Optimization device, optimization method, and optimization program

Abstract

This provides a new method for controlling the number of amplification steps in Grover adaptive search. [Solution] In an optimization device that obtains a solution from a quantum computer using Grover adaptive search, when a solution is observed in the quantum computer where the objective function value is below the threshold at a randomly selected number of amplification steps below an upper limit, the threshold is updated to the next threshold. Based on the upper limit, the next upper limit is determined, and the next number of amplification steps is randomly selected from integer values ​​below the next upper limit.

Description

[Technical Field]

[0001] This disclosure relates to an optimization apparatus, an optimization method, and an optimization program. [Background technology]

[0002] In binary optimization problems, where the goal is to find an input value x that minimizes a given objective function, Grover Adaptive Search (GAS) is known as a method that guarantees quadratic acceleration compared to brute-force search by classical computers. [Prior art documents] [Non-patent literature]

[0003] [Non-Patent Document 1] L. Giuffrida, D. Volpe, GACirillo, M. Zamboni and G. Turvani, "Engineering Grover Adaptive Search:Exploring the Degrees of Freedom for Efficient QUBO Solving," in IEEEJournal on Emerging and Selected Topics in Circuits and Systems, vol. 12, no.3, pp. 614-623, Sept. 2022, doi: 10.1109 / JETCAS.2022.3202566. [Non-Patent Document 2] Durr, C. and Hoyer, P.: A quantum algorithm for finding the minimum, arXiv preprint quant-ph / 9607014 (1996). [Non-Patent Document 3] Boyer, M., Brassard, G., Hoyer, P. and Tapp, A.: Tight bounds on quantum searching, Fortschritte derPhysik: Progress of Physics, Vol. 46, No. 4-5, pp. 493-505 (1998). [Overview of the project] [Problems that the invention aims to solve]

[0004] In Grover search, a subroutine of Grover adaptive search, the number of amplification iterations must be appropriately set according to the number of solutions in the search region. However, because Grover adaptive search narrows down the solutions where the objective function value is less than the threshold by gradually decreasing the threshold, the number of solutions at each search step is unknown, and the number of amplification iterations must be determined randomly.

[0005] Standard methods for controlling the number of amplification steps randomly select an amplification step from a natural number less than an exponentially increasing upper limit at each search step. This upper limit is reset each time the solution is updated. However, since the ideal value of the amplification step increases monotonically with solution updates, the reset can lead to the selection of an inefficient amplification step.

[0006] Non-patent document 1 describes a method for controlling the increase of the upper limit at each search step linearly rather than exponentially, but since the upper limit is reset each time the solution is updated, the above-mentioned problem cannot be solved.

[0007] The purpose of this disclosure is to provide a new method for controlling the number of amplification cycles in Grover adaptive search. [Means for solving the problem]

[0008] An optimization device according to one aspect of the present disclosure is an optimization device that obtains a solution from a quantum computer by Grover adaptive search, and includes a processor and a memory. The processor cooperates with the memory to observe a solution in which an objective function value is less than a threshold value at an amplification number randomly selected from integer values less than an upper limit value in the quantum computer. When updating the threshold value to the next threshold value, a next upper limit value is determined based on the upper limit value, and a next amplification number is randomly selected from integer values less than the next upper limit value.

[0009] An optimization method according to one aspect of the present disclosure is an optimization method that obtains a solution from a quantum computer by Grover adaptive search. In the quantum computer, a solution in which an objective function value is less than a threshold value is observed at an amplification number randomly selected from integer values less than an upper limit value. When updating the threshold value to the next threshold value, a next upper limit value is determined based on the upper limit value, and a next amplification number is randomly selected from integer values less than the next upper limit value.

[0010] An optimization program according to one aspect of the present disclosure causes a classical computer to execute the above optimization method.

[0011] These general or specific aspects may be implemented in a system, device, method, integrated circuit, computer program, or recording medium, or may be implemented in any combination of a system, device, method, integrated circuit, computer program, and recording medium.

Advantages of the Invention

[0012] According to the present disclosure, a new method for controlling the amplification number in Grover adaptive search can be provided.

