shorter the route, the fewer the number of vertices on the hardware graph to be used. Therefore, a method of obtaining the shortest route using the Dijkstra method or the like is conceivable.

Electrical configuration diagram in the first embodiment Functional configuration diagram Image diagram of partitioning an optimization problem into subproblems Flowchart showing optimization problem solution derivation processing Flowchart showing variable embedding processing Explanatory diagram of embedding variables in hardware graph (Part 1) Explanatory diagram of the process of embedding variables in the hardware graph (Part 2) Explanatory diagram of a method of embedding a complete graph into a hardware graph in the second embodiment. Explanatory diagram of the process of embedding variables in the hardware graph (Part 3) Flowchart showing hardware graph vertex reservation processing Explanatory diagram of vertex reservation processing in a chimera graph (Part 1) Explanatory diagram of vertex reservation processing in a chimera graph (Part 2) Explanatory diagram of vertex reservation processing in a chimera graph (Part 3) Explanatory diagram of vertex reservation processing in a chimera graph (Part 4) Flowchart showing optimization problem solution derivation processing of the third embodiment Description diagram of the problem graph Illustration of embedding order in problem graph Problem graph of multivalued problem expressed using binary variables in the fourth embodiment Illustration of selection method for selecting binary variables (Part 1) Explanation of the selection method for selecting binary variables (Part 2) Flowchart showing multi-valued variable embedding processing Explanatory diagram showing selection priority of binary variables Flowchart showing conversion processing to binary optimization problem in the fifth embodiment Description diagram of conversion process to binary optimization problem

Several embodiments of the variable embedding method and processing system of the present invention will be described below with reference to the drawings. In the following embodiments, portions having the same or similar functions are assigned the same reference numerals and explanations are given, and descriptions of configurations having the same or similar functions, their effects, and linkage operations are necessary. abbreviated according to

(First embodiment)
1A to 6 show explanatory diagrams of the first embodiment. Quantum Ising machine 1 shown in FIG. 1A is configured using function-specific Ising hardware 2, and configures interactions between multiple variables X as physical constraints of hardware, It is configured to simulate the same situation as the optimization problem by using dynamic properties.

The Ising type hardware 2 is composed of a quantum processor, which is dedicated hardware configured by a specific fixed architecture. This Ising type hardware 2 has vertices V (V1 to V9) arranged in a large number on the virtual hardware, and has restrictions on interaction between the vertices V (V1 to V9) of the virtual hardware. It can be represented by a hardware graph G2. In the following description, when some of these vertices V are specified, the vertices V are given suffixes as necessary. Also, some or all of the many vertices V may be collectively referred to as vertices V.

The interaction constraint between these vertices V is, for example, as shown in FIG. , and other vertices V can be configured so as to be non-bonded.

In addition, a hardware graph G2 is applied that configures a unit cell C that includes a plurality of vertices V that are coupled so as to constrain the interaction of all or part of each other, and that includes these unit cells C for a plurality of grids. However, in this case, it may be configured such that there is a constraint on the connection of vertices V between adjacent grids.

A hardware graph G2 called a chimera graph is a representative example of the hardware graph G2 having unit cells C for a plurality of grids (see the right side of FIG. 5, FIG. 9, FIG. 10, etc.). This chimera graph includes, for example, a unit cell C (eg, C11, C12, C21, C22) having n×m vertices V in the vertical and horizontal directions (corresponding to the first direction and the second direction), and the unit cell C is It is a hardware graph G2 provided with a plurality of grids in the vertical and horizontal directions. In the following description, when some of these unit cells C are specified, the unit cells C will be given suffix numbers as necessary. Also, part or all of the unit cell C may be collectively referred to as the unit cell C.

This chimera graph, as shown on the right side of FIG. The vertices V constituting the cell C are connected to the corresponding vertices V of the other vertically and horizontally adjacent unit cells C. FIG. This chimera graph is a hardware graph G2 having a constraint that other portions than the above are uncoupled. The connectivity of the chimera graph shown on the right side of FIG. 5 is detailed. In the chimeric graph, a plurality of vertices V11 to V14 in the first column are connected to a plurality of vertices V21 to V24 in the second column, respectively, a plurality of vertices V11 to V14 in the first column are not connected to each other, and a plurality of vertices V11 to V14 in the first column A plurality of grids of unit cells C11 . Also, at this time, the vertices V11 to V14 in the first column of the unit cells C11 and C12 are connected in the vertical direction, and the vertices V21 to V24 in the second column of the unit cells C11 and C21 are connected in the horizontal direction. Combined.

A computer 3 shown in FIG. 1A constitutes a variable embedding device, and is a device having a function for embedding a variable X in a form adapted to the Ising type hardware 2 of this quantum Ising machine 1 . The computer 3 is a general-purpose computer in which a CPU 4, a memory 5 such as ROM and RAM, and an input/output interface 6 are connected by a bus. The memory 5 is used as a non-transitional substantive recording medium. The computer 3 executes a variable embedding program stored in the memory 5 by the CPU 4, searches for an optimal variable X embedding method by executing various procedures, The variable X is embedded in the wear 2.

The optimization process executed by the quantum Ising machine 1 assumes a search space consisting of a Euclidean space with one or more n dimensions, and in this search space, an evaluation function H or the variable X that satisfies the condition that the evaluation function H becomes the minimum value, that is, the processing for obtaining the optimum solution. The evaluation function H is generated by a plurality of requirements and constraints and represents a mathematical function derived based on one or more n variables X, such as an arbitrary polynomial, rational function, irrational function, exponential function , logarithmic functions, and combinations of addition, subtraction, multiplication, and division. Hereinafter, each variable in the n-dimensional problem will be indicated as X1, X2, . As shown in FIG. 1B, the computer 3 has various functions such as a determination unit 7 and an embedding unit 8 as functions realized by executing programs stored in the memory 5 .

FIG. 2 shows an image of dividing a large-scale optimization problem (original problem) of the evaluation function H into subproblems, and FIG. is shown by a flow chart.

As shown in FIG. 3, the computer 3 inputs a problem graph G1 (S1). This problem graph G1 represents the interaction relationship of each variable X obtained from the requirements and constraints on the evaluation function H as a result of the analysis of the evaluation function H by the computer 3 . An example of this problem graph G1 is shown on the left side of FIG. This problem graph G1 shows an example in which the variable X1 is connected to the variables X2 to X9, and the other variables X2 to X9 are not connected to each other.

After that, the computer 3 embeds the variable X forming the problem graph G1 into the hardware graph G2 of the Ising type hardware 2 (S2). Vertices V corresponding to the same variable X must form connected subgraphs on the hardware graph G2. Moreover, the variable X that is connected in the problem graph G1 must have at least one or more connections on the hardware graph G2. If this rule is not satisfied, the quantum Ising machine 1 cannot be properly processed. For this reason, the computer 3 is required to perform the embedding process of the variable X appropriately.

FIG. 4 shows a flow chart of embedding processing of the variable X. As shown in FIG. When the computer 3 embeds the variable X in the hardware graph G2 of the quantum Ising machine 1, the computer 3 determines whether redundant allocation will occur if the variable X is selected and temporarily embedded in the hardware graph G2 (S12). , the variable X is embedded in the hardware graph G2 when duplicate assignment does not occur (S13), and the variable X is not embedded in the hardware graph G2 when duplicate assignment occurs (S14). Then, the processing of S11 to S14 is repeated until the termination condition is satisfied. The termination condition is, for example, that there is no variable X that has not been embedded, or that it is determined that the variable X cannot be additionally embedded as a result of repeating these processes S11 to S14 up to a predetermined upper limit number of times.

For example, FIG. 5 shows an example in which the computer 3 embeds the variable X of the problem graph G1 into the hardware graph G2 of the quantum Ising machine 1. In FIG. The hardware graph G2 uses the chimera graph described above as shown on the right side of FIG.

