Association determination device, control device, quantum annealing system, association determination method, control method, and recording medium

Abstract

An association determination device according to the present invention comprises an association determination means for determining a plurality of associations associating a graph, including a plurality of nodes and a plurality of edges, with an annealing machine including a plurality of quantum bit devices and a plurality of couplers each coupling the same number of quantum bit devices.

Description

Association determination device, control device, quantum annealing system, association determination method, control method, and recording medium

[0001] The present invention relates to an association determination device, a control device, a quantum annealing system, an association determination method, a control method, and a recording medium.

[0002] One method for searching for a solution to a combinatorial optimization problem is quantum annealing using the LHZ method, for example (see, for example, Patent Document 1).

[0003] Japanese Patent Application Publication No. 2022-076900

[0004] When performing quantum annealing, it is preferable that the accuracy of the solution obtained is as high (good) as possible. To this end, it is preferable to be able to avoid conditions that result in particularly poor accuracy of the solution obtained.

[0005] An example of an object of the present invention is to provide an association determination device, a control device, a quantum annealing system, an association determination method, a control method, and a recording medium that can solve the above-mentioned problems.

[0006] According to a first aspect of the present invention, an association determination apparatus includes an association determination means for determining a plurality of associations for associating a graph including a plurality of nodes and a plurality of edges with an annealing machine including a plurality of quantum bit devices and a plurality of couplers, each of which couples the same number of quantum bit devices.

[0007] According to a second aspect of the present invention, a control device includes: an association determination means for determining a plurality of associations for associating a graph including a plurality of nodes and a plurality of edges with an annealing machine including a plurality of quantum bit devices and a plurality of couplers, each of which couples the same number of quantum bit devices; and a control means for controlling the annealing machine including the quantum bit devices and the couplers in accordance with the determined associations to perform quantum annealing.

[0008] According to a third aspect of the present invention, a quantum annealing system comprises an annealing machine and a control device, wherein the annealing machine comprises a plurality of quantum bit devices and a plurality of couplers each coupling the same number of quantum bit devices, and the control device comprises association determination means for determining a plurality of associations for associating a graph including a plurality of nodes and a plurality of edges with the annealing machine, and control means for controlling the annealing machine including the quantum bit devices and the couplers in accordance with the determined associations to perform quantum annealing.

[0009] According to a fourth aspect of the present invention, a method for determining correspondences includes a computer determining multiple correspondences for a graph including a plurality of nodes and a plurality of edges to an annealing machine including a plurality of quantum bit devices and a plurality of couplers, each of which couples an equal number of quantum bit devices.

[0010] According to a fifth aspect of the present invention, a control method includes a computer determining multiple associations for associating a graph including a plurality of nodes and a plurality of edges with an annealing machine including a plurality of quantum bit devices and a plurality of couplers, each of which couples the same number of quantum bit devices, and controlling the annealing machine including the quantum bit devices and the couplers in accordance with the determined associations to perform quantum annealing.

[0011] According to a sixth aspect of the present invention, a recording medium is a recording medium storing a program that causes a computer to determine multiple associations that associate a graph including a plurality of nodes and a plurality of edges with an annealing machine that includes a plurality of quantum bit devices and a plurality of couplers that each couple the same number of quantum bit devices.

[0012] According to a seventh aspect of the present invention, a recording medium is a recording medium having recorded thereon a program that causes a computer to determine multiple associations that associate a graph including a plurality of nodes and a plurality of edges with an annealing machine that includes a plurality of quantum bit devices and a plurality of couplers that each couple the same number of quantum bit devices, and to control the annealing machine that includes the quantum bit devices and the couplers in accordance with the determined associations to perform quantum annealing.

[0013] According to the present invention, it is expected that conditions under which the accuracy of the obtained solution becomes particularly poor can be avoided when performing quantum annealing.

[0014] 1 is a diagram illustrating an example of the configuration of a quantum annealing system according to at least one embodiment; 2 is a diagram illustrating an example of the configuration of an annealing machine according to at least one embodiment; 3 is a diagram illustrating an example of the configuration of a control device according to at least one embodiment; 4 is a diagram illustrating a first example of numbering of nodes in a complete graph; 5 is a diagram illustrating an example of mapping a graph to an annealing machine using the LHZ method; 6 is a diagram illustrating a second example of numbering of nodes in a complete graph; 7 is a diagram illustrating a first example of correlation between nodes in a graph; 8 is a diagram illustrating a first example of mapping the correlation between nodes in a graph to quantum bit devices; 9 is a diagram illustrating an example of mapping a graph to an annealing machine and a result of quantum annealing; 10 is a diagram illustrating a second example of correlation between nodes in a graph; 11 is a diagram illustrating a second example of mapping the correlation between nodes in a graph to quantum bit devices; 12 is a diagram illustrating a third example of mapping the correlation between nodes in a graph to quantum bit devices; 13 is a diagram illustrating a fourth example of mapping the correlation between nodes in a graph to quantum bit devices; 14 is a diagram illustrating a fifth example of mapping the correlation between nodes in a graph to quantum bit devices; 15 is a diagram illustrating a sixth example of mapping the correlation between nodes in a graph to quantum bit devices. FIG. 1 is a diagram showing an example of the relationship between the association of a graph with an annealing machine and parameter values, and the result of quantum annealing. FIG. 2 is a diagram showing an example of the procedure of processing performed by a control device according to at least one embodiment. FIG. 3 is a diagram showing an example of the configuration of an association determination device according to at least one embodiment. FIG. 4 is a diagram showing an example of the configuration of a control device according to at least one embodiment. FIG. 5 is a diagram showing an example of the configuration of a quantum annealing system according to at least one embodiment. FIG. 6 is a diagram showing an example of the procedure of processing in an association determination method according to at least one embodiment. FIG. 7 is a schematic block diagram showing the configuration of a computer according to at least one embodiment.

[0015] The following describes embodiments of the present invention, but the following embodiments do not limit the scope of the invention as claimed. Furthermore, not all of the combinations of features described in the embodiments are necessarily essential to the solution of the invention.

[0016] 1 is a diagram showing an example of the configuration of a quantum annealing system according to at least one embodiment. In the configuration shown in Fig. 1, the quantum annealing system 1 includes an annealing machine 100 and a control device 200.

[0017] The annealing machine 100 performs quantum annealing under the control of the control device 200. Fig. 2 is a diagram showing an example of the configuration of the annealing machine 100. In the configuration shown in Fig. 2, the annealing machine 100 includes a quantum bit device 110 and a four-body coupler 120.