Brief Description of the Drawings

[0013] [Figure 1] A diagram showing the algorithm proposed in Non-Patent Document 2 [Figure 2] A graph showing an example of the transition of the amplification number by a standard method [Figure 3] A graph showing an example of the transition of the amplification number by the proposed method [Figure 4] Figure showing an algorithm example of a new GAS simulation method [Figure 5] Figure showing rank transition [Figure 6] Graph showing the relationship between the increase coefficient λ of the upper limit value and the solution performance [Figure 7] Graph showing the comparison result of the solution performance between the conventional method and the proposed method [Figure 8] Graph showing the TSP rank distribution [Figure 9] Graph showing the comparison result with the proposed method by 4-point TSP [Figure 10] Graph showing the relationship between the number of qubits and the solution performance [Figure 11] Figure showing an example of an algorithm of an improved version of GAS that abolishes the reset of the upper limit value [Figure 12] Graph showing the relationship between the number of oracle calls and the average rank of the obtained solutions [Figure 13] Block diagram showing a configuration example of the optimization system according to the present embodiment

Mode for Carrying Out the Invention

[0014] Hereinafter, embodiments of the present disclosure will be described in detail with appropriate reference to the drawings. However, a more detailed description may be omitted as necessary. For example, detailed descriptions of well-known matters and duplicate descriptions of substantially the same configurations may be omitted. This is to avoid making the following description unnecessarily redundant and to facilitate the understanding of those skilled in the art. Note that the attached drawings and the following description are provided to enable those skilled in the art to fully understand the present disclosure, and are not intended to limit the subject matter described in the claims.

[0015] (This embodiment) 0. Overview In binary optimization problems where we search for an input value x that minimizes a given objective function, Grover Adaptive Search (GAS) is known to offer quadratic acceleration compared to brute-force search by classical computers. GAS calls Grover Search (GS), a well-known basic quantum algorithm, as a subroutine to search for inputs with a function value less than a threshold. If found, it updates the threshold with that function value to narrow down the search target. To efficiently search for a solution, it is necessary to appropriately select the number of iterations (amplification counts) of the circuit that amplifies the desired state in each step of GS.

[0016] In this embodiment, we consider the number of amplification cycles and propose a new method for controlling them. Furthermore, simulation results show that this method allows for obtaining the optimal solution with less computational effort compared to standard control methods. The proposed method offers the potential to significantly reduce computation time for combinatorial optimization problems and other similar problems in the future.

[0017] 1. Introduction Combinatorial optimization problems are problems that seek inputs that minimize a given objective function, and they have a wide range of applications, such as determining personnel schedules in supply chains, formulating optimal delivery routes, and optimizing financial portfolios. Among these, quadratic unconstrained binary optimization problems (QUBO) and higher-order unconstrained binary optimization problems (HUBO) are being actively researched. These problems can always be solved by brute-force searching using conventional classical computers to find the optimal solution. However, for large-scale problems, the computational cost is enormous, making it impossible to obtain the optimal solution in a realistic amount of time. Therefore, several approaches have been proposed to solve QUBO and HUBO using quantum computing.

[0018] First, quantum annealing, also known as quantum simulated annealing, is a heuristic technique that cleverly utilizes quantum effects such as state superposition. In quantum annealing, QUBOs can be input directly, but when dealing with HUBOs, auxiliary bits must be added to convert them to QUBOs, and the increase in the number of bits negatively impacts the solution. Quantum annealing requires dedicated hardware, but services that provide this on the cloud are already commercially available, and hybrid solvers combining classical and quantum methods can solve problems with up to 20,000 variables.

[0019] Next, there are other methods such as QAOA (Quantum Approximate Optimization Algorithm) and FALQON (Feedback-based Algorithm for Quantum Optimization). These methods are said to be executable even on general-purpose quantum computers (NISQ: Noisy Intermediate-Scale Quantum device) that can handle a small number of bits and lack error correction capabilities.