At this time, the computer 3 randomly selects the variables X1 to X9 and embeds the variables X in the hardware graph G2 in order from the selected variable X. In practice, the variables X1 to X9 are randomly selected and embedded, but here, in order to facilitate understanding of the explanation, an example of embedding the variables X1 to X9 in order is shown.
At this time, the computer 3 embeds the variables X2 to X9 after embedding the variable X1. As mentioned above, variables X2-X9 must all be combined with variable X1. Therefore, the variables X2 to X6 selected before the variables X7 to X9 are embedded in the vertex V connected to the variable X1 (see the right side of FIG. 5). Here, the variables X7 to X9 cannot be combined with the variable X1 unless they are redundantly assigned to the vertex V in which the variables X2 to X6 are embedded. end without

After finishing the embedding process of the variable X, the quantum Ising machine 1 shifts to the process of FIG. 3 and executes the optimization process (S3). The quantum Ising machine 1 substitutes the variables X1 to X6 as partial variables into the evaluation function H in the optimization process, and uses the gradient method or other optimization techniques to set conditions (optimization conditions) in which the evaluation function H is lower than a predetermined value. Values of variables X1 to X6 are obtained so as to satisfy (subproblem solution derivation processing). At this time, the quantum Ising machine 1 determines the optimum values of the variables X1 to X6 under the condition that a predetermined time has passed since the start of the process or that the number of trials has been repeated a predetermined number or more, and updates the variables X1 to X6 ( S4). In this case, the evaluation value of the evaluation function H may be obtained using fixed values for the other variables X7 to X9. This solves the subproblem.

After that, the process returns to S2, the computer 3 randomly selects the variable X again in S2, embeds the selected variable X in the hardware graph G2, and the quantum Ising machine 1 performs optimization processing using these variables X. to determine the optimum values for the combination of variables X and update these variables X in S4. For the variable X not selected at this time, it is preferable to use the optimum value of the variable X (here, variables X1 to X6) obtained as the optimum value before the processing. A variable X that has never been selected should be set to a fixed value for processing. Then, the quantum Ising machine 1 determines that the end condition is satisfied under the condition that a predetermined time has elapsed since the start of processing or that the trial has been repeated a predetermined number of times or more (YES in S5), and the result of this variable X or An evaluation value of the evaluation function H and the like are output. This allows us to solve the overall optimization problem. By performing such iterative processing, the original problem can be divided into subproblems and solved as shown in FIG.

As described above, according to this embodiment, when embedding a part of all the variables X in the vertices V of the hardware graph G2, the variables X of the optimization problem are embedded in the vertices V of the hardware graph G2. It is determined whether or not overlapping assignment is required (S12), variables X7 to X9 that require overlapping assignment are not used as variables X7 to X9 of the partial problem, and variables X1 to X6 that do not require overlapping assignment are used. It is selected and embedded in the vertex V of the hardware graph G2. Therefore, for a large-scale optimization problem in which all the variables X cannot be embedded in the vertices V of the hardware graph G2, the interrelationships of the variables X of the optimization problem are expressed in the problem graph G1 to solve the problem of the large-scale optimization problem. When solving the optimization problem, the variables X1 to X9 of the problem graph G1 are divided into subproblems that can be embedded in the hardware graph G2, and the optimization of the subproblems is repeated to solve a large-scale optimization problem. be able to.
Note that FIG. 5 shows an example in which the number of variables X in the problem graph G1 is nine in order to simplify the explanation. Therefore, only some of the vertices V on the hardware graph G2 can be used. However, in practice, when solving a large-scale problem with many variables X, the computer 3 executes the embedding process described above, so that more vertices V on the hardware graph G2 can be used and a larger Note that the subproblem variable X can be embedded.

<Comparative example>
FIG. 6 shows a comparative example when, for example, the technology described in Patent Document 2 is applied and duplicate allocation is allowed. In such a case, since there are only five vertices V that connect the variable X1 to the vertex V11 of the unit cell C11 in which the variable X1 has been embedded, the variables X2 to X6 can be embedded in the hardware graph G2 without redundant assignment. However, the variables X7 to X9 are redundantly assigned and embedded in the vertices V21 to V23. If such a process is applied, it will take a long time to calculate the optimal value while resolving this duplicate allocation.

<Summary and effects of the present embodiment>
According to this embodiment, the computer 3 determines whether or not redundant assignment is required when performing the embedding process of the variable X (S12), and divides the variables X7 to X9 that require redundant assignment into partial Variables X1 to X6 that do not require redundant assignment are selected and embedded in the vertex V of the hardware graph G2 without using them as the variable X in question. For this reason, it becomes possible to perform processing without permitting overlapping assignments as shown in the prior art, and there is no need to perform processing to eliminate overlapping assignments, thereby significantly shortening the processing time.

(Second embodiment)
7 to 10 show additional explanatory diagrams of the second embodiment. In this embodiment, the computer 3 functions as a reservation unit, a selection unit, a prohibition unit, a linking unit, and a cancellation unit by executing programs stored in the memory 5 . According to the method of the first embodiment, if the problem graph G1 has a relatively small number of connections, only the variables X that do not require overlapping assignments are used, and the variables X that form a large subproblem are placed on the hardware graph G2. It can be embedded. However, as shown in FIG. 5, if there are many variables X1 connected to many variables X2 to X9 in the problem graph G1, before using up the vertices V on the hardware graph G2, The subproblem stops growing.

When embedding each variable X on the hardware graph G2, the computer 3 selects at least one vertex V included in the subgraph on the hardware graph G2 assigned to each variable X. , for a vertex V that connects with the selected vertex V on the hardware graph G2 and follows a direction that can efficiently increase the number of connections with other variables X on the hardware graph G2. It is desirable to prohibit the embedding of variables X other than the variables X, and to leave room for extending the subgraphs to which the relevant variables X are connected on the hardware graph G2. Hereinafter, this operation is called "reservation".
FIG. 8 shows a flowchart of embedding processing of the vertex V on the hardware graph G2. As shown in FIG. 8, the computer 3 first sets the reservation method (association method) by using (referring to) the method of embedding the variable X of the complete graph G1a into the hardware graph G2 (S21).

A specific reservation method, a so-called linking method, differs depending on the type of hardware graph G2. It is desirable to use a method of embedding the variable X of the complete graph G1a (see FIG. 7A), which is assumed to be the problem graph G1 and has the largest number of connections, into the hardware graph G2 (that is, use it as a reference).

FIG. 7A shows a general method of embedding the complete graph G1a in the hardware graph G2 (chimera graph) in order to facilitate understanding of this reservation method setting process. Represents connected subgraphs assigned to variables. Variables X1 to X4 of complete graph G1a are embedded in a subgraph including vertices V21 to V24 of unit cell C11. Variables X5-X8 are embedded in a subgraph containing vertices V21-V24 of unit cell C22. Variables X9-X12 are embedded in a subgraph containing vertices V21-V24 of unit cell C33.
As shown in FIG. 7A, subgraphs connected on the hardware graph G2 (graphs connected by thick lines in FIG. 7A) have a structure in which the unit cells C are connected vertically and horizontally. The computer 3 uses this method of embedding variables X1 to X12 (as a reference) to link each vertex V traced across the unit cells C in S21. In other words, the computer 3 associates a certain vertex V with the vertex V included in the unit cell C in the vertical direction and/or the horizontal direction in S21.

Regarding the vertex V21 (embedded vertex V of the variable X1) of the unit cell C11 attached in the right diagram of FIG. Vertices V12 and V22 of each unit cell C linked to the hardware graph G2 with respect to the vertex V22 (embedded vertex V of the variable X6) attached in the right figure of FIG.

The computer 3 associates the vertex V21 of the unit cell C11 on the hardware graph G2 with the vertex V11 of the unit cells C11, C12, C13, . . Further, the computer 3 links the vertex V22 of the unit cell C22 on the hardware graph G2 to the vertex V12 of the unit cells C22, C23, .

Furthermore, the computer 3 uses such a method of embedding the variable X of the complete graph G1a into the hardware graph G2 (as a reference), and uses a part of the embedding method to link the vertex V. can be An example of this is shown in FIGS. 9A, 9B, 10A and 10B. These FIGS. 9A to 10B show a chimera graph having unit cells C connecting 2×4 vertices V for a plurality of 4×4 grids as a hardware graph G2. shows an example of linking vertices V in . 9A and 9B show an example of tying to the vertex V traced in the direction straddling the vertical direction, and FIGS. 10A and 10B show an example of tying to the vertex V traced in the direction straddling the horizontal direction. ing.