[0018] The quantum bit device 110 is an element for expressing a quantum bit. The quantum bits included in the annealing machine 100 are not limited to a specific type. For example, the quantum bit device 110 may be configured using a Josephson Parametric Oscillator (JPO), but is not limited thereto.

[0019] The four-body coupler 120 couples four quantum bit devices 110. The coupling of quantum bit devices is also referred to as quantum bit device interaction. However, the coupler included in the annealing machine 100 is not limited to a four-body coupler. The annealing machine 100 can be configured in various ways, including multiple couplers that couple the same number of quantum bit devices 110.

[0020] The control device 200 controls the annealing machine 100 to perform quantum annealing. In particular, the control device 200 determines multiple associations that associate a graph representing a combinatorial optimization problem to be solved by quantum annealing with the annealing machine 100. The control device 200 then controls the annealing machine 100 to perform quantum annealing for each of the multiple determined associations, and evaluates each association based on the execution results. The control device 200 may be configured using a computer.

[0021] The control device 200 is an example of an association determination device. The device that determines the association between the combinatorial optimization problem and the annealing machine 100 and the device that controls the annealing machine 100 in accordance with the determined association may be configured as separate devices.

[0022] Fig. 3 is a diagram showing an example of the configuration of the control device 200. In the configuration shown in Fig. 3, the control device 200 includes a communication unit 210, a display unit 220, an operation input unit 230, a storage unit 280, and a processing unit 290. The processing unit 290 includes a graph acquisition unit 291, an association unit 292, and a control unit 293.

[0023] The communication unit 210 communicates with other devices such as the annealing machine 100. For example, the communication unit 210 transmits a control signal for controlling the annealing machine 100 to the annealing machine 100. The communication unit 210 also receives a signal indicating the result of quantum annealing from the annealing machine 100. The communication unit 210 may also receive a combinatorial optimization problem from a device that stores the combinatorial optimization problem to be solved by quantum annealing.

[0024] The display unit 220 has a display screen such as a liquid crystal panel or an LED (Light Emitting Diode) panel, and acquires various images. For example, the display unit 220 may display multiple associations between graphs and the annealing machine 100 determined by the control device 200, as well as an evaluation for each association.

[0025] The operation input unit 230 includes input devices such as a keyboard and a mouse, and accepts user operations. For example, the operation input unit 230 may accept user operations for making various settings related to quantum annealing, such as setting the number of iterations of quantum annealing.

[0026] The storage unit 280 stores various types of data. For example, the storage unit 280 may store various types of data related to quantum annealing, such as the associations determined by the control device 200. The storage unit 280 is configured using a storage device included in the control device 200.

[0027] The processing unit 290 performs various processes by controlling each unit of the control device 200. The functions of the processing unit 290 are performed, for example, by a CPU (Central Processing Unit) included in the control device 200 reading and executing a program from the storage unit 280.

[0028] The graph acquisition unit 291 acquires a graph indicating the combinatorial optimization problem to be solved by quantum annealing. For example, the communication unit 210 may receive a graph indicating the combinatorial optimization problem to be solved by quantum annealing (a graphical representation of the combinatorial optimization problem) from another device. The graph acquisition unit 291 may then extract the graph from the signal received by the communication unit 210. A graph indicating the combinatorial optimization problem to be solved by quantum annealing may also be simply referred to as a graph.

[0029] The association unit 292 determines multiple associations of graphs with the annealing machine 100. In particular, the association unit 292 determines multiple associations of graphs with the annealing machine 100 so as to include two associations that are not symmetrical to each other. Furthermore, the association unit 292 evaluates the associations based on the results of quantum annealing using the determined associations. The association unit 292 is an example of an association determination means.

[0030] The control unit 293 controls the annealing machine 100 in accordance with each of the associations determined by the association unit 292. As a result, the control unit 293 causes the annealing machine 100 to perform quantum annealing for each of the associations determined by the association unit 292. The control unit 293 corresponds to an example of control means.

[0031] The association between a graph and an annealing machine performed by the association unit 292 will be further described. The following describes an example in which the association unit 292 associates a graph with the annealing machine 100 using the Lechner-Hauke-Zoller (LHZ) method. However, the method by which the association unit 292 performs the association is not limited to a specific method, and various methods are possible that allow for multiple associations, including two associations that are not symmetrical to each other.

[0032] Figure 4 shows a first example of node numbering in a complete graph. A complete graph is a graph in which all nodes are directly connected by edges. Figure 4 shows an example of a complete graph with five nodes, with the nodes numbered 1 through 5.

[0033] In the LHZ method, a complete graph is associated with an annealing machine. The energy function (Hamiltonian) of a combinatorial optimization problem expressed by a complete graph is expressed as in Equation (1).

[0034]

[0035] σ ｉ denotes a binary variable in a combinatorial optimization problem. ｉ takes the value of 1 or -1. ｉｊ is a two-variable term J ｉｊ σ ｉ σ ｊ The coefficient J ｉｊ is the variable σ ｉ and σ ｊ It can be considered as a coefficient showing the correlation with h ｊ is a one-variable term h ｊ σ ｊ The coefficient hｊ is the variable σ ｊ The coefficient J can be considered as an offset for ｉｊ and the value of the coefficient h ｊ The value of is determined depending on the combinatorial optimization problem.

[0036] The energy function H shown in equation (1) is transformed into equation (2).

[0037]

[0038] Here, h ｊ =J ０ｊ Also, σ ０ = 1. The energy function H shown in equation (2) is rewritten as equation (3).

[0039]

[0040] σ （ｉ，ｊ） denotes a binary variable that takes the value 1 or -1. （ｉ，ｊ） = σ ｉ σ ｊ N is a positive integer indicating the maximum value of j. A sub-equation indicating a penalty for making it difficult for contradictions to occur in the variable values ​​is added to the energy function indicated by equation (3), resulting in an energy function as indicated by equation (4).

[0041]

[0042] C denotes a parameter for adjusting the value of the penalty given to the energy function H when a contradiction occurs in the variable values. When quantum annealing is performed, the value of the parameter C can be reflected in the coupling strength of the quantum bit devices by the coupler. Hereinafter, the parameter for adjusting the value of the penalty given to the energy function when a contradiction occurs in the variable values ​​will also be referred to as the parameter for adjusting the coupling strength of the coupler.