[0020] Finally, there is Grover Adaptive Search (GAS), which is the main method dealt with in this embodiment. In GAS, to find the optimal solution, the objective function value of the best solution at each search step is used as a threshold, and Grover Search (GS) is repeatedly performed on solutions below that threshold to find better solutions. GAS has an advantage over quantum annealing in that it can directly solve HUBO. Furthermore, while the methods mentioned above may only yield approximate solutions (local optima), GAS has the remarkable property of being able to obtain the optimal solution for any problem. However, because it requires deep quantum circuits, a fault-tolerant quantum computer (FTQC) capable of error correction is necessary for its execution. It has been theoretically proven that GAS can be quadratically accelerated compared to classical exhaustive search. For this reason, it is considered a promising approach to combinatorial optimization once FTQC is realized in the future.

[0021] One of the challenges in operating GAS is controlling the number of amplification cycles (details in Section 2.2). While several methods have been proposed besides the one in the original GAS paper (Non-Patent Literature 2), it remains unclear which is the optimal method. Optimizing the control of the number of amplification cycles is expected to reduce the constant overhead of computation, making this a crucial issue for applying GAS.

[0022] In this embodiment, we examine the simulation results when using the control method described in the original GAS paper (Non-Patent Literature 2) to clarify the conditions under which calculations can be performed more efficiently, and propose a new method for controlling the number of amplification cycles. Furthermore, we introduce a GAS simulation method that operates faster than state vector calculations and verify the effectiveness of the proposed method. In addition, we propose a GAS framework that takes into account the termination conditions, considering the simulation results when using the proposed method.

[0023] The configuration of this embodiment is as follows. Chapter 2 explains GS and GAS as theoretical background. Chapter 3 then considers the number of amplification cycles and proposes a new control method. Chapter 4 explains the GAS simulation method and verifies the effectiveness of the proposed method. Finally, Chapter 5 presents the conclusions.

[0024] 2. Theoretical background 2.1 Grover Search This section will first summarize the main component of GAS, which is GS. In GS, the probability amplitude of the state corresponding to the desired search target is amplified, and then the quantum state is measured. The following three operators are used to construct this quantum state.

[0025] [ka]

[0026] [ka] This operator inverts the sign of only desired states and needs to be prepared individually depending on the problem. In GAS, an oracle is used to invert the sign of states below a certain threshold. It is known that GAS oracles can be efficiently constructed by embedding a function value in the phase of the auxiliary bits, extracting it using the inverse quantum Fourier transform, and comparing it with the threshold.

[0027] The dimension of the search space is N=2 n This section provides a detailed explanation of GS when n is the number of bits and t is the number of elements to be searched. [ka] Let's assume that we first create a superposition of all N possible states using the Hadamard gate.

[0028]

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[0029] [ka]

[0030]

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[0031] [ka]

[0032]

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[0033] [ka]

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[0034] [ka]

[0035]

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[0036] [ka]

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[0037]

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[0038] In practice, the number of elements to search, t, is often unknown. In the case of GAS, t corresponds to the number of inputs for which the objective function value is below a threshold. Since it is impossible to know t accurately without exhaustively searching the distribution of solutions, it is difficult to select the ideal number of amplification iterations. In cases where t is unknown in this way, following the method proposed in Non-Patent Document 3, the expected number of oracle calls required to obtain the desired state is at most,

number

[0039] The method involves repeatedly performing the above GS (Gross Scale) operation, and randomly selecting the number of amplification steps from an integer value less than the upper limit, which increases exponentially with each repetition. In GAS, which will be described in detail in the next section, this method is applied each time the threshold is updated.

[0040] 2.2 Grover Adaptive Search In GAS, a certain objective function is achieved by sampling quantum states.

number

[0041]

number

[0042] x is a binary variable vector of length n that can be represented by n bits. The quantum state to be sampled is constructed by amplifying the probability amplitude of the desired state from an initial state which is a superposition of all possible solutions, similar to the Grover search described above. Generally, the number of oracle calls used to determine whether a given bit sequence x is the target of the search is used as an evaluation metric for search algorithms. When finding the optimal solution by exhaustive search, it is necessary to apply the oracle once to every bit sequence of length n, so the order of the number of oracle calls is O(N) (N=2). n )

[0043] On the other hand, according to Non-Patent Document 2, the number of oracle calls required to obtain the optimal solution with a probability of 1 / 2 or more in GAS is at most:

number

[0044] This shows that GAS can achieve quadratic acceleration on the order of computational complexity compared to classical brute-force search.