As shown in FIG. 9A, the computer 3 connects the unit cells C12 continuously in the adjacent direction (vertical direction in this case) corresponding to the vertex V11 (corresponding to the first vertex) of the unit cell C11 selected as the base point. Each vertex V11 (corresponding to the second vertex) of C13, C14, . Correspondingly, it may be linked to the vertex V13 (corresponding to the second vertex) of the unit cells C31, C32, and C34 that are continuously connected in the adjacent direction (vertical direction in this case).

Further, as shown in FIG. 10A, the computer 3 selects the vertex V21 (corresponding to the first vertex) of the unit cell C11 selected as the base point, and calculates each of the unit cells C21, C31, and C41 that are continuously connected in the horizontal direction. Alternatively, as shown in FIG. 10B, the computer 3 selects the vertex V22 (corresponding to the second vertex) of the unit cell C23 selected as the base point and connects Each vertex V22 (corresponding to the second vertex) of unit cells C13, C33, and C43 that are continuously connected in the horizontal direction may be linked.

Then, the computer 3 selects the variable X (S22 in FIG. 8), determines whether or not redundant assignment is required (S23), and does not embed the variable X when redundant assignment is required (S24). However, even if the variable X is embedded when duplicate assignment is not required (S25), one or more of the vertices V in which the variable X is embedded are selected (S26), and the selected vertex V is used as a base point. Another vertex V is reserved (S27). Then, the computer 3 determines whether or not to cancel (cancel) the prohibition of embedding to the reserved vertex V (S28). It is determined whether or not the condition is satisfied (S30), and if the condition is not satisfied, the process returns to S22, the variable X is selected again, and the processes of S23 to S30 are executed.

For example, if the problem graph G1 has the configuration on the left side of FIG. 7B, and the computer 3 embeds the variable X1 in the vertex V11 of the unit cell C11 on the hardware graph G2, for example, in S21, FIG. When the vertices V11 of the unit cells C12, C13, and C14 are linked as shown, the vertices V11 of the unit cells C12, C13, . (corresponding to the second vertex) can be reserved for embedding the variable X1 (S26-S27).

Therefore, for example, as shown on the right side of FIG. 7B, when the vertex V11 of the unit cell C11 is embedded with the variable X1 on the hardware graph G2, the vertex V11 of the unit cell C12 is reserved and secured for the variable X1. will be At this time, only the variable X1 is allowed to be embedded in the vertex V11 of the unit cell C12 until all the variables X2 to X9 are embedded.
After that, when the computer 3 selects the variables X2 to X9 in this order and embeds them in the vertex V of the hardware graph G2 after embedding the variable X1, the unit cell C11 The variables X2 to X5 are embedded in order in the vertices V21 to V24 connected to the vertex V11 of .

Next, when the computer 3 embeds the variable X6 in the hardware graph G2, the vertex V11 of the unit cell C12 is reserved for the variable X1, and embedding the variable X6 in the vertex V11 of the unit cell C12 is prohibited. ing. Therefore, the computer 3 embeds the variable X1 in the vertex V11 of the unit cell C12 and embeds the variable X6 in the vertex V21 of the unit cell C12. Thus, by embedding the variable X1 at the vertex V11 of the unit cell C12, the variable X that can be combined with the variable X1 can be increased, and the variables X7 to X9 are respectively embedded at the vertices V22 to V24 of the unit cell C12. be able to.

As described above, by embedding the variable X while leaving room for extending the subgraph above the hardware graph G2, it is possible to embed the variables X1 to X9 without redundant assignment. In this example, only the reservation of the variable X1 is focused, but when additionally embedding the variables X2 to X9, the reservation of the vertex V is executed for all the variables X in S25 to S27. Then, the computer 3 preferably cancels the reservation of the variable X when the reservation should be canceled (S28-S29). As a result, it is possible to efficiently grow subproblems even in the problem graph G1 having a large number of connections.

After that, although not shown in FIG. 8, the quantum Ising machine 1 executes optimization processing in the same manner as in S3 of FIG. . In this embodiment, nine variables X1 to X9 have been used to simplify the explanation. For this reason, a simple example has been shown in which all variables X1 to X9 can be embedded in the hardware graph G2 in one routine embedding process. The scale problem (original problem) is to be solved.

At this time, as described in the first embodiment, a partial problem is solved by embedding some of the variables X in one embedding process. Even in such a case, if the embedding method described in this embodiment is applied, more variables X can be embedded in vertices V than in the first embodiment. As a result, hardware resources can be effectively used, and the optimum value calculation time can be shortened.

<Summary and effects of the present embodiment>
As described above, according to this embodiment, the vertices V of the hardware graph G2 are linked (the reservation method is set) based on the embedding method for embedding the complete graph G1a in the hardware graph G2. (S21), at least one or more vertices V connected by subgraphs are selected from the vertices V on the hardware graph G2 (S26), and when each variable X is embedded in the hardware graph G2, this linkage is performed. (reservation method), the vertex V11 of the unit cell C12, which is connected from the vertex V11 of the selected unit cell C11 and increases the connection with the other variables X2 to X9, is embedded in the variable X1 in S25. By making a corresponding reservation, the embedding of variables X2 to X9 other than the embedding variable X1 is prohibited (S27).

For this reason, there is room for extending the coupling of the variable X on the hardware graph G2. In addition, since the vertices V to be linked are predetermined based on the embedding method of the complete graph G1a with the highest connectivity density in the problem graph G1, even if the original problem of the optimization problem is a problem with high connectivity density, Even if there is, it is possible to expand and grow the partial problem. For this reason, when solving the entire large-scale problem (original problem) by repeatedly solving the partial problems, even when compared to the method of the first embodiment, when solving the partial problems one time, the variable X is used to solve more vertices V can be embedded in the original problem, reducing the time to solve the original problem. Although an example in which a chimera graph is applied as the hardware graph G2 has been shown, hardware graphs G2 with other structures may be used, and the vertices V to be linked are determined by embedding the complete graph G1a in each hardware graph G2. should be decided based on

(Third Embodiment)
11 to 13 show additional explanatory diagrams of the third embodiment. In the present embodiment, as a result of the computer 3 deriving the correlation of the variables X of the evaluation function H, as shown in FIG. to explain.
Further, the computer 3 and the quantum Ising machine 1 execute the embedding process of the variable X of the first embodiment, the linking process on the hardware graph G2 and the reservation process described in the second embodiment. It is explained as a premise. In this embodiment, the computer 3 executes programs stored in the memory 5 to function as a prohibition unit, a cancellation unit, and a selection unit.

In the initial state, the computer 3 inputs the problem graph G1 (S31), randomly selects the variable X of the problem graph G1 (S32), and embeds it in the hardware graph G2 (S33). At this time, on the hardware graph G2, as described in the second embodiment, the vertex V of another unit cell C corresponding to the vertex V is reserved and secured. For example, when the hardware graph G2 is composed of the chimera graph shown in FIG. 9 or 10, when the variable Xa is embedded in the vertex V11 of the unit cell C11, Vertices V11 of unit cells C12, C13, . As a result, the vertex V11 of the unit cells C12, C13, .

After that, the computer 3 selects the embedded variable Xa in the problem graph G1 (S34), selects the non-embedded variable Xb coupled to this embedded variable Xa (S35), and selects the variable Xb. It is embedded in the hardware graph G2 (S36). FIG. 12 shows candidates for the embedded variable Xa and the variable Xb to be added during processing. As shown in FIG. 12, when the computer 3 selects the variable X on the problem graph G1, the computer 3 selects the variable Xb linked to the embedded variable Xa in S35 and embeds it in the hardware graph G2 in S36. At this time, as shown in S12 to S14 in FIG. 4, the computer 3 does not embed duplicate allocations in the hardware graph G2, and embeds duplicate allocations in the hardware graph G2.

