Optimization device, optimization method, and optimization program

Abstract

Provided is an optimization device that uses Grover Adaptive Search to obtain a solution from a quantum computer, wherein, when a solution is observed in which an objective function value is less than a threshold value at an amplification frequency randomly selected from integer values less than an upper limit value in the quantum computer, and the threshold value is updated to the next threshold value, the next upper limit value is determined on the basis of the upper limit value, and the next amplification frequency is randomly selected from integer values less than said next upper limit value.

Description

Optimization device, optimization method, and optimization program

[0001] The present disclosure relates to an optimization device, an optimization method, and an optimization program.

[0002] In a binary optimization problem in which an input value x that minimizes an objective function is searched for, Grover Adaptive Search (GAS) is known, which is guaranteed to be quadratically accelerated compared to a full search using a classical computer.

[0003] 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.Durr, C. and Hoyer, P.: A quantum algorithm for finding the minimum, arXiv preprint quant-ph / 9607014 (1996).Boyer, M., Brassard, G., Hoyer, P. and Tapp, A.: Tight boundson quantum searching, Fortschritte derPhysik:Progress of Physics, Vol. 46, No. 4-5, pp. 493-505 (1998).

[0004] In Grover search, a subroutine of Grover adaptive search, the number of amplifications must be appropriately set according to the number of solutions contained in the search region. However, because Grover adaptive search narrows down the solutions whose 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 amplifications must be determined randomly.

[0005] In the standard method for controlling the number of amplifications, the number of amplifications is randomly selected from natural numbers less than an upper limit that increases exponentially at each search step. This upper limit is reset each time the solution is updated. However, since the ideal value of the number of amplifications increases monotonically with each solution update, resetting the number of amplifications results in an inefficient selection.

[0006] Non-Patent Document 1 discloses a method for controlling the increase in the upper limit value for each search step linearly rather than exponentially, but the upper limit value is reset each time the solution is updated, so the above-mentioned problem cannot be solved.

[0007] An object of the present disclosure is to provide a novel method for controlling the number of amplifications in Grover adaptive search.

[0008] An optimization device according to one aspect of the present disclosure is an optimization device that obtains a solution from a quantum computer using Grover adaptive search, and is equipped with a processor and memory. The processor cooperates with the memory to determine, when a solution is observed in the quantum computer where the objective function value is less than a threshold value at an amplification number randomly selected from integer values ​​less than an upper limit value, and when updating the threshold to the next threshold value, the next upper limit value is determined based on the upper limit value, and the 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 for obtaining a solution from a quantum computer using Grover adaptive search, wherein a solution is observed in the quantum computer where the objective function value is less than a threshold value at an amplification count randomly selected from integer values ​​less than an upper limit value, and when updating the threshold to the next threshold value, the next upper limit value is determined based on the upper limit value, and the next amplification count 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 comprehensive or specific aspects may be realized as a system, an apparatus, a method, an integrated circuit, a computer program, or a recording medium, or may be realized as any combination of a system, an apparatus, a method, an integrated circuit, a computer program, and a recording medium.

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

[0013] Graph showing an example of the transition of the number of amplifications using a standard method. Graph showing an example of the transition of the number of amplifications using the proposed method. Graph showing an algorithm for one example of a new GAS simulation method. Graph showing rank transition. Graph showing the relationship between the upper limit increase coefficient λ and solution-finding performance. Graph showing the results of a comparison of solution-finding performance between the conventional method and the proposed method. Graph showing TSP rank distribution. Graph showing the results of a comparison with the proposed method using four-point TSP. Graph showing the relationship between the number of qubits and solution-finding performance. Graph showing an example of an improved GAS algorithm that does not reset the upper limit. Graph showing the relationship between the number of oracle calls and the average rank of the obtained solutions. Block diagram showing an example of the configuration of an optimization system according to this embodiment.