[0043] In the LHZ method, as shown in equation (4), the quantum bit σ corresponds to the combination of two variables in a combinatorial optimization problem. （ｉ，ｊ） Then, in order to implement a sub-formula that indicates a penalty for making it difficult for contradictions to occur in the variable values, the qubits are coupled by four-body couplers.

[0044] For example, consider the case where the energy function of a combinatorial optimization problem to be solved by quantum annealing is expressed as in equation (5).

[0045]

[0046] Equation (5) is a combinatorial optimization problem represented by a complete graph with five variables, as in the example of Figure 4, and is ｊ is the energy function when the offset shown in equation (5) is 0. For the energy function shown in equation (5), the combination of two variables is replaced with one quantum bit, and further, a sub-equation indicating a penalty to make it less likely that contradictions will occur in the variable values ​​is provided, resulting in an energy function such as equation (6).

[0047]

[0048] Fig. 5 is a diagram showing an example of mapping a graph to an annealing machine using the LHZ method, where the energy function is expressed as in Equation (6), and shows an example of mapping the graph in Fig. 4 to an annealing machine.

[0049] In the example of FIG. 5, the white circles represent the variable σ in Equation (6). （ｉ，ｊ） The notation (ij) where two numbers are enclosed in parentheses, such as (12), (13), ..., indicates the variable σ in Equation (6). （ｉ，ｊ） indicates the qubit device to which

[0050] In this way, associating a variable corresponding to a combination of two nodes with a quantum bit can be considered as associating an edge connecting two nodes with a quantum bit. Specifically, during quantum annealing, the variable σ （ｉ，ｊ） The qubit device to which the coefficient J ｉｊ It is conceivable to input a control signal according to the value of the coefficient J ｉｊ The value of can be considered as the value attached to the edge connecting the i-th node and the j-th node.

[0051] A square indicates a qubit device that is a fixed bit (a qubit with a fixed value). The notation "+1" indicates that the qubit device is controlled as a fixed bit with a value of 1. A black circle indicates a four-body coupler. A model of a qubit device to which a complete graph is associated using the LHZ method, as in the example of Figure 5, is also called an LHZ model.

[0052] Note that equation (5) is the same as equation (1) except for the variable σ ｊ The number of j is 5, and for j = 1, 2, ..., 5, h ｊ = 0, that is, the variable σ ｊ Offset h ｊ This is an equation in which the value of is set to 0. As a result, the variable σ as shown in equation (2) ０ The energy function is the energy function of equation (6) which corresponds to equation (4) without the need to introduce

[0053] On the other hand, if the offset value is not 0, the variable σ ０ After introducing , we can obtain an energy function of the form of equation (4): This gives us an LHZ model corresponding to a complete graph with one more node than the case with an offset value of 0.

[0054] Figure 6 is a diagram showing a second example of node numbering in a complete graph. Figure 6 shows an example in which the nodes of a complete graph having five nodes, like the example in Figure 4, are numbered differently from those in Figure 4. In the example in Figure 4, the nodes are numbered in the order 1, 2, 3, 4, and 5, going clockwise as one faces the diagram. On the other hand, in the example in Figure 6, the nodes are numbered in the order 1, 3, 5, 2, and 4, going clockwise as one faces the diagram.

[0055] When a complete graph having five nodes is associated with the annealing machine shown in Fig. 5, the correspondence between the quantum bits and the edges may differ depending on how the nodes are numbered. For example, in the example of Fig. 6, the edge connecting the first node and the fourth node in the example of Fig. 4 is associated with the quantum bit device (12) in Fig. 5.

[0056] In this way, different associations of graphs with annealing machines can result in different quantum annealing results. Therefore, the association unit 292 determines multiple associations of graphs with annealing machines 100.

[0057] By performing quantum annealing on each of the multiple associations determined by the association unit 292 and selecting the solution with the highest accuracy from the obtained solutions, it is possible to avoid an association that will result in the solution with the lowest accuracy. In this respect, it is expected that the control device 200 can avoid conditions that will result in the solution with particularly low accuracy when performing quantum annealing.

[0058] The control unit 293 may control the annealing machine 100 to perform quantum annealing for each of the multiple associations determined by the association unit 292. The association unit 292 may then evaluate the accuracy of the solution obtained by quantum annealing for each association, and present the solution with the best evaluation to the user.

[0059] The associating unit 292 may evaluate that the accuracy of the solution is higher as the evaluation indicated by the value of the energy function is higher. For example, in the case of the energy function H of Equation (6), the associating unit 292 may evaluate that the accuracy of the solution is higher as the value of the energy function H is smaller.

[0060] Alternatively, the correspondence unit 292 may be configured to evaluate, among solutions that satisfy the constraints of the combinatorial optimization problem, the more highly evaluated the value of the energy function is, the more accurate the solution is.The correspondence unit 292 may determine whether a solution satisfies the constraints of the combinatorial optimization problem by using a subexpression that calculates a penalty when a contradiction occurs in variable values.

[0061] For example, in the case of the energy function H of equation (6), the relevant sub-equation is −[Σ ｉ＝１ ３ σ （ｉ，ｉ＋１） σ （ｉ，ｉ＋２） σ （ｉ＋１，ｉ＋２） +Σ ｉ＝１ ２ Σ ｊ＝ｉ＋２ ４σ （ｉ，ｊ） σ （ｉ，ｊ＋１） σ （ｉ＋１，ｊ） σ （ｉ＋１，ｊ＋１） When the value of this subexpression is the minimum value of −6, the association unit 292 may determine that the solution satisfies the constraints in the combinatorial optimization problem.

[0062] Alternatively, the control device 200 may present a plurality of associations to the user. The user may then use a quantum annealing system or another device to perform quantum annealing for each of the associations presented by the control device 200. The user may then adopt the solution with the highest accuracy from among the solutions obtained by quantum annealing.

[0063] The following will further explain why the accuracy of the solution obtained may differ depending on the mapping of the graph to the annealing machine. Figure 7 is a diagram showing a first example of the correlation between nodes in a graph. The graph in Figure 7 represents a combinatorial optimization problem whose energy function is expressed as in Equation (5).

[0064] In the example of FIG. 7, the coefficient J associated with one edge of the complete graph having five nodes is ｉｊ The value of the coefficient J ｉｊ The value of coefficient J is -1. ｉｊ If the value of is +1, the variable σ ｉ and σ ｊ When the values ​​of J and J are the same, ｉｊ σ ｉ σ ｊ The value of the energy function H becomes +1, and the value of the energy function H becomes relatively small. ｉｊ If the value of is -1, the variable σ ｉ and σ ｊ When the values ​​of J and J are different, ｉｊ σ ｉ σ ｊ becomes +1, and the value of the energy function H becomes relatively small.