[0045] The details of the algorithm proposed in Non-Patent Document 2 are shown in Figure 1 as Algorithm1. m and r are the upper limit of the number of amplification cycles in each iteration and the selected number of amplification cycles, respectively. [ka] R is the cumulative number of oracle calls and represents the computational cost up to each search step. The general flow of Algorithm1 shown in Figure 1 is to "repeatedly perform a Grover search targeting solutions below the threshold, and if a solution below the threshold is observed, update the threshold with its objective function value to narrow down the solution space." Generally, the number of solutions below the threshold is unknown, so the ideal number of amplification steps r is expressed in equation (12). opt This is also unclear. Several methods for selecting the number of amplification cycles have been proposed in addition to the one used in Algorithm1, and in this report, we have devised a new method for selecting the number of amplification cycles as one of our main contributions. The details are described in Section 3.1.

[0046] 3. Proposed method 3.1 Method for controlling the number of amplification cycles: No upper limit reset. First, we will explain the most standard method for controlling the number of amplification steps, as seen in Algorithm1 and other papers dealing with Qiskit libraries and GAS. In this method, the number of amplification steps r is randomly selected from integer values ​​less than the upper limit m. m starts at an initial value of 1 and is multiplied by λ every 3-12 loops of Algorithm1, and is reset to 1 each time a solution below the threshold is observed and the threshold is updated (double-lined portion in Algorithm1). This is because GS, using the method described in Non-Patent Document 3 mentioned in 2.1, is applied at each threshold.

[0047] Figure 2 shows an example of the changes in the number of amplification cycles (long dashed line in Figure 2) and the upper limit (solid line in Figure 2) in a GAS simulation. Figure 2 is a graph showing an example of the changes in the number of amplification cycles using a standard method.

[0048] By randomly selecting the number of amplification iterations in this way, the probability amplitude of a solution below the threshold can be amplified to some extent and then measured by repeating the loop.

[0049] Here, when the threshold is updated, the ideal number of amplification cycles r opt (The short dotted line in Figure 2) will always increase. This is because if the threshold decreases, the number of solutions below the threshold also decreases, which corresponds to a decrease in t in equation (12). Therefore, by resetting the upper limit, m and r opt The gap widens, and it is presumed that amplification processing in some search steps is wasted. Therefore, in this embodiment, we propose a new method for controlling the number of amplification cycles that eliminates the resetting of the upper limit in the above method.

[0050] Figure 3 shows an example of the progression of the number of amplification cycles using the proposed method. Figure 3 is a graph showing an example of the progression of the number of amplification cycles using the proposed method.

[0051] In Figure 3, if we look at the short dotted line representing the ideal number of amplification cycles and the solid line representing the upper limit of the number of amplification cycles, we can see that the difference is kept small compared to Figure 2, which shows the case when the upper limit is reset. Thus, the proposed method allows r opt Improved responsiveness to the system and more efficient oracle calls are expected to lead to a reduction in the total number of oracle calls (and thus computational load) in the entire algorithm.

[0052] Other proposed methods include linearly increasing the upper limit while keeping the upper limit reset, and selecting the number of amplification cycles according to a pre-optimized pattern. A comparison of these methods will be discussed in Chapter 4.

[0053] 3.2 Termination Conditions The termination condition for GAS is the cumulative number of oracle calls.

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[0054]

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[0055] 4. Numerical experiments and analysis This chapter analyzes the control method for the number of amplifications described in Section 3.1, from the perspective of convergence to the optimal solution and the number of oracle calls. First, Section 4.1 introduces a faster and more memory-efficient GAS simulation method. Next, Section 4.2 evaluates the performance of the proposed method in comparison with conventional methods. Section 4.3 compares the proposed method with conventional methods and methods from previous studies, using TSP as an example, to clarify the superiority of the proposed method. Section 4.4 shows that the proposed method is effective for problems of various scales. Finally, Section 4.5 considers the optimal termination conditions based on the above simulation results.