[0014] Hereinafter, embodiments of the present disclosure will be described in detail with appropriate reference to the drawings. However, more detailed description than necessary may be omitted. For example, detailed descriptions of well-known matters and redundant descriptions of substantially identical configurations may be omitted. This is to avoid unnecessary redundancy in the following description and to facilitate understanding by those skilled in the art. Note that the accompanying 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] (Present Embodiment) 0. Overview In binary optimization problems that search for an input value x that minimizes a certain objective function, Grover Adaptive Search (GAS), which is guaranteed to be quadratically accelerated compared to a full search by a classical computer, is known. GAS calls Grover Search (GS), a well-known basic quantum algorithm, as a subroutine to search for an input that results in a function value less than a threshold, and if found, updates the threshold with that function value to narrow down the search targets. To efficiently search for a solution, it is necessary to appropriately select the number of iterations (number of amplifications) of the circuit that amplifies the desired state in GS at each step.

[0016] In this embodiment, we consider this amplification number and propose a new method for controlling the amplification number. Simulation results show that this method can obtain an optimal solution with less computational effort than the standard control method. The proposed method has the potential to significantly reduce the computation time for combinatorial optimization problems in the future.

[0017] 1. Introduction Combinatorial optimization problems are problems that require inputs that minimize a certain objective function. They have a wide range of applications, including supply chain personnel scheduling, optimal delivery route planning, and financial portfolio optimization. Among these, the quadratic optimization problem with unconstrained binary variables (QUBO: Quadratic Unconstrained Binary Optimization) and the higher-order optimization problem with unconstrained binary variables (HUBO: Higher-Order Unconstrained Binary Optimization) are actively researched. Even with conventional classical computers, optimal solutions can always be obtained by searching for solutions through a brute force approach. However, for large-scale problems, the computational costs are enormous, making it impossible to obtain optimal solutions within a realistic timeframe. Therefore, several approaches have been proposed for solving QUBO and HUBO using quantum computing.

[0018] First, quantum annealing, also known as simulated quantum annealing, is a heuristic technique that skillfully utilizes quantum effects such as state superposition. Quantum annealing can input QUBO as is, but when dealing with HUBO, auxiliary bits must be added to convert it to QUBO, and the increase in the number of bits has a negative impact on the solution. While dedicated hardware is required to use quantum annealing, services that provide this on the cloud are already commercially available, and a hybrid solver with classical and quantum algorithms can solve problems with up to 20,000 variables.

[0019] Next, there are also methods such as the Quantum Approximate Optimization Algorithm (QAOA) and the Feedback-based Algorithm for Quantum Optimization (FALQON). These methods are said to be executable on general-purpose quantum computers (NISQ: Noisy Intermediate-Scale Quantum devices) that can handle a small number of bits and do not have error correction capabilities.

[0020] Finally, there is Grover Adaptive Search (GAS), which is the main focus of this embodiment. In GAS, to find the optimal solution, the objective function value of the best solution at each search step is set as a threshold, and Grover Search (GS) is repeatedly performed, searching for solutions below that threshold, to search for a better solution. GAS has an advantage over quantum annealing in that it can solve HUBO problems as they are. Furthermore, while the previously mentioned methods may only be able to obtain approximate solutions (locally optimal solutions), GAS has the distinct property of being able to obtain optimal solutions for any problem. However, since deep quantum circuits are required, a fault-tolerant quantum computer (FTQC) capable of error correction is required for execution. It has been theoretically proven that GAS can be quadratically accelerated compared to classical full search. For this reason, when FTQC becomes a reality in the future, it is thought to be a promising approach to combinatorial optimization.

[0021] One of the challenges in operating GAS is how to control the parameter known as the number of amplifications (see Section 2.2 for details). In addition to the method proposed in the original GAS paper (Non-Patent Document 2), several other methods have been proposed, but it is not yet clear which is the optimal method. Optimizing the method for controlling the number of amplifications is expected to reduce the constant overhead of computational complexity, making this an important issue in applying GAS.