[0065] 8 is a diagram showing a first example of mapping the correlation between nodes of the graph to the qubit device, showing the mapping of the graph shown in FIG. 7 to the annealing machine shown in FIG.

[0066] In this case, the ten edges of the graph in FIG. 7 are associated with the ten quantum bit devices of the annealing machine in FIG. 5. If any one of the ten quantum bit devices has a coefficient value J ｉｊ = +1 is assigned to the other qubit devices, and the coefficient value J ｉｊ =-1 is associated with the digits [0] to [9] in FIG. 8, and ten possible associations are possible.

[0067] Figure 9 is a diagram showing an example of the relationship between the mapping of graphs to annealing machines and the results of quantum annealing. Figure 9 shows the success rate obtained in experiments for each of the mappings from [0] to [9] in Figure 8. The success rate here refers to the proportion of times that an optimal solution was obtained out of the number of times quantum annealing was performed. Here, the success rate is used as an index value for the accuracy of the solution.

[0068] In the experiment shown in Figure 9, the success rate was calculated for each of the various values ​​of parameter C in equation (6) for each of the associations from [0] to [9], and the value of parameter C and the success rate that resulted in the highest success rate are shown. The horizontal axis of the graph in Figure 9 represents the value of parameter C, and the vertical axis represents the success rate.

[0069] As shown in Fig. 9, when two correspondences are symmetrical to each other, such as [0] and [9] in Fig. 8, these two correspondences have the same success rate. In the example of Fig. 8, [0] and [9], [1] and [8], [2] and [6], and [4] and [7] are all line-symmetrical to each other, and have the same success rate in the example of Fig. 9.

[0070] In this way, it was found that the accuracy of the solution obtained may differ depending on the association of the graph with the annealing machine, and that when the associations of the graphs with the annealing machines are symmetrical to each other, the accuracy of the solution will be similar. Therefore, the association unit 292 may determine multiple associations for associating the graph with the annealing machine 100 so that the two associations include associations that are not symmetrical to each other.

[0071] As a result, at least one of the associations determined by the association unit 292 will be an association other than the association that results in the lowest accuracy of the solution. In the example of Fig. 9, at least one of the associations determined by the association unit 292 will be an association other than [0] or [9], and an association other than the association that results in the lowest success rate will be obtained. In this respect, the control device 200 can more reliably avoid conditions that result in particularly poor accuracy of the obtained solution.

[0072] It was also discovered that the accuracy of the obtained solution may vary depending on the value of the parameter that adjusts the coupling strength of the coupler. Therefore, the control unit 293 may control the annealing machine 100 to perform quantum annealing for each of the multiple values ​​of the parameter that adjusts the coupling strength of the coupler for each association determined by the association unit 292. The association unit 292 may then evaluate the accuracy of the solution for each association and each parameter value, and present the best-evaluated combination of association and parameter value, along with the resulting solution, to the user (the person requesting quantum annealing). In this regard, the control device 200 can avoid conditions that result in particularly poor accuracy of the obtained solution, even for the value of the parameter that adjusts the coupling strength of the coupler.

[0073] 10 is a diagram showing a second example of the correlation between nodes of a graph. In the example of FIG. 10, coefficients J associated with four edges that form a closed loop of four nodes are ｉｊ The value of the coefficient J ｉｊ10 differs from the example in Figure 7 in that the value of is -1. In other respects, the example in Figure 10 is similar to the example in Figure 7. There are 15 possible ways to associate the graph shown in Figure 10 with the annealing machine shown in Figure 5.

[0074] Fig. 11 is a diagram showing a second example of the mapping of the correlation between nodes in the graph to quantum bit devices. Fig. 11 shows one of the mappings of the graph shown in Fig. 10 to the annealing machine shown in Fig. 5. Experiments were conducted to calculate the success rate (the rate at which optimization was achieved) for each of 15 mappings of the graph shown in Fig. 10 to the annealing machine shown in Fig. 5 and for each value of parameter C in equation (6). As a result, the highest success rate of 0.886 was obtained for the mapping shown in Fig. 11. The value of parameter C at that time was 1.43.

[0075] Fig. 12 is a diagram showing a third example of mapping the correlation between nodes of a graph to quantum bit devices. Fig. 12 also shows one of the mappings of the graph shown in Fig. 10 to the annealing machine shown in Fig. 5. Of the 15 mappings of the graph shown in Fig. 10 to the annealing machine shown in Fig. 5, the one with the smallest maximum success rate was the mapping shown in Fig. 12. The maximum success rate was 0.367, and the value of parameter C at that time was 0.96.

[0076] Fig. 13 is a diagram showing a fourth example of mapping the correlation between nodes of a graph to quantum bit devices. Fig. 13 also shows one of the mappings of the graph shown in Fig. 10 to the annealing machine shown in Fig. 5. The mapping shown in Fig. 13 is symmetrical to the mapping shown in Fig. 12, and resulted in the same success rate and parameter values ​​as in Fig. 12.

[0077] In the examples of Figures 11 to 13, the maximum success rate is 0.886 - 0.367 = 0.519, a large difference of more than 50%. As such, the correspondence between the graph and the annealing machine can result in a large difference in the accuracy of the obtained solution. In response to this, the correspondence unit 292 determines multiple correspondences, and the control unit 293 controls the annealing machine 100 to perform quantum annealing for each of the multiple correspondences, thereby making it possible to select correspondences that result in relatively high solution accuracy. In this regard, the control device 200 is expected to be able to avoid conditions that result in particularly poor accuracy of the obtained solution.

[0078] 14 is a diagram showing a fifth example of mapping the correlation between nodes of a graph to a quantum bit device. Fig. 14 shows an example of mapping a complete graph having five nodes to the quantum annealing apparatus of Fig. 5, where the correlation between the nodes of the graph is represented by real values.

[0079] The correlation between the nodes of the graph takes the values ​​[0.13, -0.52, 0.81, -0.94, -0.83, -0.02, -1.00, 0.63, 0.97, -0.95]. In the example of Figure 14, the number of possible associations of the graph to the annealing machines is the same as the number of numbers assigned to the five nodes, resulting in 5! = 120 possible associations. Of the 120 possible associations, the association shown in Figure 14 achieved the highest success rate.