Computer 3 also determines whether or not the embedding process for all variables Xb linked to embedded variable Xa has been completed (S37). The variable Xb is selected in S35 and the variable Xb is embedded in S36 until the variable Xb coupled to Xa disappears. Then, when the variable Xb disappears (YES in S37), the computer 3 releases the prohibition of embedding in the vertex V reserved in association with the embedded variable Xa (S38). That is, in S33, embedding of other variables X than the embedded variable Xa is prohibited for a specific vertex V of the hardware graph G2. It was prohibited to embed another variable X in the vertex V reserved for the embedded variable Xa selected in S34. be able to.

Therefore, the variable Xb or Xc not yet embedded at this point can be embedded in the hardware graph G2 using the vertex V whose reservation was released (canceled) in S38, and the hardware graph G2 can be used efficiently. . Then, the computer 3 repeats these processes of S34 to S38 until the end condition is satisfied. The end condition at this time is that the embedded variable Xa disappears, that the non-embedded variables Xb and Xc disappear, or that a predetermined number of trials or more have been repeated.

A specific example of the processing contents of these S35 to S39 is shown in FIG. As shown in the center of FIG. 13, computer 3 selects one embedded variable Xa and selects variable Xb coupled to this embedded variable Xa. After embedding, the reservation of the vertex V of the hardware graph G2 corresponding to the selected embedded variable Xa is cancelled. Then, as shown on the right side of FIG. 13, the computer 3 selects another embedded variable Xa, and further selects a non-embedded variable Xb coupled to this other embedded variable Xa. As a result, the unembedded variables Xb and Xc can be embedded while canceling the reservation set in the hardware graph G2. You can increase the number of entries.

After that, after the quantum Ising machine 1 executes the optimization process related to the partial problem (S40), the computer 3 updates the variable X based on the processing result of the quantum Ising machine 1 (S41), and sets the termination condition. It is determined whether or not the condition is satisfied (S42). If the end condition is not satisfied, the process returns to S32, and the computer 3 repeats the process from newly selecting the variable X of the problem graph G1. Then, the computer 3 outputs the solution of the variable X when the termination condition of S42 is satisfied (S43). As the end condition of S42, the same condition as the end condition of S5 of the first embodiment may be used, so the explanation is omitted.

<Summary and effects of the present embodiment>
According to this embodiment, the computer 3 selects one embedded variable Xa, and after completing the embedding process for all variables Xb linked to the embedded variable Xa on the problem graph G1, I have canceled the reservation. As a result, the embedding process of the variable Xb can be executed while efficiently resolving the reservation in the hardware graph G2, and the hardware resources can be effectively used by reducing the number of reserved vertices V. FIG. Moreover, more variables Xb and Xc can be embedded in the hardware graph G2 in one partial problem solution derivation process.

In addition, when the computer 3 selects the variable X to be additionally embedded, it selects from among the non-embedded variables Xb linked to the already embedded embedded variable Xa. It eliminates the selection of an independent variable Xc that does not interact with the variable Xa. As a result, the variable X can be embedded in the hardware graph G2 while leaving the structure of the problem graph G1, which is difficult to optimize due to frustration.

(Fourth embodiment)
14 to 18 show additional explanatory diagrams of the fourth embodiment. In this embodiment, a method of embedding variables when expressing a multivalued problem using one-hot display will be described. The description will be omitted, and the description will focus on the parts that differ from the above-described embodiment. The computer 3 has various functions as an invalidation processing unit and a conversion unit as functions realized by executing the programs stored in the memory 5 .

The one-hot display method is known as a representative method for binary representation of a large-scale problem using multivalued variables S ₁ to S _N . Hereinafter, some or all of the multi-valued variables S ₁ to S _N will be referred to as multi-valued variables S _i as required. When multi-valued variables S ₁ to S _N are one-hot displayed using binary variables x ₁ ^(q) to x _N ^(q) _, binary variables x _i It can be realized by providing ( ^q) to x _i ^(q) . However, q=1 to Q. In the following, some or all of the binary variables x _i ^(q) to x _i ^(q) will be referred to as binary variables x _i as required. As a simple example, the evaluation function H ₀ of the one-dimensional Potts model is shown in equation (1), but the evaluation function H ₀ is not limited to this example.

The evaluation function H ₀ in equation (1) is a multi-valued variable S _i of 1 to Q so that the evaluation function H ₀ is the smallest under the constraint that it can take Q states from 1 to Q. It represents an optimization problem in which one state is selected from Q states by a multi-valued variable S _i .
The following equation (2) represents a transformation equation obtained by transforming the evaluation function H ₀ of equation (1) using the binary variable x _i based on the one-hot constraint. FIG. 14 shows a problem graph G1 of an optimization problem related to the evaluation function H ₀ of formula (2). When the optimization problem related to the evaluation function H ₀ is one-hot displayed, there are N×Q binary variables x _i for expressing the evaluation function H ₀ as shown in FIG.

The first term on the right side of the evaluation function H ₀ in equation (2 _{) is binary variables x i (} ^q) _, It is a term that is accumulated when x _j ^(q) satisfies the condition that the hot state value is "1", and indicates the cost part of the multi-valued optimization problem represented by equation (1). where j=i+1 or i−1. In the first term on the right side of this equation (2), when the interaction coefficient J _i of the adjacent binary variables x _i and x _j is positive, the evaluation function H ₀ is lowered to enable further optimization. there is Further, when the interaction coefficient J _i in the first term on the right side of the equation (2) is negative, the sign in the equation (2) may be changed so as to reduce the evaluation function H ₀ . Also, the formula may be changed so that the higher the evaluation function H ₀ is, the more optimized it is.

The one-hot constraint indicates a constraint that only one hot state value "1" appears in each binary variable x _i representing each multi-valued variable S _i , and the second term on the right side of the evaluation function H ₀ is represents this constraint. The second term on the right side of the evaluation function H ₀ is set so that the evaluation function H ₀ is the lowest when only one binary variable x _i among the multi-valued variables S _i is set to the hot state value “1”. Shows additional constraint terms. λ in the second term on the right side of equation (2) is a parameter that represents the ratio of the influence of the first term on the right side and the second term on the right side in equation (2). can be increased, and the constraint condition that only one binary variable x _i has a hot state value of “1” can be strengthened. Conversely, by setting the parameter λ small, the first can increase the effect of the term.

When subproblems of an optimization problem containing one-hot constraints are embedded in the hardware graph G2, the search for the optimal solution must be chosen such that the subproblem contains as many solutions as possible that satisfy the one-hot constraints. cannot be executed. For example, as shown in FIG. 15, when the computer 3 partially selects binary variables x _i that are all set to the cold state value “0” in one multi-valued variable S _i , There is only one solution that satisfies the one-hot constraint as shown. See partial selection variable Sa of binary variable x _i shown in FIG. Therefore, even if the quantum Ising machine 1 executes the optimization process, the optimal solution of the partial problem is fixed to the solution corresponding to the binary variables x _i all of which are set to the cold state value "0". In other words, in this case, it becomes impossible to search for a better solution for the overall binary variable x _i .

Therefore, in a series of processes in which the optimization process of the binary variable x _i is repeated, as shown in _{FIG .} It is desirable to choose the subproblems to necessarily contain binary variables x _i that have the value "1". Computer 3 selects the binary variables x _i as variables of the subproblem to include the binary variables x _i with the hot state value "1" so that the subproblem contains many states that satisfy the one-hot constraint. can be configured as Refer to the partial selection variables Sb1, Sb2, and Sb3 of the binary variable x _i shown in FIG. As a result, subproblem optimization can efficiently search for a more accurate solution that satisfies the one-hot constraint.

A method of embedding the multi-valued variable S _i will be described below with reference to FIGS. 3 and 17. FIG.
First, the computer 3 inputs the problem graph G1 in S1 of FIG. 3, and executes the embedding process of the binary variable x _i in S2. FIG. 17 shows a flowchart of embedding processing of the binary variable x _i corresponding to FIG. 4 of the above-described embodiment. When the computer 3 embeds the binary variables x _i in the hardware graph G2 of the quantum Ising machine 1, it randomly selects one multi-valued variable S _i from the multi-valued variables S ₁ to S _N and selects The binary variable x _i assigned to the multi-valued variable S _i is embedded (S61). Here, an example of selecting the i-th multivalued variable S _i is shown. It should be noted that the computer 3 selects the multi-valued variable S _i in S61, and the multi-valued variable S _j selected after returning to S61 after judging NO in the end condition of S70, which will be described later, is filled in the problem graph G1. The non-embedded multi-valued variable S _j (where j is _i −1 or i+1) adjacent to and connected to the embedded multi-valued variable S i is targeted.