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

[0023] This embodiment is structured as follows. Chapter 2 explains GS and GAS as the theoretical background. Chapter 3 then considers the number of amplifications and proposes a new control method. Chapter 4 explains the GAS simulation method and verifies the effectiveness of the proposed method. Finally, Chapter 5 provides a conclusion.

[0024] 2. Theoretical Background 2.1 Grover Search In this section, we first summarize GS, a major component of GAS. In GS, the probability amplitude of the state corresponding to the desired search target is amplified before measuring the quantum state. The following three operators are used to construct this quantum state:

[0025]

[0026]

[0027] is an operator that inverts the sign of only the desired state and must be prepared individually depending on the problem. GAS uses an oracle that inverts the sign of states below a certain threshold. It is known that a GAS oracle can be efficiently constructed by embedding a function value in the phase of the auxiliary bit, extracting it using an inverse quantum Fourier transform, and comparing it with a threshold.

[0028] The dimension of the search space is N = 2 ｎ (n: number of bits) and the number of search targets is t.

[0029]

[0030] First, create a superposition of all N possible states using a Hadamard gate.

[0031]

[0032]

[0033]

[0034]

[0035]

[0036]

[0037]

[0038]

[0039]

[0040]

[0041]

[0042] The probability of observing either ２ Therefore, the ideal number of amplifications that allows the desired state to be observed with a probability of 1 is expressed by the following formula.

[0043]

[0044] In practice, the number of search targets, t, is often unknown. In the case of GAS, t corresponds to the number of inputs for which the objective function value is less than the threshold. Since t cannot be accurately known without exhaustively determining the distribution of solutions, it is difficult to select the ideal number of amplifications. In this case where t is unknown, if the method proposed in Non-Patent Document 3 is followed, the expected number of oracle calls required to obtain the desired state is at most

[0045]

[0046] It has been shown that

[0047] The method is to repeat the above GS many times and randomly select the number of amplifications from integer values ​​less than the upper limit that increases exponentially according to the number of repetitions. In GAS, which will be described in detail in the next section, this method is applied every time the threshold is updated.

[0048] 2.2 Grover Adaptive Search In GAS, a certain objective function is searched by sampling the quantum state.

[0049]

[0050] Generate a bit string that is a candidate for the argument (=optimal solution) that gives the minimum value for

[0051]

[0052] 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 that is a superposition of all possible solutions, as in the Grover search described above. The number of calls to the oracle that determines whether a certain bit string x is a search target is generally used as an evaluation index for a search algorithm. When finding the optimal solution by exhaustive search, the oracle must be applied once for every bit string of length n, so the order of the number of calls to the oracle is O(N) (N = 2 ｎ )

[0053] On the other hand, according to Non-Patent Document 2, in GAS, the number of oracle calls required to obtain the optimal solution with a probability of 1 / 2 or more is at most

[0054]

[0055] It has been shown that:

[0056] This shows that GAS can accelerate the computational complexity by a factor of two compared to classical exhaustive search.

[0057] Details of the algorithm proposed in Non-Patent Document 2 are shown in Fig. 1 as Algorithm 1. m and r are the upper limit of the number of amplifications in each iteration and the selected number of amplifications, respectively.

[0058]

[0059] R is the cumulative number of oracle calls, and represents the computational cost up to each search step. The general flow of Algorithm 1 shown in Figure 1 is "repeatedly execute Grover search to search for solutions below the threshold, and if a solution below the threshold is observed, update the threshold with the objective function value to narrow down the solution space." In general, since the number of solutions below the threshold is unknown, the ideal amplification number r expressed by Equation (12) is ｏｐｔ It is also unclear. Several methods for selecting the number of amplifications have been proposed other than the one adopted in Algorithm 1, and as one of the major contributions of this report, we have devised a new method for selecting the number of amplifications. Details of this method are described in Section 3.1.