[0080] 15 is a diagram showing a sixth example of mapping the correlations between nodes in a graph to quantum bit devices. Of the 120 mappings, the mapping shown in FIG. 15 had the smallest maximum success rate.

[0081] Figure 16 is a diagram showing an example of the relationship between the association of graphs with annealing machines, parameter values, and quantum annealing results. Figure 16 shows the maximum success rate and the value of parameter C at that time for each of the 120 associations described above. The horizontal axis of the graph in Figure 16 represents the value of parameter C, and the vertical axis represents the success rate.

[0082] In the example of Fig. 16, in the association with the largest (highest) maximum success rate, the maximum success rate is approximately 1. On the other hand, in the association with the smallest (lowest) maximum success rate, the maximum success rate is less than 0.5. In this way, the association can result in a difference of 50% or more in success rate.

[0083] Therefore, as described above, the association unit 292 may determine multiple associations, and the control unit 293 may control the annealing machine 100 to perform quantum annealing for each of the multiple associations. This makes it possible to select an association that provides a relatively accurate solution. In this respect, the control device 200 is expected to be able to avoid conditions that result in particularly poor accuracy in the resulting solution.

[0084] 16, the optimal value of parameter C (the value of the parameter that adjusts the coupling strength of the coupler) varies depending on the association. Therefore, as described above, the control unit 293 may control the annealing machine 100 to perform quantum annealing for each of the multiple values ​​of the parameter that adjusts the coupling strength of the coupler for each association determined by the association unit 292. The association unit 292 may then evaluate the accuracy of the solution for each association and each parameter value, and present the best-evaluated combination of association and parameter value, along with the resulting solution, to the user (the person requesting quantum annealing). In this regard, the control device 200 can avoid conditions that result in particularly poor accuracy in the solution obtained, even for the value of the parameter that adjusts the coupling strength of the coupler.

[0085] 17 is a diagram showing an example of the procedure of processing performed by the control device 200. In the processing of FIG. 17, the associating unit 292 determines one association of the graph with the annealing machine 100 (step S11). In the second or subsequent executions of step S11, the associating unit 292 determines an association that is different from the previously determined association.

[0086] Next, the association unit 292 determines whether the association determined in step S11 is asymmetric with any of the previously determined associations (step S12). If it is determined that any of the previously determined associations is symmetric with the association determined this time (step S12: NO), the process returns to step S11.

[0087] On the other hand, if it is determined that the correspondence determined this time is asymmetric with any of the correspondences determined previously (step S12: YES), the processing unit 290 starts a loop L11 in which processing is performed for each value of the parameter that adjusts the coupling strength of the coupler (step S13).

[0088] In the processing of loop L11, the control unit 293 controls the annealing machine 100 to perform quantum annealing based on the association determined by the association unit 292 (step S14). Next, the processing unit 290 performs termination processing of loop L11 (step S15). Specifically, the processing unit 290 determines whether quantum annealing based on the currently determined association has been performed for all parameter values ​​that are the targets of execution in loop L11. If it is determined that there are parameter values ​​for which quantum annealing has not yet been performed, the processing unit 290 continues processing loop L11 for the parameter values ​​for which quantum annealing has not been performed. On the other hand, if it is determined that quantum annealing has been performed for all parameter values, the processing unit 290 terminates loop L11.

[0089] When loop L11 is completed, the processing unit 290 determines whether a matching termination condition is met (step S16). The matching termination condition here is not limited to a specific condition. For example, the matching termination condition may be a condition that loop L11 has been processed for a predetermined number of matchings that are not symmetrical to each other. Alternatively, the matching termination condition may be a condition that the energy function value is smaller than a predetermined threshold (a good evaluation is obtained).

[0090] If the processing unit 290 determines that the association termination condition is not met (step S16: NO), the process returns to step S11. On the other hand, if the processing unit 290 determines that the association termination condition is met, the processing unit 290 controls the display unit 220 to display the results of quantum annealing (step S17). For example, the association unit 292 may evaluate the accuracy of the solution for each association and for each parameter value. Then, the association unit 292 may display the best-evaluated combination of association and parameter value, as well as the solution at that time, on the display unit 220. After step S17, the control device 200 terminates the process of FIG. 17.

[0091] The associating unit 292 may perform machine learning of the association between the graph and the annealing machine 100. For example, the associating unit 292 may be configured to include a machine learning model such as a neural network. The associating unit 292 may then perform machine learning of the association between the graph and the annealing machine 100 using a training dataset that is training data that combines a graph representing a combinatorial optimization problem with an association that can obtain an optimal solution.

[0092] As described above, the associating unit 292 determines multiple associations that associate a graph including multiple nodes and multiple edges with an annealing machine 100 that includes multiple quantum bit devices 110 and multiple couplers that each couple the same number of quantum bit devices 110. The four-body coupler 120 is an example of a coupler.

[0093] According to the control device 200, it is expected that the correspondences determined by the correspondence unit 292 will include correspondences other than the correspondence that results in the lowest accuracy of the solution. In this respect, it is expected that the control device 200 can avoid conditions that will result in particularly poor accuracy of the solution obtained when performing quantum annealing.

[0094] Furthermore, the associating unit 292 determines multiple associations for associating graphs with the annealing machine 100 such that the two associations for associating graphs with the annealing machine 100 include associations that are not symmetrical with each other. According to the control device 200, it is expected that at least one of the associations determined by the associating unit 292 will be an association other than the association that results in the lowest accuracy of the solution. In this respect, the control device 200 can more reliably avoid conditions that result in particularly poor accuracy of the obtained solution.

[0095] The associating unit 292 also determines multiple associations that associate a graph including multiple nodes and multiple edges with an annealing machine 100 that includes multiple quantum bit devices 110 and multiple couplers that each couple the same number of quantum bit devices. The four-body coupler 120 is an example of a coupler. The control unit 293 controls the annealing machine 100 that includes the quantum bit devices 110 and the couplers in accordance with the determined associations to perform quantum annealing.

[0096] According to the control device 200, it is expected that the associations determined by the association unit 292 will include associations other than the association that results in the lowest accuracy of the solution. In this respect, it is expected that the control device 200 will be able to avoid conditions that will result in particularly poor accuracy of the solution when performing quantum annealing. According to the control device 200, it is expected that a relatively high-accuracy solution will be obtained by the control unit 293 performing quantum annealing in accordance with the associations determined by the association unit 292.