Since there are Q binary variables x _i from 1 to Q in the i-th multi-valued variable S _i , the computer 3 determines the embedding order of the binary variables x _i in the hardware graph G2 in S62. do. The order of priority with which the computer 3 embeds the binary variables x _i in the hardware graph G2 is desirably determined based on the following constraints.
A binary variable x _i to be embedded first in a certain multi-valued variable S _i by the computer 3 is adjacent to and connected to an already embedded binary variable x _j (where j is i−1 or i+1). is preferably a binary variable x _i . The computer 3 gives the highest priority to the binary variable x _i whose hot state value is “1” among the binary variables x _i that satisfy this combination condition, and gives the highest priority to the binary variable x whose cold state value is “0”. You should choose _i as the next priority.

In addition, the computer 3 _gives the highest priority to the binary variable x _{i whose} hot state value is “1” and the cold It is desirable to give the next priority to the binary variables x _i that have a state value of "0". The computer 3 preferentially selects the binary variable x _i that is adjacent to and connected to the embedded binary variable x _j among the binary variables x _i that satisfy the condition of the cold state value “0” as the embedding target. It is preferable to select the binary variable x _i adjacent to and not connected to the embedded binary variable x _j as the next priority. A specific example of priority will be described later.

After the computer 3 selects the binary variables x _i in the order of priority described above in S63, in S64 the computer 3 determines whether or not redundant allocation occurs when embedding the binary variables x _i in the hardware graph G2. When the computer 3 embeds the target binary variable x _i in the hardware graph G2, if duplicate assignment is required, the computer 3 does not embed it in the hardware graph G2 in S65. Conversely, computer 3 embeds in hardware graph G2 in S66 if duplicate allocation is not required. If the computer 3 does not require redundant allocation, it reserves another vertex V of the hardware graph G2 with the vertex V in which the binary variable _xi is embedded as the base point, as described in the above embodiment.

Here, the computer 3 determines not to embed the target binary variable x _i in S65 if duplicate assignment is required when embedding the target binary variable x _i in the hardware graph G2. If the status value is "1", it is preferable to invalidate, ie cancel, the embedding of the selected multi-valued variable S _i in S69, and proceed to the embedding process of the next multi-valued variable S _j . That is, if a binary variable x _i with a hot state value of "1" cannot be embedded in the hardware graph G2, only a binary variable x _i with a cold state value of "0" can be embedded. As a result, the multivalued variable S _i whose subproblem contains only one state that satisfies the one-hot constraint can be excluded from the subproblem.

Then, in S67, the computer 3 determines whether or not the embedding process for all the binary variables x _i assigned to the multi-valued variable S _i selected in S61 has been completed _. The processes of S63 to S66 are repeated until the embedding process of .

When the computer 3 finishes embedding all the binary variables x _i assigned to the selected multi-valued variable S _i , when two or more of all the binary variables x _i can be embedded, , the process proceeds directly to S70 to continue the process. However, when the computer 3 can only embed one or less binary variables x _i among the selected multi-valued variables S _i , i.e., binary variables x _i indicating a hot state value of "1" or a cold state value of "0". , invalidates, that is, cancels the embedding of all the multi-valued variables S _i selected in S69. At this time, the computer 3 removes the already-embedded binary variables x _i from the vertices V of the hardware graph G2, and cancels the reservation of the other vertices V, so that the hardware resources of the quantum Ising machine 1 are cancel the reservation of As a result, the multi-valued variables S _i that do not satisfy the one-hot constraint can be excluded from the variables representing the subproblems.

The computer 3 determines in S70 whether or not the end condition is satisfied, and sets another multi-valued variable S _j (where j is i−1 or i+1) adjacent to or connected to the multi-valued variable S _i until the end condition is satisfied. Repeat the process from where you choose. When the computer 3 can no longer connect the binary variable x _j representing the multi-valued variable S _j to the binary variable x _i of the previously embedded multi-valued variable S _i on the hardware graph G2, S70 , the embedding of other multi-valued variables S _j is terminated.

After the computer 3 completes the embedding process for all multi-valued variables S _i , the quantum Ising machine 1 executes the optimization process at S3 in FIG. The quantum Ising machine ₁ optimizes the subproblems embedded in the optimization process using the gradient method or other optimization techniques, and uses the binary variables x _i to find the value of At this time, the quantum Ising machine 1 determines and updates the optimum value of the binary variable x _i under the condition that a predetermined time has passed since the start of processing or that the trial has been repeated a predetermined number of times or more (S4). In this case, it is preferable to obtain the evaluation value of the evaluation function H ₀ using a fixed value such as an optimal solution or an initial value obtained so far as another non-embeddable binary variable x _i . This allows us to solve the subproblem.

Thereafter, the computer 3 returns to S2, selects the multi-valued variable S _i again in S2, and embeds the binary variable x _i in the selected multi-valued variable S _i in the hardware graph G2. The Ising machine 1 executes optimization processing using these binary variables x _i , determines the optimum value of the combination of the binary variables x _i , and updates these binary variables x _i in S4. For the binary variable x _i not selected at this time, it is preferable to use the optimum value of the binary variable x _i obtained as the optimum value before the processing. Binary variables x _i in multi-valued variables S _i that have never been selected should be set to fixed values for processing.

Then, the quantum Ising machine 1 judges that the termination condition is satisfied (YES in S5) under the condition that a predetermined time has passed since the start of processing or that the trial has been repeated a predetermined number of times or more, and these multi-valued variables S _i or the evaluation value of the evaluation function H ₀ is output. This allows us to solve the overall optimization problem. By performing iterative processing in this way, the original problem can be divided into subproblems and solved.

<Specific example of embedding method in step S62>
A specific example of step S62 when embedding the binary variable x _i in the hardware graph G2 will be described with reference to FIG. A simple example with Q=4 is shown here. Although the computer 3 embeds the binary variables x ₁ ⁽¹⁾ and x ₁ ⁽²⁾ out of the multi-valued variables S ₁ in the hardware graph G2 of the quantum Ising machine 1, the other binary variables x ₁ ^{(3 )} , x ₁ ⁽⁴⁾ are embedded in the hardware graph G2, duplicate assignments occur and the embedding is disabled. In FIG. 18, the binary variables x ₁ ⁽¹⁾ and x ₁ ⁽²⁾ already embedded in the vertex V of the hardware graph G2 are denoted by "Z1".

It is then assumed that the embedded binary variable x ₁ ⁽²⁾ has a hot state value of "1". In FIG. 18, the computer 3 computes the binary variables x ₂ ⁽¹⁾ , x ₂ ( ₂ ⁾ , x ₂ ⁽³⁾ in the multi-valued variable S ₂ adjacently connected to the multi-valued variable S 1 . , x ₂ ⁽⁴⁾ are shown as priorities "1st" to "4th".

As described in S62 above, the _binary variable x2 to be embedded _first is preferably the _binary variable x2 that is linked to the embedded binary variable x1. Among the _binary variables x2, it is desirable to give the highest priority to the _binary variable x2 having a hot state value of "1". In the case of the example shown in FIG. 18, the binary variables x ₂ ⁽¹⁾ and x ₂ ⁽²⁾ that are linked to the embedded binary variable x ₁ are both cold state values “0”. Either of the value variables x ₂ ⁽¹⁾ and x ₂ ⁽²⁾ may be selected, but here an example of randomly selecting the binary variable x ₂ ⁽¹⁾ is shown.