[0060] 3. Proposed Method 3.1 Method for Controlling the Number of Amplifications: No Upper Limit Reset First, we will explain the most standard method for controlling the number of amplifications, as described in Algorithm 1, and adopted in papers dealing with Qiskit libraries and GAS. In this method, the number of amplifications r is randomly selected from integer values ​​less than the upper limit m of the number of amplifications. m starts with an initial value of 1 and is multiplied by λ for each loop 3-12 of Algorithm 1. Each time a solution less than the threshold is observed and the threshold is updated, m is reset to 1 (the double-lined portion in Algorithm 1). This is because GS according to the method in Non-Patent Document 3 mentioned in Section 2.1 is applied to each threshold.

[0061] Figure 2 shows an example of the results of changes in the number of amplifications (long dashed line in Figure 2) and the upper limit (solid line in Figure 2) in the GAS simulation. Figure 2 is a graph showing an example of changes in the number of amplifications using a standard method.

[0062] By randomly selecting the number of amplifications in this way, by repeating the loop, it is possible to measure the probability amplitude of a solution below the threshold after amplifying it to some extent.

[0063] Here, when the threshold is updated, the ideal number of amplifications r ｏｐｔ (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 It is presumed that the difference between the maximum and minimum values ​​widens, resulting in the amplification process being wasted in some search steps. Therefore, in this embodiment, a new method for controlling the number of amplifications is proposed, which does not require resetting the upper limit value in the above method.

[0064] An example of the transition of the number of amplifications using the proposed method is shown in Figure 3. Figure 3 is a graph showing an example of the transition of the number of amplifications using the proposed method.

[0065] In Figure 3, if we look at the short dotted line indicating the ideal number of amplifications and the solid line indicating the upper limit of the number of amplifications, we can see that the difference is kept small compared to Figure 2, where the upper limit is reset. ｏｐｔIt is expected that improved tracking and efficient oracle calls will lead to a reduction in the number of oracle calls (= amount of calculation) throughout the algorithm.

[0066] Other methods proposed include linearly increasing the upper limit without resetting it, and selecting the number of amplifications according to a pre-optimized pattern. A comparison with these methods will be made in Chapter 4.

[0067] 3.2 Termination Condition The termination condition for GAS is when the cumulative number of oracle calls reaches

[0068]

[0069] is to exceed.

[0070]

[0071] is a value set based on a value that minimizes the expected number of oracle calls required to find the optimal solution. Therefore, even if the optimal solution is already found with a sufficiently high probability under this termination condition, there is a possibility that the algorithm will not terminate. In this embodiment, we propose a termination condition that takes into account the simulation results of the proposed method in Chapter 4. This is expected to avoid the redundant processing mentioned above and realize a highly efficient GAS that completes the search in a shorter time.

[0072] 4. Numerical Experiments and Analysis In this chapter, we analyze the method for controlling 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, in Section 4.1, we introduce a faster and more memory-efficient GAS simulation method. Next, in Section 4.2, we evaluate the performance of the proposed method in comparison with conventional methods. In Section 4.3, we compare the proposed method with conventional methods and methods from previous research using TSP as an example, and clarify the superiority of the proposed method. In Section 4.4, we demonstrate the effectiveness of the proposed method for problems of various scales. Finally, in Section 4.5, we consider the optimal termination conditions based on the above simulation results.

[0073] 4.1 Simulation Method From a practical standpoint, GAS has merit when it is necessary to find an optimal solution to a combinatorial optimization problem with a large number of bits that cannot be solved in a realistic time using an exhaustive search on an existing classical computer. Therefore, it is thought that GAS will be used for problems with at least 60 bits.

[0074] On the other hand, classical computers cannot efficiently simulate quantum states, and 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 number of amplifications and thresholds in GAS, it is necessary to calculate the quantum state vector for each sampling, making it difficult to evaluate performance based on the length of the calculation time.