[0056] 4.1 Simulation Method From a practical standpoint, the advantages of GAS become apparent when we need to find the optimal solution to a combinatorial optimization problem with a number of bits that cannot be solved in a realistic amount of time using brute-force search on existing classical computers. Therefore, it is reasonable to assume that GAS will be used for problems with at least 60 bits or more.

[0057] On the one hand, classical computers cannot efficiently simulate quantum states. In the case of FTQC algorithms such as GAS, even with the computing power of supercomputers, the scale is limited to about 40 bits. Furthermore, in the case of algorithms that perform sampling while changing quantum circuit parameters such as the amplification times and thresholds of GAS, since the calculation of the quantum state vector is required for each sampling, it becomes difficult to evaluate the performance from the length of the calculation time.

[0058] Therefore, in order to clarify the scalability of the solution performance of GAS with respect to the number of bits, the present embodiment provides a new simulation method of GAS as shown in Algorithm2 of FIG. 4. In this method, in a problem where the solution distribution is known, sampling of quantum states is treated as a simple probability process. As a result, the calculation of the state vector is no longer necessary, and the behavior of GAS can be simulated quickly and with less memory.

[0059] In this simulation method, a "rank" is introduced to represent which is the smallest objective function value to be returned. N = 2 n As shown in the figure, ranks from 1 to N are assigned to each bit string of a certain solution candidate, and 1 is considered as the rank of the optimal solution. For the sake of convenience, the same rank assignment is also performed when the same function value is obtained from two or more inputs. When simulating various amplification times control methods, the generator A of the amplification times can be designed according to the method.

[0060] The initial rank ρ0 is randomly selected from integers from 1 to N, and ρ is the lowest rank among the solution candidates sampled up to the i-th search step. i At this time, at the i + 1 step of GAS, the solution candidates with ranks less than ρ i become the search targets of Grover search. From Equation (9), the probability of observing any one of them is sin 2 ((2r + 1)θ). Algorithm2 simulates this behavior of GAS. With probability sin​​​The random variable b is zero at ((2r+1)θ). i Using b i When = 1, ρ i +1 to ρ i Update by randomly selecting an integer between 1 and less than, b i When = 0, ρ i +1 to ρ i Leave it as is.

[0061] ρ i The probability that = 1 will be called the success probability, and in Section 4.2 and beyond, we will compare the methods by examining the relationship between the number of oracle calls and the success probability.

[0062] In Algorithm2, the objective function

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[0063] If the same function value is obtained from multiple inputs, the rank ρ in the next search step is... i The method for selecting +1 (item 10 in Algorithm2) needs to be changed. An example of this case is shown in Figure 5 using a simplified function. Figure 5 shows the rank transition when f(x)=x0+x1 and ρ0=4. Note that in Figure 5, T=(2r+1)θ.

[0064] 4.2 Evaluation of the proposed method In Algorithm1, the standard method for selecting the number of amplification cycles is known to yield the best solution performance when the upper limit amplification ratio λ = 1.34. To investigate the optimal value of λ even in the case of the proposed method without reset, we first performed 10,000 simulations on a uniform rank distribution problem with n=40 bits, varying λ in increments of 0.05 from 1.05 to 1.5. The benchmark results are shown in Figure 6. Figure 6 is a graph showing the relationship between the upper limit increase coefficient λ and the solution performance.

[0065] In the graph shown in Figure 6, the horizontal axis represents the number of oracle calls.

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[0066] In Figure 6, the data points in the upper left indicate that the optimal solution was reached with less computation. Comparing the performance of λ based on the number of oracle calls at the point where a 99% success rate was achieved, the case with λ = 1.1 showed the best performance.

[0067] Based on the above results, we compared the performance of the Algorithm1 method using a value of λ=1.34 and the proposed method using a value of λ=1.1. The results are shown in Figure 7. Figure 7 is a graph showing the comparison of the solution-finding performance of the conventional method and the proposed method.

[0068] In Figure 7, the solid line in the upper left shows the experimental data for the proposed method. When comparing the number of oracle calls required to obtain the optimal solution with a 99% probability, the proposed method reaches the optimal solution in approximately 22% fewer calls than the conventional method. This means that the computational complexity has been reduced by 22%, which supports the superiority of the proposed method.