As for the binary variables x _i to be embedded secondly and thereafter, the binary variables x _i with the hot state value “1” are given the highest priority, and the binary variables x _i with the cold state value “0” are given the highest priority. should be the next priority. In the example shown in FIG. 18, since the binary variable x ₂ ⁽⁴⁾ has the hot state value “1”, the binary variable x ₂ ⁽ 4) with the hot state value “1” is used as the second embedded variable. set to target. Further, the computer 3 preferentially selects the binary variable x _i linked to the embedded binary variable x _j among the binary variables x _i satisfying the condition of the cold state value “0” as the embedding target. . Therefore, in the example shown in FIG. 18, the binary variable x ₂ ⁽²⁾ is the third embedding target. The computer 3 also selects the binary variable x _i that is not linked to the embedded binary variable x _j as the next priority. In the example shown in FIG. 18, the binary variable x ₂ ⁽³⁾ is the final embedding target. It is desirable to embed the binary variables x _i in such order.

<Summary and effects of the present embodiment>
According to this embodiment, when solving a large-scale optimization problem in which an evaluation function H ₀ defined by a multi-valued variable S _i is expressed using a binary variable x _i by a one-hot constraint, the subproblem variable When _selecting to include at least some of the binary variables x _i in the multi-valued variables S _i as are selected and embedded in the hardware graph G2. As a result, each multi-valued variable S _i includes a state that satisfies a plurality of one-hot constraints in the subproblem, making it possible to efficiently search for a more accurate solution.

Further, when the computer ₃ additionally embeds the _binary variable x2 of the multi-valued variable S2 that is combined on the problem graph with the already embedded multi _- valued variable S1, the computer ₃ expresses the multi-valued variable S2. Among the binary variables x ₂ , the binary variable x ₂ ⁽¹⁾ connected to the embedded binary variable x ₁ ⁽¹⁾ on the problem graph G1 is preferentially embedded. Then, a subproblem can be constructed so as to include as many interactions as possible between the multi-valued variables S _i and S _j that are connected on the problem graph G1.

Further, when the computer ₃ additionally embeds the _binary variable x2 of the multi-valued variable S2 that is combined on the problem graph with the already embedded multi _- valued variable S1, the computer ₃ expresses the multi-valued variable S2. Among the binary variables x ₂ , the binary variable x ₂ ⁽⁴⁾ that satisfies the hot state value “1” is preferentially embedded. Then, the binary variable x ₂ ⁽⁴⁾ having a hot state value of "1" can be preferentially embedded in the hardware graph G2.

When the computer 3 determines that the binary variable x _i with the hot state value “1” cannot be embedded, the computer 3 embeds the entire multi-valued variable S _i represented by the binary variable x _i . is invalidated, and the previously embedded binary variable x _i is removed from the vertex V of the hardware graph G2. As a result, the multivalued variable S _i having only one state satisfying the one-hot constraint can be excluded from the subproblem. Moreover, the reservation of the hardware resource of the quantum Ising machine 1 can be cancelled.

Further, when the computer 3 determines that two or more of all the binary variables x _i of the selected multi-valued variables S _i cannot be embedded, the computer 3 _determines that the multi-valued variables S We invalidate the entire embedding of _i and remove the previously embedded binary variable x _i from the vertex V of the hardware graph G2. A multi-valued variable S i in which both a hot state value of “1” and a cold state value of “0” cannot be embedded can be excluded from the subproblem, and a multi-valued variable S _i that has only one state that satisfies the one-hot constraint in the subproblem. The value variables S _i can be eliminated from the subproblem.

(Fifth embodiment)
19 and 20 show additional explanatory diagrams of the fifth embodiment. Although the evaluation function H ₀ may be configured as in Equation (2) using the parameter λ, defining the ratio of the influence of the second term on the right side of Equation (2) as a variable using the parameter λ is a quantum Ising machine 1 makes it more likely that solutions that do not satisfy the one-hot constraint will also be derived as solutions to the overall optimization problem. Moreover, since the parameter λ is adjusted to the optimum value, there is a possibility that the amount of calculation processing will increase.

When such a situation is assumed, before the computer 3 implements the partial problem in the quantum Ising machine 1, the multi-valued state "1" to "Q" of the multi-valued variable S _i is selected from two options 2 It is better to convert it to a choice optimization problem. By having the computer 3 perform conversion processing to a binary optimization problem in advance, it is possible to implement a partial problem in the quantum Ising machine 1 excluding solutions for the multi-valued variables S _i that do not satisfy the one-hot constraint. In this case, the constraint corresponding to the second term on the right side of formula (2) can be eliminated, and the parameter λ in the second term on the right side of formula (2) need not be defined as a variable. The computer 3 and the quantum Ising machine 1 can obtain the optimum solution for the binary variable x _i by executing the optimization process of the cost part corresponding to the first term on the right side of the equation (2), and the calculation processing amount is can be significantly reduced.

For example, the computer 3 uses the procedure shown in FIG. 19 to maintain the current multi-valued states “1” to “Q” for all multi-valued variables S _i , or maintain the current multi-valued states “1” to “Q”. , is converted into a binary optimization problem that selects , and then the subproblems related to the binary optimization problem are implemented in the quantum Ising machine 1 to execute the optimization process.

As shown in FIG. 19, the computer 3 sequentially selects the multivalued variables S ₁ to S _N in S51, and selects the transition destination candidate of the hot state value "1" among the multivalued variables S _i selected in S52. randomly select. The computer 3 repeats the processing of steps S51 to S53 for all the multivalued variables S ₁ to S _N , so that, as shown in S54, the current multivalued states of the multivalued variable S _i to It can be converted into a two-choice optimization problem in which "Q" is maintained or the current multi-valued state "1" to "Q" is transitioned.

A specific example is shown in FIG. Assuming that the multi-valued variable S _i is set to the multi-valued state "1" at the time of executing the processing shown in FIG. Randomly select one of If the computer 3 selects the multi-valued state "3" as the transition destination, the multi-valued variable S _i is set to either the current multi-valued state "1" or the transition destination multi-valued state "3". The decision is entrusted to the quantum Ising machine 1 . As a result, the multi-valued variable S _i can be converted into a two-choice optimization problem in which the multi-valued variable S i maintains the current multi-valued state "1" or transitions to the multi-valued state "3". This binary optimization problem is defined by a binary variable y _i , eg a binary variable y _i =0 represents maintenance and a binary variable y _i =1 represents transition.

Further, if the computer 3 recognizes that the evaluation function H ₀ is more optimized when the multi-valued variables S _i and S _j connected on the problem graph G1 are set to the same multi-valued state, then At least one of the current multi-valued state "1" and the multi-valued state "3" of the transition destination of the multi-valued variable S _i is combined with one multi-valued variable S _i of another multi-valued variable S _j It is desirable to construct the binary optimization problem so that the state is the same as at least one of the current multi-valued state and the multi-valued state of the transition destination. Then, when the quantum Ising machine 1 optimizes the binary optimization problem, it is possible to efficiently obtain the binary variable x _i that optimizes the evaluation function H ₀ .

In other words, using the binary state values of the binary variables x _i instead of the expression of the multi-valued variables S _i , the binary variables _{x j} The binary variable x _i is either the current binary state value (hot state value '1' or cold state value '0') or this so that the binary optimization problem contains states that result in the same binary state value is the same as at least one of the current binary state value of the adjacently coupled binary variable x _j or the binary state value of the transition destination different from this It is better to select so that

Conversely, when the computer 3 recognizes that the evaluation function H ₀ becomes smaller when the adjacent multi-valued variables S _i and S _j connected on the problem graph G1 are set to different multi-valued states, Either or both of the current multi-valued state “1” and the transition destination multi-valued state “3” of one multi-valued variable S _i is the current multi-valued state and transition of another adjacent multi-valued variable S _j . It is desirable to construct the binary optimization problem such that the multi-valued state is different from both of the previous multi-valued states. Then, when the quantum Ising machine 1 optimizes the binary optimization problem, the binary variable x _i that satisfies the conditions for optimizing the evaluation function H ₀ can be obtained with high accuracy.

In other words, the _binary state value of the binary variable x _i is different from the binary variable _{x j connected} on the problem graph G1. The binary variable x _i is either the current binary state value (hot state value '1' or cold state value '0') or a different transition destination, such that the binary state value is included in the binary optimization problem. is different from both the current binary state value and the transition destination binary state value of the adjacently connected binary variable x _j .