[0075] Therefore, in this embodiment, in order to clarify the scalability of GAS solution performance with respect to the number of bits, we provide a new GAS simulation method, as shown in Algorithm 2 in Fig. 4. In this method, for problems with known solution distributions, the sampling of quantum states is treated as a simple stochastic process. This eliminates the need for state vector calculation, making it possible to simulate GAS behavior quickly and with minimal memory.

[0076] In this simulation method, a "rank" is introduced to indicate the smallest objective function value that is returned. N=2 ｎ A rank from 1 to N is assigned to each bit string of the possible solutions, with 1 being the rank of the optimal solution. For convenience, a similar rank is assigned even when the same function value is obtained from two or more inputs. When simulating various methods for controlling the number of amplifications, the generator A for the number of amplifications can be designed according to the method.

[0077] Initial rank ρ ０ is randomly selected from integers from 1 to N, and the lowest rank among the solution candidates sampled up to the i-th search step is ρ ｉ In this case, the rank at the i+1 step of GAS is ρ ｉThe Grover search targets candidate solutions less than . From equation (9), the probability of observing any of them is sin ２ ((2r+1)θ). Algorithm 2 simulates this behavior of GAS. Probability sin ２ ((2r+1)θ) is 1, probability cos ２ Random variable b that becomes 0 at ((2r+1)θ) ｉ Using b ｉ When ρ = 1, ｉ +1 to ρ ｉ Randomly select and update an integer from the integers less than 1 and greater, b ｉ When ρ = 0, ｉ +1 to ρ ｉ Leave it as it is.

[0078] ρ ｉ = 1 is called the success probability, and in Section 4.2 and subsequent sections, we compare the methods by examining the relationship between the number of oracle calls and the success probability.

[0079] In Algorithm 2, the objective function

[0080]

[0081] This corresponds to the case where the optimal solution is found using GAS for a problem where there is a one-to-one correspondence between the input bit string and the function value, as shown in the figure. This is called a uniform rank distribution problem.

[0082] If multiple inputs give the same function value, the rank ρ in the next search step ｉ The method of selecting +1 (10 in Algorithm 2) needs to be changed. An example of this is shown in Figure 5 using a simple function. Figure 5 shows the case where f(x) = x ０ +x １ , ρ ０ 5 shows the transition of rank when T=(2r+1)θ.

[0083] 4.2 Evaluation of the Proposed Method It is known that the standard method for selecting the number of amplifications used in Algorithm 1 achieves the best solution performance when the upper limit amplification factor λ is set to 1.34. First, to investigate the optimal value of λ even when no reset is used, which is the case with the proposed method, we performed a 10,000-shot simulation for a problem with a uniform rank distribution with n = 40 bits, varying λ from 1.05 to 1.5 in increments of 0.05. The results of the benchmark are shown in Figure 6. Figure 6 is a graph showing the relationship between the upper limit increase factor λ and solution performance.

[0084] In the graph shown in Figure 6, the horizontal axis represents the number of oracle calls.

[0085]

[0086] The vertical axis represents the probability of obtaining an optimal solution of rank 1 when the number of iterations is set as the termination condition. The shading of the graph represents the value of the parameter λ.

[0087] The data in the upper left of Figure 6 indicates that the optimal solution is converged to with less computational effort. When comparing the merits of λ based on the number of oracle calls at the point where a 99% success rate is achieved, λ = 1.1 showed the best performance.

[0088] Based on the above results, we compared performance by using λ = 1.34 for Algorithm 1 and λ = 1.1 for the proposed method. The results are shown in Figure 7. Figure 7 is a graph showing the comparison of solution performance between the conventional method and the proposed method.

[0089] In Figure 7, the solid line showing the experimental data for the proposed method is in the upper left. When comparing the "number of oracle calls required to obtain an 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 amount of calculation has been reduced by 22%, a result that supports the superiority of the proposed method.