[0069] 4.3 Performance comparison with previous studies We will refer to the method of linearly increasing the upper limit as "Linear," and the method of calculating the probability distribution of rank ρ for a uniform rank distribution, pre-generating a sequence of optimal amplification counts, and then executing GAS as "Fixed." The latter "Fixed" method has the problem that the computational cost required to generate the sequence of optimal amplification counts becomes enormous as the problem size increases. Therefore, we used the small-scale Traveling Salesperson Problem (TSP) for comparison.

[0070] When solving TSP using GAS, the following QUBO function of the TSP is used as the objective function.

[0071]

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[0072] In equation (14), d ij x is the distance between cities i and j. t i is a binary variable that is 1 when city i is visited at time t. In equation (14), the first term represents the distance traveled, the second term represents the constraint that each city is visited only once, and the third term represents the constraint that multiple cities are not visited at the same time. ij Figure 8 plots the rank distribution for λ=2 in four TSPs where cities are located on the vertices of a square. Figure 8 is a graph showing the rank distribution of TSPs. In the graph shown in Figure 8, the horizontal axis is the index (0 represents the optimal solution) when groups of inputs with the same function value are sorted in order of function value, and the vertical axis is the number of inputs included in each group.

[0073] Figure 9 shows the results of comparing each control method for the number of amplification steps using this rank distribution. Figure 9 is a graph showing the comparison results with the proposed method using 4-point TSP. For the Linear method, simulations were performed while changing the increment coefficient of the upper limit, similar to Figure 6, and the results when the best value was selected are shown in Figure 9. As shown in Figure 9, the proposed method consistently has a higher success rate than the conventional method and the Linear method.

[0074] While Fixed has some areas where its solution performance surpasses the proposed method, in most cases the proposed method yielded better results. This is likely because Fixed is optimized for a uniform rank distribution, resulting in poorer solution performance when solving specific problems such as TSP. In addition, as mentioned earlier, for large-scale problems, the computational cost of the optimal number of amplification iterations must be considered, making the proposed method the best choice.

[0075] 4.4 Performance scalability relative to the number of bits This section presents the results of an investigation into how the solution performance of the proposed method changes when the number of bits in the problem is varied. For a uniform rank distribution, the performance was examined for the number of bits from 10 to 60 in increments of 5, with λ = 1.1 fixed. Figure 10 shows a graph illustrating the relationship between the number of qubits and the solution performance. As shown in Figure 10, it was confirmed that there is almost no change in the success probability beyond 20 bits. Therefore, it can be inferred that the proposed method is applicable to larger-scale problems.

[0076] 4.5 Discussion: Termination Conditions In Algorithm1, the number of oracle calls R is

number

[0077] Based on the results so far, executing GAS with this termination condition results in a redundant algorithm that calls the oracle and performs amplification processing when the probability of success is almost 1.

[0078] Considering the above, an improved version of Algorithm1's GAS, Algorithm3, is shown in Figure 11. This improved version eliminates the reset of the upper limit, allowing for the selection of an efficient number of amplification cycles while avoiding redundant processing.

[0079] Termination conditions

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[0080] Furthermore, looking at the relationship between the number of oracle calls and the average rank of the obtained solutions, as shown in Figure 12,

number

[0081] Therefore, if it is sufficient to be somewhat close to the optimal solution, you can set a smaller termination condition.

[0082] 5. Conclusion In this embodiment, we investigated the number of amplification iterations for GAS solving QUBO and HUBO problems such as combinatorial optimization problems, and proposed a new method for controlling the number of amplification iterations that eliminates the reset of the upper limit. By comparing the results of GAS simulations using the control method from previous studies, we were able to confirm the superiority of the proposed method. Furthermore, by performing simulations while changing the number of bits, we demonstrated that the proposed method is effective even for larger-scale problems. Based on these simulation results, we proposed an improved GAS framework in Algorithm3. This framework can be applied when solving QUBO and HUBO problems with GAS, and is expected to yield optimal solutions more quickly.