The computer 3 implements a subproblem of the binary optimization problem defined by the binary variable y _i by embedding it in the hardware graph G2. Quantum Ising machine 1 then optimizes subproblems of the embedded binary optimization problem to determine binary variables y _i . The computer 3 updates the binary variable x _i by referring to the output of the binary variable y _i by the quantum Ising machine 1 that minimizes the subproblem, the current binary state of each variable, and the binary state of the transition destination. do. By generating a binary optimization problem using the computer 3 and the Ising machine 1 and repeating update processing of the binary variables x _i , it is possible to obtain the optimum solution for the entire binary variables x _i .

When optimizing the multi-valued variable S _i of the multi-valued problem, the computer 3 converts the multi-valued state "1" to "Q" by the multi-valued variable S _i into a two-choice optimization problem. , partially embeds the binary variables y _i of the binary optimization problem into the vertices V of the hardware graph G2. By the computer 3 executing the conversion process to the binary optimization problem in advance, it is possible to construct a partial problem excluding the solutions of the multi-valued variables S _i that do not satisfy the one-hot constraint. can be significantly reduced. Furthermore, since the subproblem only contains states that satisfy the one-hot constraint, there is no need to adjust the parameter λ that determines the strength of the one-hot constraint. In particular, whether to maintain the current multi-valued state “1” to “Q” (binary variable y _i =0) or transition the current multi-valued state “1” (binary variable y _i =1), It is desirable to transform it into a binary optimization problem that selects .

Further, when the evaluation function H ₀ is more optimized when the multi-valued variables S _i and S _j that are adjacently connected on the problem graph G1 are in the same multi-valued state, one multi-valued variable S _i At least one of the current multi-valued state and transition destination multi-valued state of S j is the same as the current multi-valued state or transition destination multi-valued state of another adjacent multi-valued variable S _j It is desirable to construct a binary optimization problem. As a result, the evaluation function H ₀ can be further optimized when the quantum Ising machine 1 optimizes the binary optimization problem.

Further, when the evaluation function H ₀ is further optimized when the multi-valued variables S _i and S _j that are adjacently connected on the problem graph G1 are set to different multi-valued states, one multi-valued variable S _i Either or both of the current multi-valued state and the transition destination multi-valued state are different from both the current multi-valued state and the transition destination multi-valued state of another adjacent multi-valued variable S _j It is desirable to construct the binary optimization problem as follows. As a result, the evaluation function H ₀ can be further optimized when the quantum Ising machine 1 optimizes the binary optimization problem.

(Other embodiments)
The present disclosure is not limited to the above-described embodiments, and can be modified or expanded, for example, as follows. Although the hardware graph G2 has been described using a chimera graph, it is not limited to this.

In the fourth embodiment, a one-hot constraint in which only one hot state value "1" (equivalent to the first value) is provided has been described as an example. equivalent) may be applied to a one-cold constraint provided with only one. In the fourth embodiment, the formula (2) is exemplified as the evaluation formula of the evaluation function H ₀ , but when one-cold display is applied, the formula in the second term is converted as in the following formula (3) good to do

Any binary variable x _i can be used as long as the binary variable x _i can be represented such that the first value is different from all other second values. When the multi-valued variable S _i is displayed in one-hot mode, the hot state value "1" corresponds to the first value and the cold state value "0" corresponds to the second value. When the multi-valued variable S _i is displayed one-cold, the cold state value "0" corresponds to the first value and the hot state value "1" corresponds to the second value.

It is also possible to combine the configurations and processing contents of the above-described embodiments. In addition, the reference numerals in parentheses in the claims indicate the corresponding relationship with the specific means described in the embodiment described above as one aspect of the present invention, and the technical scope of the present invention is defined by It is not limited. A mode in which part of the above embodiment is omitted as long as the problem can be solved can also be regarded as an embodiment. In addition, all conceivable aspects can be regarded as embodiments as long as they do not deviate from the essence of the invention specified by the language in the claims.

Moreover, although the present invention has been described in accordance with the above-described embodiments, it is understood that the present invention is not limited to such embodiments or structures. The present invention includes various modifications and modifications within the equivalent range. In addition, various combinations and configurations, as well as other combinations and configurations including one, more, or less elements thereof, are within the scope and spirit of this disclosure.

In the drawing, 1 is a quantum Ising machine (dedicated hardware), 3 is a computer (reservation unit, prohibition unit, cancellation unit, selection unit, invalidation processing unit, conversion unit), 7 is a determination unit, 8 is an embedding unit, V, V11-14 and V21-V24 indicate vertices.

Claims (24)