[0090] 4.3 Performance Comparison with Previous Research 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, generating a sequence of optimal amplification counts in advance, and then executing GAS as "Fixed." The latter "Fixed" method has the problem that the amount of calculation required to generate a sequence of optimal amplification counts becomes enormous as the problem scale increases. For this reason, a small-scale traveling salesman problem (TSP) was used for comparison.

[0091] When solving the TSP using GAS, the following QUBO function of the TSP is used as the objective function.

[0092]

[0093] In formula (14), d ｉｊ is the distance between cities i and j, and x ｔ ｉ is a binary variable that is 1 if city i is visited at time t. In equation (14), the first term represents the travel distance, the second term represents the constraint that each city must be visited only once, and the third term represents the constraint that multiple cities must not be visited at the same time. ｉｊ Figure 8 shows a plot of the rank distribution for a four-city TSP where the cities are at the vertices of a square, with λ=2. Figure 8 is a graph showing the rank distribution of the TSP. In the graph shown in Figure 8, the horizontal axis represents 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 represents the number of inputs included in each group.

[0094] Figure 9 shows the results of a comparison of each amplification count control method using this rank distribution. Figure 9 is a graph showing the results of a comparison with the proposed method using 4-point TSP. For Linear, simulations were performed while changing the increase coefficient of the upper limit value, as in Figure 6, and Figure 9 shows the results when the best value was selected. As shown in Figure 9, the proposed method always has a higher success probability than the conventional method and Linear.

[0095] Although there are some areas where the solution performance of Fixed is superior to that of the proposed method, in most cases the proposed method produced better results. This is thought to be because Fixed is a method optimized for uniform rank distribution, and therefore its solution performance deteriorates when solving specific problems such as the TSP. In addition, as mentioned above, for large-scale problems, the computational cost of the optimal number of amplifications must be taken into account, so it can be said that using the proposed method is the best option.

[0096] 4.4 Performance Scalability with Bit Count We present the results of an investigation into how solution performance changes when the number of bits in the problem is changed using the proposed method. Figure 10 shows the data obtained by fixing λ = 1.1 and examining performance by varying the number of bits from 10 to 60 in increments of 5 for a uniform rank distribution. Figure 10 is a graph showing the relationship between the number of quantum bits and solution performance. As shown in Figure 10, we were able to confirm that there is almost no change in the success probability after 20 bits. Therefore, we infer that the proposed method can be applied to larger-scale problems.

[0097] 4.5 Discussion: Termination Condition In Algorithm 1, the number of oracle calls R is

[0098]

[0099] Exceeding this was the termination condition.

[0100] From the results so far, if GAS is executed under this termination condition, the success probability will be close to 1, resulting in a redundant algorithm that calls the oracle and performs amplification processing.

[0101] Considering the above, an improved version of the GAS of Algorithm 1, which does not reset the upper limit so that redundant processing can be avoided and an efficient number of amplifications can be selected, is shown as Algorithm 3 in FIG.

[0102] Termination condition

[0103]

[0104] is the number of oracle calls that will give an optimal solution with a 99% probability when solving a 40-bit uniform rank distribution problem.

[0105] Also, looking at the relationship between the number of oracle calls and the average rank of the obtained solutions shown in Figure 12,

[0106]

[0107] Even if we use this as the termination condition, the seventh best solution is obtained on average.

[0108] Therefore, if it is sufficient to get close to the optimal solution to some extent, the termination condition may be set to a lower value.

[0109] 5. Conclusion In this embodiment, we considered the number of amplifications in GAS for solving combinatorial optimization problems such as QUBO and HUBO, and proposed a new method for controlling the number of amplifications that eliminates the resetting of the upper limit. We compared the results of GAS simulations using control methods from previous studies and confirmed 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 Algorithm 3. This framework can be applied to solving problems such as QUBO and HUBO using GAS, and is expected to obtain optimal solutions more quickly.

[0110] <System Configuration> FIG. 13 is a block diagram showing an example of the configuration of an optimization system 1 according to this embodiment.