[0083] <System Configuration> Figure 13 is a block diagram showing an example configuration of the optimization system 1 according to this embodiment. Optimization system 1 is a system that performs the Grover adaptive search described above, and consists of a quantum computer 10 and an optimization device 20, which is an example of a classical computer. The quantum computer 10 and the optimization device 20 are connected by a predetermined communication network and can send and receive information from each other.

[0084] The quantum computer 10 includes a quantum circuit device 11 and a measurement device 12. The quantum circuit device 11 is a device that generates quantum states related to the Grover search described above. The measurement device 12 is a device that measures the quantum states in the quantum circuit device 11. The measurement device 12 may transmit the measurement results (i.e., candidate solutions) to the optimization device 20.

[0085] The optimization device 20 comprises a processor 21 and a memory 22. The processor 21 works in cooperation with the memory 22 to execute a predetermined program, thereby realizing the functions according to this embodiment. The processor 21 may be referred to by other terms such as Central Processing Unit (CPU), Large Scale Integration (LSI), or controller. The memory 22 stores the program and data necessary to realize the functions according to this embodiment. The memory 22 may be composed of a volatile storage medium and / or a non-volatile storage medium.

[0086] The optimization device 20 may include, as a function of this embodiment, a threshold update processing unit 31 and an amplification count control unit 32.

[0087] As described above, when the threshold update processing unit 31 obtains a solution from the measuring device 12 in which the objective function value is less than the threshold, it updates the threshold to the next threshold based on the objective function value of that solution and sets that next threshold in the quantum circuit device 11.

[0088] The amplification count control unit 32 determines the next upper limit by multiplying the upper limit m by λ, regardless of whether the objective function value of the candidate solution obtained from the measuring device 12 is below a threshold, and randomly selects the next amplification count r from integer values ​​less than that next upper limit. λ may be a value between 1.05 and 1.2, and preferably 1.1. Here, if the value is between 1.05 and 1.2, it corresponds to an amplification ratio that achieves a 99% success rate even within the range shown in Figure 6. In other words, this range of values ​​can be said to be a range in which it has been experimentally proven that it is possible to achieve a 99% success rate while suppressing the number of oracle calls. The amplification count control unit 32 sets the selected next amplification count r in the quantum circuit device 11.

[0089] The processor 21 may terminate the Grover adaptive search described above when the cumulative value of the amplification count r, i.e., the cumulative value of the number of oracle calls R, becomes greater than the number of oracle calls required to obtain the optimal solution with a 99% probability. The number of oracle calls required to obtain the optimal solution with a 99% probability is:

number

[0090] Alternatively, the processor 21 may terminate the Grover adaptive search described above when the cumulative value of the amplification count r, i.e., the cumulative value R of the number of oracle calls, exceeds a predetermined value. This predetermined value may be determined by the user based on, for example, the relationship between the number of oracle calls and the average rank of the solution obtained for that number of oracle calls, as shown in Figure 12. For example, if the user is satisfied with obtaining a solution that is somewhat close to the optimal solution, even if it is not the optimal solution, the Grover adaptive search can be terminated in a shorter time by setting the predetermined value to a smaller value than the value at which the optimal solution is obtained with a 99% probability.

[0091] (Summary of this embodiment) The following technologies are disclosed based on the above description of this embodiment.

[0092] <Technology 1> In an optimization device 20 that obtains a solution from a quantum computer 10 using Grover adaptive search, the device includes a processor 21 and a memory 22. The processor 21 works in cooperation with the memory 22 to determine the next upper limit based on the current upper limit when a solution below a threshold is observed in the quantum computer 10 at a randomly selected number of amplification steps below an upper limit, and when updating the threshold to the next threshold, the processor 21 randomly selects the next number of amplification steps from an integer below the next upper limit. This allows the optimal solution to be reached with less computation compared to conventional methods that reset (initialize) the upper limit each time the threshold is updated.

[0093] <Technology 2> In the optimization apparatus 20 described in Technology 1, the processor 21 sets the next upper limit to λ times the upper limit, where λ is between 1.05 and 1.2. This allows the optimal solution to be reached with less computation compared to conventional methods that reset (initialize) the upper limit each time the threshold is updated.