A hardware graph (G2) representing the interaction between vertices (V) is constructed by a specific fixed architecture, using dedicated hardware (1) for the optimization problem to embed all the variables of the optimization problem. In the case of solving a large-scale problem that cannot be embedded, when the interaction of the variables of the optimization problem is expressed in a problem graph (G1) and solved, the variables of the problem graph are converted to the hardware graph of the dedicated hardware A variable embedding method used when solving the large-scale problem by dividing it into subproblems that can be embedded in vertices and repeating optimization processing of the subproblems,
when embedding at least a portion of all the variables into the vertices of the hardware graph;
a step of determining (S12) whether the variables of the optimization problem require redundant assignment to the vertices of the hardware graph;
a step of selecting the variables that do not require overlapping assignment and embedding them in the vertices of the hardware graph without using the variables that require overlapping assignment as variables of the partial problem (S13);
A variable embedding method with a step (S21) of linking the vertices of the hardware graph based on an embedding method when embedding the complete graph (G1a) in the hardware graph;
selecting at least one or more of the vertices from subgraphs connected on the hardware graph in which the variable is embedded when embedding the variable on the hardware graph (S26); When,
When embedding the variables (X1, Xa) in the hardware graph, based on the contents of the linkage, combine from the selected vertices and combine with the other variables (X2 to X9, Xb, Xc) embedding the other variables (X2 to X9, X) other than the embedding variable (X1, Xa) by reserving the vertex that increases the coupling of the embedding variable (X1, Xa) 2. The variable embedding method according to claim 1, further comprising a step of prohibiting (S27). After executing the process of prohibiting the embedding of the other variables (X), the embedding process of all the variables (Xb) connected to the embedded variables (Xa) on the problem graph is completed. 3. The variable embedding method according to claim 2, further comprising a step of canceling said reservation later (S37, S38). When the hardware graph is composed of a chimera graph including unit cells (C, C11 . Corresponding to the other unit cells (C12, C13, . . . , C31, C32, C34, . 4. The variable embedding method according to claim 2 or 3, further comprising a step (S22) of reserving the vertices obtained as second vertices (V11, V13). When selecting the variables to be added and embedded in the problem graph,
5. A variable embedding method according to any one of claims 1 to 4, wherein the variable is selected from among the non-embedded variables linked to the embedded variables that have already been embedded. When selecting the variable to be added and embedded in the problem graph and embedding it in the hardware graph,
Select one of the embedded variables (Xa) already embedded (S34), select the variable (Xb) linked to the selected embedded variable, and complete the embedding process ( S35 to S37), selecting another embedded variable (S34), and repeating the embedding process of the variable (Xb) linked to the selected embedded variable (Xa). The variable embedding method according to any one of . The large-scale problem is constrained such that only one first value is different from all other second values, such as a one-hot constraint or a one-cold constraint It is a multi-valued problem applying multi-valued variables (S ₁ : S _N ) represented by binary variables (x ₁ ⁽¹⁾ to x ₁ ^(Q) : x _N ⁽¹⁾ to x _N ^(Q) ). ,
the binary variable set as the first value when selecting to include at least part of the binary variables from the multi-valued variables as the variables of the subproblem and embedding them in the vertices of the hardware graph 7. The variable embedding method according to any one of claims 1 to 6, wherein the binary variables are selected so as to be included in the variables and embedded in the vertices of the hardware graph. A binary variable (x ₂ ) of the multi-valued variable (S ₂ ) that is linked on the problem graph to the multi-valued variable (S ₁ ) already embedded can be added to the vertices of the hardware graph. When embedding
Among the binary variables expressing the multivalued variable, the binary variable linked to the embedded binary variable on the problem graph, or the condition that is the first value 8. The variable embedding method according to claim 7, wherein the binary variable satisfying the following is preferentially embedded. If it is determined that the binary variable set as the first value cannot be embedded (NO in S65a), or if two or more of all the binary variables representing the selected multivalued variable are embedded If it is determined that the embedding is not possible (NO in S68), the embedding of the entire multi-valued variable represented by the binary variable is invalidated, and the previously embedded binary variable is removed from the partial problem. (S69) The variable embedding method according to claim 7 or 8. The large-scale problem is constrained such that only one first value is different from all other second values, such as a one-hot constraint or a one-cold constraint It is a multi-valued problem applying multi-valued variables (S ₁ : S _N ) represented by binary variables (x ₁ ⁽¹⁾ to x ₁ ^(Q) : x _N ⁽¹⁾ to x _N ^(Q) ). ,
When optimizing the multi-valued variable (S _i ) of the multi-valued problem, the multi-valued state of the multi-valued variable is converted into a binary optimization problem in which a binary variable (y _i ) is used to select two alternatives. After doing (S51 to S54),
A variable embedding method for embedding the binary variable as the variable at the vertex of the hardware graph using the variable embedding method according to any one of claims 1 to 6. The two options of maintaining the current multi-valued state for each multi-valued variable of the multi-valued problem, or transitioning to another multi-valued state randomly selected for each multi-valued variable (S51, S52) 11. The variable embedding method according to claim 10, wherein the problem is converted into a binary optimization problem. When the evaluation function for evaluating the multi-valued problem is optimized when the multi-valued variables (S _i , S _j ) connected on the problem graph have the same value, one multi-valued variable (S _i ) at least one of the current multi-valued state and the other multi-valued state of the transition destination is combined with the one multi-valued variable (S _i ) _. Construct the binary optimization problem so that the state is the same as at least one of the multi-valued state of or the multi-valued state of the transition destination,
When the evaluation function is optimized when the multi-valued variables connected on the problem graph have different values, the current multi-valued state of one multi-valued variable or the other multi-valued state of the transition destination Either or both of the above-mentioned binary optimization problems are different from both the current multi-valued state and the transition destination multi-valued state of the other multi-valued variable coupled with the one multi-valued variable. 12. The variable embedding method according to claim 11, wherein A hardware graph (G2) representing the interaction between vertices (V) is constructed by a specific fixed architecture, using dedicated hardware (1) for the optimization problem to embed all the variables of the optimization problem. In the case of solving a large-scale problem that cannot be embedded, when the interaction of the variables of the optimization problem is expressed in a problem graph (G1) and solved, the variables of the problem graph are converted to the hardware graph of the dedicated hardware A processing system for embedding variables used when solving the large-scale problem by dividing the problem into subproblems that can be embedded in vertices and repeating optimization processing of the subproblems,
when embedding at least a portion of all the variables into the vertices of the hardware graph;
a determination unit (7, S12) that determines whether or not it is necessary to duplicately assign variables of the optimization problem to vertices of the hardware graph;
An embedding unit (8, S13) that selects the variables that do not require overlapping assignment and embeds them in vertices of the hardware graph without using the variables that require overlapping assignment as variables of the partial problem. and a processing system comprising: a linking unit (S21) that links vertices of the hardware graph based on an embedding method when embedding the complete graph (G1a) in the hardware graph;
When embedding the variable in the hardware graph, the selection unit (S26 )When,
When embedding the variables (X1, Xa) in the hardware graph, based on the contents of the linkage, combine from the selected vertices and combine with the other variables (X2 to X9, Xb, Xc) embedding the other variables (X2 to X9, X) other than the embedding variable (X1, Xa) by reserving the vertex that increases the coupling of the embedding variable (X1, Xa) a prohibition unit (S27) for prohibiting
14. The processing system of claim 13, further comprising: After executing the process of prohibiting the embedding of the other variables (X), the reservation is performed after the embedding process of all the variables (Xb) connected to the embedded variables (Xa) on the problem graph is completed. Resolving unit (S37, S38) for resolving
15. The processing system of claim 14, further comprising: When the hardware graph is composed of a chimera graph including unit cells (C, C11 . Corresponding to the other unit cells (C12, C13, . . . , C31, C32, C34, . 16. The processing system according to claim 14 or 15, further comprising a reservation unit (S22) for reserving the vertices obtained as second vertices (V11, V13). When selecting the variables to be added and embedded in the problem graph,
17. A processing system according to any one of claims 13 to 16, wherein it selects among the non-embedded variables linked to the already embedded embedded variables. When selecting the variable to be added and embedded in the problem graph and embedding it in the hardware graph,
Select one of the embedded variables (Xa) already embedded (S34), select the variable (Xb) linked to the selected embedded variable, and complete the embedding process ( S35 to S37), selecting another embedded variable (S34), and repeating the embedding process of the variable (Xb) linked to the selected embedded variable (Xa). A processing system according to any one of Claims 1 to 3. The large-scale problem is a binary variable (x ₁ ⁽¹⁾ ∼ x ₁ ^(Q) : x _N ⁽¹⁾ to x _N ^(Q) ) is a multi-valued problem to which a multi-valued variable (S ₁ : S _N ) is applied,
When the embedding unit selects to include at least some of the binary variables among the multi-valued variables as the variables of the subproblem and embeds them in the vertices of the hardware graph, the embedding unit includes the first value and 19. The processing system according to any one of claims 13 to 18, wherein the binary variables are selected and embedded in the vertices of the hardware graph so as to include the binary variables that are set. The embedding part is
A binary variable (x ₂ ) of the multi-valued variable (S ₂ ) that is linked on the problem graph to the multi-valued variable (S ₁ ) already embedded can be added to the vertices of the hardware graph. When embedding
Among the binary variables expressing the multivalued variable, the binary variable linked to the embedded binary variable on the problem graph, or the condition that is the first value 20. The processing system of claim 19, wherein said binary variables that satisfy: If it is determined that the binary variable set as the first value cannot be embedded (NO in S65a), or if two or more of all the binary variables representing the selected multivalued variable are embedded If it is determined that the embedding is not possible (NO in S68), embedding by the embedding unit of the entire multi-valued variable represented by the binary variable is invalidated, and the previously embedded binary variable is replaced. 21. The processing system according to claim 19 or 20, further comprising an invalidation processing unit (S69) for removing from said partial problem. The large-scale problem is constrained such that only one first value is different from all other second values, such as a one-hot constraint or a one-cold constraint It is a multi-valued problem applying multi-valued variables (S ₁ : S _N ) represented by binary variables (x ₁ ⁽¹⁾ to x ₁ ^(Q) : x _N ⁽¹⁾ to x _N ^(Q) ). ,
When optimizing the multi-valued variable (S _i ) of the multi-valued problem, the multi-valued state of the multi-valued variable is converted into a binary optimization problem in which a binary variable (y _i ) is used to select two alternatives. A conversion unit (S51 to S54) for
After the transforming unit transforms the multi-valued problem into the binary optimization problem, the embedding unit uses the processing system according to any one of claims 13 to 18 to add the vertices of the hardware graph to the A processing system that embeds a binary variable as said variable. The two options of maintaining the current multi-valued state for each multi-valued variable of the multi-valued problem, or transitioning to another multi-valued state randomly selected for each multi-valued variable (S51, S52) 23. The processing system of claim 22, which converts to a binary optimization problem. When the evaluation function for evaluating the multi-valued problem is optimized when the multi-valued variables connected on the problem graph have the same value, the current multi-valued variable (S _i ) of the one multi-valued variable (S i ) At least one of the state and the other multi-valued state of the transition destination is the current multi-valued state or the multi-valued state of the transition destination of another multi-valued variable (S _j ) that is combined with the one multi-valued variable Construct the binary optimization problem so that it is in the same state as at least one or more of
When the evaluation function is optimized when the multi-valued variables connected on the problem graph have different values, the current multi-valued state of one multi-valued variable or the other multi-valued state of the transition destination Either or both of the above-mentioned binary optimization problems are different from both the current multi-valued state and the transition destination multi-valued state of the other multi-valued variable coupled with the one multi-valued variable. 24. The processing system of claim 23, wherein the processing system comprises:

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