[0111] The optimization system 1 is a system that performs the Grover adaptive search described above, and includes 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 via a predetermined communication network, and can send and receive information to and from each other.

[0112] 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 a quantum state related to the Grover search described above. The measurement device 12 is a device that measures the quantum state in the quantum circuit device 11. The measurement device 12 may transmit the measurement results (i.e., solution candidates) to the optimization device 20.

[0113] The optimization device 20 includes a processor 21 and a memory 22. The processor 21 executes a predetermined program in cooperation with the memory 22 to realize the functions according to this embodiment. The processor 21 may be interpreted as other terms such as a central processing unit (CPU), a large scale integration (LSI), or a controller. The memory 22 stores programs and data for realizing the functions according to this embodiment. The memory 22 may be configured by a volatile storage medium and / or a non-volatile storage medium.

[0114] The optimization device 20 may have a threshold update processing unit 31 and an amplification count control unit 32 as functions according to this embodiment.

[0115] As described above, when the threshold update processing unit 31 obtains a solution from the measurement 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 the next threshold to the quantum circuit device 11.

[0116] Regardless of whether the objective function value of the solution candidate acquired from the measurement device 12 is less than the threshold, the amplification count control unit 32 multiplies the upper limit m by λ to determine the next upper limit, and randomly selects the next amplification count r from integer values ​​less than the next upper limit. λ may be a value between 1.05 and 1.2, preferably 1.1. Here, a value between 1.05 and 1.2 corresponds to an amplification ratio that achieves a 99% success rate even within the range shown in FIG. 6 . In other words, this value range can be said to be a range that has been experimentally proven to be capable of achieving a 99% success rate while suppressing the number of oracle calls. The amplification count control unit 32 sets the selected next amplification count r to the quantum circuit device 11.

[0117] The processor 21 may terminate the Grover adaptive search described above when the cumulative value of the number of amplifications r, i.e., the cumulative value R of the number of oracle calls, becomes larger than the number of oracle calls required to obtain an optimal solution with a 99% probability. The number of oracle calls required to obtain an optimal solution with a 99% probability is given by

[0118]

[0119] where N may be the number of candidate solutions, i.e., the dimension of the search space.

[0120] Alternatively, the processor 21 may terminate the Grover adaptive search described above when the cumulative value of the number of amplifications 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 the relationship between the number of oracle calls and the average rank of the solutions obtained for that number of oracle calls, as shown in FIG. 12 . For example, if the user is satisfied with obtaining a solution that is 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 value smaller than the value that results in the optimal solution being obtained with a 99% probability.

[0121] (Summary of the Present Embodiment) The above description of the present embodiment discloses the following techniques.

[0122] <Technology 1> An optimization device 20 that obtains a solution from a quantum computer 10 using Grover adaptive search includes a processor 21 and a memory 22, and the processor 21 cooperates with the memory 22 to determine, when a solution below a threshold is observed in the quantum computer 10 at an amplification count that is randomly selected from integer values ​​that are less than an upper limit, and when updating the threshold to a next threshold, the next upper limit is determined based on the upper limit, and the next amplification count is randomly selected from integer values ​​that are less than the next upper limit.

[0123] This makes it possible to reach an optimal solution with a smaller amount of calculation compared to the conventional method in which the upper limit value is reset (initialized) every time the threshold is updated.

[0124] <Technology 2> In the optimization device 20 described in Technology 1, the processor 21 sets the next upper limit value to λ times the upper limit value, where λ is between 1.05 and 1.2.

[0125] This makes it possible to reach an optimal solution with a smaller amount of calculation compared to the conventional method in which the upper limit value is reset (initialized) every time the threshold is updated.

[0126] <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.

[0127] This reduces redundant calls to the oracle.

[0128] <Technology 4> In the optimization device 20 described in Technology 3, the number of oracle calls required to obtain an optimal solution with a probability of 99% is:

[0129]

[0130] where N is the number of candidate solutions.