[0097] Furthermore, the control unit 293 performs quantum annealing for each of the determined associations based on each of multiple values ​​of the parameter that adjusts the coupling strength of the coupler. According to the control device 200, it is possible to avoid conditions that particularly deteriorate the accuracy of the obtained solution, even with regard to the value of the parameter that adjusts the coupling strength of the coupler.

[0098] Second Embodiment Fig. 18 is a diagram showing an example of the configuration of an association determination device according to at least one embodiment. In the configuration shown in Fig. 18, an association determination device 610 includes an association determination unit 611.

[0099] With this configuration, the association determination unit 611 determines multiple associations that associate a graph including multiple nodes and multiple edges with the annealing machine 100 that includes multiple quantum bit devices and multiple couplers that each couple the same number of quantum bit devices. The association determination unit 611 is an example of an association determination means.

[0100] According to the association determination device 610, it is expected that the associations determined by the association determination unit 611 will include associations other than the association that results in the lowest accuracy of the solution. In this respect, it is expected that the association determination device 610 can avoid conditions that result in particularly poor accuracy of the solution obtained when performing quantum annealing.

[0101] 19 is a diagram illustrating an example of the configuration of a control device according to at least one embodiment. In the configuration illustrated in FIG. 19, a control device 620 includes an association determination unit 621 and a control unit 622.

[0102] With this configuration, the association determiner 621 determines multiple associations that associate a graph including multiple nodes and multiple edges with an annealing machine including multiple quantum bit devices and multiple couplers that each couple the same number of quantum bit devices. The controller 622 controls the annealing machine including the quantum bit devices and the couplers in accordance with the determined associations to perform quantum annealing. The association determiner 621 is an example of association determination means. The controller 622 is an example of control means.

[0103] According to the control device 620, it is expected that the associations determined by the association determination unit 621 will include associations other than the association that results in the lowest accuracy of the solution. In this respect, it is expected that the control device 620 will be able to avoid conditions that will result in particularly poor accuracy of the solution when performing quantum annealing. According to the control device 620, it is expected that a relatively high accuracy solution will be obtained by the control unit 622 performing quantum annealing in accordance with the associations determined by the association determination unit 621.

[0104] <Fourth embodiment> Fig. 20 is a diagram showing an example of the configuration of a quantum annealing system according to at least one embodiment. In the configuration shown in Fig. 20, a quantum annealing system 630 includes an annealing machine 631 and a control device 634. The annealing machine 631 includes a plurality of quantum bit devices 632 and a plurality of couplers 633. The control device 634 includes an association determination unit 635 and a control unit 636.

[0105] In this configuration, the multiple couplers 633 each couple the same number of quantum bit devices 632. The association determiner 635 determines multiple associations that associate a graph including multiple nodes and multiple edges with the annealing machine 631. The controller 636 controls the annealing machine 631 including the quantum bit devices 632 and the couplers 633 in accordance with the determined associations to perform quantum annealing. The association determiner 635 is an example of association determination means. The controller 636 is an example of control means.

[0106] According to the quantum annealing system 630, it is expected that the associations determined by the association determination unit 635 will include associations other than the association that results in the lowest accuracy of the solution. In this respect, it is expected that the quantum annealing system 630 will be able to avoid conditions that will result in particularly poor accuracy of the solution when performing quantum annealing. According to the quantum annealing system 630, it is expected that a relatively high accuracy solution will be obtained by the control unit 636 performing quantum annealing in accordance with the associations determined by the association determination unit 635.

[0107] Fifth Embodiment Fig. 21 is a diagram showing an example of a processing procedure in an association determination method according to at least one embodiment. The association determination method shown in Fig. 21 includes determining an association (step S611).

[0108] In determining the correspondence (step S611), the computer determines multiple correspondences that correspond a graph including multiple nodes and multiple edges to an annealing machine 100 that includes multiple quantum bit devices and multiple couplers that each couple the same number of quantum bit devices.

[0109] According to the correspondence determination method shown in Fig. 21, it is expected that the determined correspondences will include correspondences other than the correspondence that will result in the lowest accuracy of the solution. In this respect, it is expected that the correspondence determination method shown in Fig. 21 can avoid conditions that will particularly deteriorate the accuracy of the obtained solution when performing quantum annealing.

[0110] Sixth Embodiment Fig. 22 is a diagram showing an example of a processing procedure in a control method according to at least one embodiment. The control method shown in Fig. 22 includes determining an association (step S621) and performing control (step S622).

[0111] In determining the association (step S621), the computer determines multiple associations that associate a graph including multiple nodes and multiple edges with an annealing machine including multiple quantum bit devices and multiple couplers that each couple the same number of quantum bit devices. In performing control (step S622), the computer controls the annealing machine including the quantum bit devices and the couplers in accordance with the determined associations to perform quantum annealing.

[0112] According to the control method shown in Fig. 22, it is expected that the determined correspondences will include correspondences other than the correspondence that will result in the lowest accuracy of the solution. In this respect, it is expected that the control method shown in Fig. 22 will be able to avoid conditions that will particularly deteriorate the accuracy of the obtained solution when performing quantum annealing. According to the control method shown in Fig. 22, it is expected that a relatively high-accuracy solution will be obtained by the control unit 636 performing quantum annealing in accordance with the correspondence determined by the correspondence determination unit 635.

[0113] 23 is a schematic block diagram showing the configuration of a computer according to at least one embodiment. In the configuration shown in FIG. 23, a computer 700 includes a CPU (Central Processing Unit) 710, a main memory device 720, an auxiliary memory device 730, an interface 740, and a non-volatile recording medium 750.

[0114] One or more of the above-described control device 200, association determination device 610, control device 620, and control device 634, or a part thereof, may be implemented in the computer 700. In this case, the operation of each of the above-described processing units is stored in the auxiliary storage device 730 in the form of a program. The CPU 710 reads the program from the auxiliary storage device 730, loads it into the main storage device 720, and executes the above-described processing in accordance with the program. The CPU 710 also allocates storage areas in the main storage device 720 corresponding to each of the above-described storage units in accordance with the program. Communication between each device and other devices is executed by the interface 740, which has a communication function, and performs communication under the control of the CPU 710.

[0115] When the control device 200 is implemented in a computer 700, the operations of the processing unit 290 and each of its units are stored in the form of a program in an auxiliary storage device 730. The CPU 710 reads the program from the auxiliary storage device 730, loads it into the main storage device 720, and executes the above-described processing in accordance with the program.