[0094] <Technology 3> In the optimization device 20 described in Technology 1 or 2, the processor 21 terminates the Grover adaptive search when the cumulative value R of the number of amplifications becomes greater than the number of oracle calls required to obtain the optimal solution with a 99% probability. This helps to suppress redundant oracle calls.

[0095] <Technology 4> In the optimization device 20 described in Technology 3, the number of oracle calls required to obtain the optimal solution with a 99% probability is:

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[0096] <Technology 5> In the optimization apparatus 20 described in Technology 1 or 2, the processor 21 terminates the Grover adaptive search when the cumulative value R of the number of amplification cycles exceeds a predetermined value, the predetermined value being determined by the user based on the relationship between the number of oracle calls and the average rank of the solution obtained for that number of oracle calls. This allows, for example, users who are satisfied with a solution that is somewhat close to the optimal solution, even if it is not the absolute best solution, to complete the Grover adaptive search in a shorter time.

[0097] <Technology 6> As an optimization method for obtaining a solution from a quantum computer using Grover adaptive search, a solution below the threshold is observed in quantum computer 10 at an amplification count randomly selected from integer values ​​below the upper limit. When updating this threshold to the next threshold, the next upper limit is determined based on the current upper limit, and the next amplification count is randomly selected from integer values ​​below the next upper limit. This allows the optimal solution to be reached with less computation compared to conventional methods that reset (initialize) the upper limit each time the threshold is updated.

[0098] <Technology 7> The optimization program causes a classical computer (e.g., optimization device 20) to execute the optimization method described in Technology 6. This allows the optimal solution to be reached with less computation compared to conventional methods that reset (initialize) the upper limit each time the threshold is updated.

[0099] While embodiments have been described above with reference to the attached drawings, this disclosure is not limited to such examples. It is clear to those skilled in the art that various modifications, alterations, substitutions, additions, deletions, and equivalents can be conceived within the scope of the claims, and these are also understood to fall within the technical scope of this disclosure. Furthermore, the components of the embodiments described above may be combined in any way without departing from the spirit of the invention. [Industrial applicability]

[0100] The technology disclosed herein is useful for optimization devices, optimization methods, and optimization programs that perform Grover adaptive search. [Explanation of Symbols]

[0101] 1. Optimization System 10 Quantum Computers 11 Quantum circuit device 12 Measuring device 20 Optimization device 21 processors 22 memory 31 Threshold update processing unit 32 Amplification Count Control Unit

Claims

1. An optimization device that obtains solutions from a quantum computer using Grover adaptive search, The system comprises a processor and memory, and the processor cooperates with the memory, In the quantum computer, when a solution is observed in which the objective function value is less than the threshold at an amplification count randomly selected from integer values ​​less than the upper limit, and the threshold is updated to the next threshold, the next upper limit is determined based on the upper limit, and the next amplification count is randomly selected from integer values ​​less than the next upper limit. Optimization device.

2. The processor sets the next upper limit to λ times the upper limit, where λ is between 1.05 and 1.

2. The optimization apparatus according to claim 1.

3. The processor terminates the Grover adaptive search when the cumulative number of amplifications exceeds the number of oracle calls required to obtain the optimal solution with a 99% probability. The optimization apparatus according to claim 1.

4. The number of oracle calls required to obtain the optimal solution with a 99% probability is: [Math 1] And N is the number of candidate solutions. The optimization apparatus according to claim 3.

5. The processor terminates the Grover adaptive search when the cumulative value of the amplification cycles exceeds a predetermined value. The predetermined value is determined by the user based on the relationship between the number of oracle calls and the average rank of the solution obtained for that number of oracle calls. The optimization apparatus according to claim 1.

6. This is an optimization method that obtains solutions from quantum computers using Grover adaptive search, In the quantum computer, when a solution is observed in which the objective function value is less than the threshold at an amplification count randomly selected from integer values ​​less than the upper limit, and the threshold is updated to the next threshold, the next upper limit is determined based on the upper limit, and the next amplification count is randomly selected from integer values ​​less than the next upper limit. Optimization method.

7. The optimization method described in claim 6 is made to be executed by a classical computer. Optimization program.

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