[0131] This reduces redundant calls to the oracle.

[0132] <Technology 5> 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 a predetermined value, where the predetermined value is determined by the user based on the relationship between the number of oracle calls and the average rank of the solutions obtained for that number of oracle calls.

[0133] This allows the Grover adaptive search to be completed in a shorter time, for example, when the user only needs to obtain a solution that is close to the optimal solution, even if it is not the optimal solution.

[0134] <Technology 6> As an optimization method for obtaining a solution from a quantum computer using Grover adaptive search, a solution below a threshold is observed in the quantum computer 10 at an amplification count randomly selected from integer values ​​below an upper limit value, and when updating the threshold to the next threshold value, the next upper limit value is determined based on the upper limit value, and the next amplification count is selected randomly from integer values ​​below the next upper limit value.

[0135] This makes it possible to reach an optimal solution with a smaller amount of calculation compared to the conventional method in which the upper limit value is reset (initialized) every time the threshold is updated.

[0136] <Technology 7> The optimization program causes a classical computer (for example, the optimization device 20) to execute the optimization method described in Technology 6.

[0137] This makes it possible to reach an optimal solution with a smaller amount of calculation compared to the conventional method in which the upper limit value is reset (initialized) every time the threshold is updated.

[0138] Although the embodiments have been described above with reference to the accompanying drawings, the present disclosure is not limited to such examples. It is clear that a person skilled in the art can conceive of various modifications, alterations, substitutions, additions, deletions, and equivalents within the scope of the claims, and it is understood that these also fall within the technical scope of the present disclosure. Furthermore, the components in the above-described embodiments may be combined in any manner without departing from the spirit of the invention.

[0139] The disclosures of the specification, drawings and abstract contained in Japanese Patent Application No. 2023-104409, filed on June 26, 2023, are incorporated herein by reference in their entirety.

[0140] The techniques of the present disclosure are useful for optimization devices, optimization methods, optimization programs, etc. that perform Grover adaptive search.

[0141] REFERENCE SIGNS LIST 1 Optimization system 10 Quantum computer 11 Quantum circuit device 12 Measurement device 20 Optimization device 21 Processor 22 Memory 31 Threshold update processing unit 32 Amplification number control unit

Claims

1. An optimization device that obtains a solution from a quantum computer using Grover adaptive search, comprising a processor and a memory, wherein the processor cooperates with the memory to: when a solution is observed in the quantum computer where the objective function value is less than a threshold at a number of amplifications randomly selected from integer values ​​less than an upper limit value, and when updating the threshold to a next threshold value, determine a next upper limit value based on the upper limit value, and randomly select the next number of amplifications from integer values ​​less than the next upper limit value.

2. The optimization device according to claim 1, wherein the processor sets a next upper limit value that is λ times the upper limit value, where λ is between 1.05 and 1.

2.

3. The optimization device according to claim 1, wherein the processor terminates the Grover adaptive search when the cumulative value of the number of amplifications becomes greater than the number of oracle calls required to obtain an optimal solution with a 99% probability.

4. The number of oracle calls required to obtain the optimal solution with 99% probability is The optimization device according to claim 3 , wherein N is the number of candidate solutions.

5. The optimization device of claim 1, wherein the processor terminates the Grover adaptive search when the cumulative value of the number of amplifications becomes greater than a predetermined value, the predetermined value being determined by a user based on a relationship between the number of oracle calls and the average rank of the solution obtained for that number of oracle calls.

6. An optimization method for obtaining a solution from a quantum computer using Grover adaptive search, wherein a solution in which the objective function value is less than a threshold is observed in the quantum computer at a number of amplifications randomly selected from integer values ​​less than an upper limit, and when updating the threshold to a next threshold, a next upper limit is determined based on the upper limit, and the next number of amplifications is randomly selected from integer values ​​less than the next upper limit.

7. An optimization program for causing a classical computer to execute the optimization method according to claim 6.

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