[0116] Furthermore, the CPU 710 allocates a storage area for the storage unit 280 in the main storage device 720 in accordance with the program. Communication with other devices by the communication unit 210 is achieved by the interface 740 having a communication function and performing communication under the control of the CPU 710. Display of images by the display unit 220 is achieved by the interface 740 having a display device and displaying various images under the control of the CPU 710. Reception of user operations by the operation input unit 230 is achieved by the interface 740 having an input device and receiving user operations under the control of the CPU 710.

[0117] When the association determination device 610 is implemented in the computer 700, the operation of the association determination unit 611 is stored in the form of a program in the auxiliary storage device 730. The CPU 710 reads the program from the auxiliary storage device 730, loads it into the main storage device 720, and executes the above-described processing in accordance with the program.

[0118] Furthermore, the CPU 710, in accordance with the program, allocates a storage area in the main storage device 720 for the association determination device 610 to perform processing. Communication between the association determination device 610 and other devices is performed by the interface 740, which has a communication function, and performs communication under the control of the CPU 710. Interaction between the association determination device 610 and a user is performed by the interface 740, which has an input device and an output device, presenting information to the user via the output device under the control of the CPU 710 and accepting user operations via the input device.

[0119] When the control device 620 is implemented in the computer 700, the operations of the association determination unit 621 and the control unit 622 are stored in the form of a program in the auxiliary storage device 730. The CPU 710 reads the program from the auxiliary storage device 730, loads it into the main storage device 720, and executes the above-described processing in accordance with the program.

[0120] Furthermore, the CPU 710 allocates a storage area in the main storage device 720 for the control device 620 to perform processing in accordance with the program. Communication between the control device 620 and other devices is performed by the interface 740 having a communication function and performing communication under the control of the CPU 710. Interaction between the control device 620 and a user is performed by the interface 740 having an input device and an output device, presenting information to the user via the output device under the control of the CPU 710, and accepting user operations via the input device.

[0121] When the control device 634 is implemented in the computer 700, the operations of the association determination unit 635 and the control unit 636 are stored in the form of a program in the auxiliary storage device 730. The CPU 710 reads the program from the auxiliary storage device 730, loads it into the main storage device 720, and executes the above-described processing in accordance with the program.

[0122] Furthermore, the CPU 710 allocates a storage area in the main storage device 720 for the control device 634 to perform processing in accordance with the program. Communication between the control device 634 and other devices is performed by the interface 740 having a communication function and performing communication under the control of the CPU 710. Interaction between the control device 634 and a user is performed by the interface 740 having an input device and an output device, presenting information to the user via the output device under the control of the CPU 710, and accepting user operations via the input device.

[0123] One or more of the above-described programs may be recorded on nonvolatile recording medium 750. In this case, interface 740 may read the programs from nonvolatile recording medium 750. Then, CPU 710 may directly execute the programs read by interface 740, or may temporarily store the programs in main storage device 720 or auxiliary storage device 730 and then execute them.

[0124] Note that a program for executing all or part of the processing performed by control device 200, association determination device 610, control device 620, and control device 634 may be recorded on a computer-readable recording medium, and the program recorded on the recording medium may be loaded into a computer system and executed to perform the processing of each unit. Note that the term "computer system" herein includes hardware such as an operating system (OS) and peripheral devices. Furthermore, the term "computer-readable recording medium" refers to portable media such as flexible disks, optical magnetic disks, read-only memories (ROMs), and compact disc read-only memories (CD-ROMs), as well as storage devices such as hard disks built into computer systems. The program may be for implementing part of the aforementioned functions, or may be capable of implementing the aforementioned functions in combination with a program already recorded in the computer system.

[0125] Although the embodiments of the present invention have been described above in detail with reference to the drawings, the specific configuration is not limited to these embodiments and includes designs within the scope of the present invention. Furthermore, the above-described embodiments may be combined with other embodiments as appropriate.

[0126] A part or all of the above-described embodiments can be described as, but not limited to, the following supplementary notes.

[0127] (Supplementary Note 1) An association determination device comprising: an association determination means for determining a plurality of associations for associating a graph including a plurality of nodes and a plurality of edges with an annealing machine including a plurality of quantum bit devices and a plurality of couplers each coupling the same number of quantum bit devices.

[0128] (Supplementary Note 2) The association determination device according to Supplementary Note 1, wherein the association determination means determines a plurality of associations for associating the graph with the annealing machine such that two associations for associating the graph with the annealing machine include associations that are not symmetrical with each other.

[0129] (Supplementary Note 3) A control device comprising: an association determination means for determining a plurality of associations for associating a graph including a plurality of nodes and a plurality of edges with an annealing machine including a plurality of quantum bit devices and a plurality of couplers each coupling the same number of quantum bit devices; and a control means for controlling the annealing machine including the quantum bit devices and the couplers in accordance with the determined associations to perform quantum annealing.

[0130] (Supplementary Note 4) The control device according to Supplementary Note 3, wherein the association determination means determines a plurality of associations for associating the graph with the annealing machine such that two associations for associating the graph with the annealing machine include associations that are not symmetrical to each other.

[0131] (Supplementary Note 5) The control device according to Supplementary Note 3 or Supplementary Note 4, wherein the control means performs quantum annealing for each of the determined plurality of associations based on each of a plurality of values ​​of a parameter that adjusts the coupling strength of the coupler.

[0132] (Supplementary Note 6) A quantum annealing system comprising an annealing machine and a control device, wherein the annealing machine comprises a plurality of quantum bit devices and a plurality of couplers each coupling the same number of quantum bit devices, and the control device comprises: association determination means for determining a plurality of associations for associating a graph including a plurality of nodes and a plurality of edges with the annealing machine, and control means for controlling the annealing machine including the quantum bit devices and the couplers in accordance with the determined associations to perform quantum annealing.

[0133] (Supplementary Note 7) The quantum annealing system according to Supplementary Note 6, wherein the association determination means determines a plurality of associations for associating the graph with the annealing machine such that two associations for associating the graph with the annealing machine include associations that are not symmetric with each other.

[0134] (Supplementary Note 8) The quantum annealing system according to Supplementary Note 6 or Supplementary Note 7, wherein the control means performs quantum annealing for each of the determined plurality of associations based on each of a plurality of values ​​of a parameter that adjusts the coupling strength of the coupler.

[0135] (Supplementary Note 9) A method for determining correspondences, the method including: determining, by a computer, a plurality of correspondences for associating a graph including a plurality of nodes and a plurality of edges with an annealing machine including a plurality of quantum bit devices and a plurality of couplers, each of which couples the same number of quantum bit devices.

[0136] (Supplementary Note 10) The association determination method according to Supplementary Note 9, comprising the computer determining a plurality of associations for associating the graph with the annealing machine such that the two associations for associating the graph with the annealing machine include associations that are not symmetrical to each other.

[0137] (Supplementary Note 11) A control method comprising: a computer determining a plurality of associations for associating a graph including a plurality of nodes and a plurality of edges with an annealing machine including a plurality of quantum bit devices and a plurality of couplers, each of which couples the same number of quantum bit devices; and controlling the annealing machine including the quantum bit devices and the couplers in accordance with the determined associations to perform quantum annealing.

[0138] (Supplementary Note 12) The control method according to Supplementary Note 11, comprising the computer determining a plurality of associations for associating the graph with the annealing machine, such that the two associations for associating the graph with the annealing machine include associations that are not symmetrical with each other.

[0139] (Supplementary Note 13) The control method according to Supplementary Note 11 or Supplementary Note 12, comprising the computer performing quantum annealing for each of the determined plurality of associations based on each of a plurality of values ​​of a parameter that adjusts the coupling strength of the coupler.

[0140] (Supplementary Note 14) A recording medium storing a program that causes a computer to execute the following: determining multiple associations that associate a graph including multiple nodes and multiple edges with an annealing machine that includes multiple quantum bit devices and multiple couplers that each couple the same number of quantum bit devices.

[0141] (Supplementary Note 15) The recording medium described in Supplementary Note 14, wherein the program causes the computer to execute the following: determining multiple ways of associating the graph with the annealing machine, such that two ways of associating the graph with the annealing machine include associating ways that are not symmetrical with each other.

[0142] (Supplementary Note 16) A recording medium having recorded thereon a program that causes a computer to execute the following steps: determining a plurality of associations that associate a graph including a plurality of nodes and a plurality of edges with an annealing machine including a plurality of quantum bit devices and a plurality of couplers that each couple the same number of quantum bit devices; and controlling the annealing machine including the quantum bit devices and the couplers in accordance with the determined associations to perform quantum annealing.

[0143] (Supplementary Note 17) The recording medium described in Supplementary Note 16, wherein the program causes the computer to execute: determining multiple ways of associating the graph with the annealing machine, such that two ways of associating the graph with the annealing machine include asymmetric associations.

[0144] (Supplementary Note 18) The recording medium according to Supplementary Note 16 or Supplementary Note 17, wherein the program causes the computer to execute the following: performing quantum annealing for each of the determined multiple associations based on each of multiple values ​​of a parameter that adjusts the coupling strength of the coupler.

[0145] The present invention may be applied to an association determination device, a control device, a quantum annealing system, an association determination method, a control method, and a recording medium.

[0146] 1 Quantum annealing system 100, 631 Annealing machine 110, 632 Quantum bit device 120 Four-body coupler 200, 620, 634 Control device 210 Communication unit 220 Display unit 230 Operation input unit 280 Storage unit 290 Processing unit 291 Graph acquisition unit 292 Correspondence unit 293, 622, 636 Control unit 610 Correspondence determination device 621, 635 Correspondence determination unit 633 Coupler

Claims

1. An association determination device comprising an association determination means for determining multiple associations for associating a graph including multiple nodes and multiple edges with an annealing machine including multiple quantum bit devices and multiple couplers each coupling the same number of quantum bit devices.

2. The association determination device according to claim 1, wherein the association determination means determines a plurality of associations for associating the graph with the annealing machine such that the two associations for associating the graph with the annealing machine include associations that are not symmetrical with each other.

3. A control device comprising: an association determination means for determining a plurality of associations for associating a graph including a plurality of nodes and a plurality of edges with an annealing machine including a plurality of quantum bit devices and a plurality of couplers each coupling the same number of quantum bit devices; and a control means for controlling the annealing machine including the quantum bit devices and the couplers in accordance with the determined associations to perform quantum annealing.

4. The control device according to claim 3, wherein the association determination means determines a plurality of associations for associating the graph with the annealing machine such that the two associations for associating the graph with the annealing machine include associations that are not symmetrical with each other.

5. The control device according to claim 3 or 4, wherein the control means performs quantum annealing for each of the determined plurality of associations based on each of a plurality of values ​​of a parameter that adjusts the coupling strength of the coupler.

6. A quantum annealing system comprising an annealing machine and a control device, wherein the annealing machine comprises a plurality of quantum bit devices and a plurality of couplers each coupling the same number of quantum bit devices, and the control device comprises: association determination means for determining a plurality of associations for associating a graph including a plurality of nodes and a plurality of edges with the annealing machine, and control means for controlling the annealing machine including the quantum bit devices and the couplers in accordance with the determined associations to perform quantum annealing.

7. The quantum annealing system according to claim 6, wherein the association determination means determines multiple associations for associating the graph with the annealing machine such that two associations for associating the graph with the annealing machine include associations that are not symmetrical with each other.

8. The quantum annealing system according to claim 6 or claim 7, wherein the control means performs quantum annealing for each of the determined plurality of associations based on each of a plurality of values ​​of a parameter that adjusts the coupling strength of the coupler.

9. A method for determining correspondences, the method comprising: a computer determining a plurality of correspondences for associating a graph including a plurality of nodes and a plurality of edges with an annealing machine including a plurality of quantum bit devices and a plurality of couplers, each of which couples the same number of quantum bit devices.

10. A control method comprising: a computer determining a plurality of associations that associate a graph including a plurality of nodes and a plurality of edges with an annealing machine including a plurality of quantum bit devices and a plurality of couplers, each of which couples the same number of quantum bit devices; and controlling the annealing machine including the quantum bit devices and the couplers in accordance with the determined associations to perform quantum annealing.

11. A recording medium storing a program that causes a computer to execute the following: determining multiple associations that associate a graph including multiple nodes and multiple edges with an annealing machine that includes multiple quantum bit devices and multiple couplers that each couple the same number of quantum bit devices.

12. A recording medium having recorded thereon a program that causes a computer to execute the following steps: determine multiple associations that associate a graph including multiple nodes and multiple edges with an annealing machine including multiple quantum bit devices and multiple couplers that each couple the same number of quantum bit devices; and control the annealing machine including the quantum bit devices and the couplers in accordance with the determined associations to perform quantum annealing.

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