論文

Vehicle routing problems (VRPs) can be solved as optimization problems. Practical applications of the VRPs are involved in various areas including manufacturing, supply chain, and tourism. Conventional approaches using von Neumann computers obtain good approximate solutions to the optimization problems, but conventional approaches show disadvantages of computation costs in large-scale or complex problems due to the combinatorial explosion. Oppositely, Ising machines or quantum annealing machines are non-von Neumann computers that are designed to solve complex optimization problems. In this paper, we propose an Ising-machine based approach for the vehicle routing problem with balanced pick-up (VRPBP). The development of the VRPBP is motivated by postal items pick-up services in the real-world. Our approach includes various features of VRP variants. We propose a 2-phase approach to solve the VRPBP and key elements in each phase are mapped onto quadratic unconstrained binary optimization (QUBO) forms. Specifically, the first phase belongs to the clustering phase which is an extension to the knapsack problem with additional distance and load balancing concerns. The second phase is mapped to the traveling salesman problem. Experimental results of our approach are evaluated in terms of solution quality and computation time compared with conventional approaches.

Ising machines have recently attracted much attention because they are expected to solve combinatorial optimization problems efficiently. We focus on an Ising machine whose algorithm is based on parallel tempering (PT), and experimentally evaluate the performance of the Ising machine for MIN-CUT problems. Experimental results show that the Ising machine outperforms a famous graph partitioning solver in terms of the quality of solution and the time-to-target-solution.

Quantum annealing machines are next-generation computers for solving combinatorial optimization problems. Although physical simulations are one of the most promising applications of quantum annealing machines, a method how to embed the target problem into the machines has not been developed except for certain simple examples. In this study, we focus on a method of representing real numbers using binary variables, or quantum bits. One of the most important problems for conducting physical simulation by quantum annealing machines is how to represent the real number with quantum bits. The variables in physical simulations are often represented by real numbers but real numbers must be represented by a combination of binary variables in quantum annealing, such as quadratic unconstrained binary optimization (QUBO). Conventionally, real numbers have been represented by assigning each digit of their binary number representation to a binary variable. Considering the classical annealing point of view, we noticed that when real numbers are represented in binary numbers, there are numbers that can only be reached by inverting several bits simultaneously under the restriction of not increasing a given Hamiltonian, which makes the optimization very difficult. In this work, we propose three new types of real number representation and compared these representations under the problem of solving linear equations. As a result, we found experimentally that the accuracy of the solution varies significantly depending on how the real numbers are represented. We also found that the most appropriate representation depends on the size and difficulty of the problem to be solved and that these differences show a consistent trend for two annealing solvers. Finally, we explain the reasons for these differences using simple models, the minimum required number of simultaneous bit flips, one-way probabilistic bit-flip energy minimization, and simulation of ideal quantum annealing machine.

Topology optimization is an essential tool in computational engineering, for example, to improve the design and efficiency of flow channels. At the same time, Ising machines, including digital or quantum annealers, have been used as efficient solvers for combinatorial optimization problems. Beyond combinatorial optimization, recent works have demonstrated applicability to other engineering tasks by tailoring corresponding problem formulations. In this study, we present a novel Ising machine formulation for computing design updates during topology optimization with the goal of minimizing dissipation energy in flow channels. We explore the potential of this approach to improve the efficiency and performance of the optimization process. To this end, we conduct experiments to study the impact of various factors within the novel formulation. Additionally, we compare it to a classical method from the literature using the number of optimization steps and the final values of the objective function as indicators of the time intensity of the optimization and the performance of the resulting designs, respectively. Our findings show that the proposed update strategy can accelerate the topology optimization process while producing comparable designs. However, it tends to be less exploratory, which may lead to lower performance of the designs. These results highlight the potential of incorporating Ising formulations for optimization tasks but also show their limitations when used to compute design updates in an iterative optimization process. In conclusion, this work provides an efficient alternative for design updates in topology optimization and enhances the understanding of integrating Ising machine formulations in engineering optimization.

Ising machines are specialized solvers for combinatorial optimization problems (COPs), which are typically formulated as quadratic unconstrained binary optimization (QUBO) models. To enhance their performance, various methods have been proposed to construct subQUBOs by reducing the number of variables in the original QUBO. This study proposes a general subQUBO construction method applicable to a wide range of problems. The variables are clustered based on their interaction structure using the K-means clustering. A subQUBO is constructed from a randomly selected cluster and a pre-obtained tentative solution and then solved. This process of cluster selection and subQUBOs is performed iteratively. We evaluated the performance of the method on the quadratic assignment problem and compared it with two baselines: one that randomly reduces variables and another that directly solves the original QUBO without any variable reduction. Experimental evaluations demonstrated that the proposed method achieves high-accuracy solutions within the same computation time, particularly for large problem sizes.

Efficient and adaptive placement of virtualized RAN functions is crucial to satisfy the diverse and dynamic Quality of Service (QoS) requirements in 5G networks. While blackbox optimization methods such as Factorization Machine with Annealing (FMA) have shown promise for combinatorial optimization problems with limited data, their adaptability is limited in dynamic environments due to their reliance on historical data. In this paper, we propose an enhanced FMA approach that incorporates a Transformer-based time-series prediction model to calibrate the surrogate parameters at each time step, enabling adaptation to dynamic environments. Numerical experiments in simulated RAN environments demonstrate that the proposed calibration achieves a 0.50% improvement in mean QoS satisfaction ratio over that of standard FMA when a moderate calibration coefficient is used, providing the best tradeoff between adaptability and robustness.

Ising machines, quantum annealing (QA) machines, are promising for solving combinatorial optimization problems at high speed. Problems are formulated as Ising models where optimal solutions correspond to ground states, but stochastic noise often causes deviations, necessitating error reduction strategies. The quantum annealing correction (QAC) model introduces penalty spins to favor consistent solutions across replicas and has shown effectiveness in QA, but its behavior under simulated annealing (SA) remains insufficiently understood. This study systematically evaluates the stacked model, a recently proposed error reduction method, against QAC-based and simple parallelization approaches in SA. Our results demonstrate that, even for large-scale systems, the stacked model consistently outperforms alternatives as the number of replicas increases, maintaining relatively high success probability through spontaneous replica alignment via structured coupling. These findings highlight the stacked model as a scalable and error reduction framework for Ising-based optimization.

Ising machines, including quantum annealing machines, are promising next-generation computers for combinatorial optimization problems. Due to hardware limitations, most Ising-type hardware can only solve functions expressed in linear or quadratic terms of binary variables. Therefore, problems with higher-order terms require order reduction, which increases the number of variables and constraints and may degrade solution quality. This study evaluates the effectiveness of directly solving such problems without order reduction by using the polynomial unconstrained binary optimization (PUBO) solver implemented in Toshiba's Simulated Quantum-inspired Bifurcation Machine+ (SQBM+). We compare its performance against a conventional quadratic unconstrained binary optimization (QUBO) solver on the same hardware platform. As benchmarks, we use a vehicle routing problem with distance balancing and the low autocorrelation binary sequence (LABS) problem, both of which naturally include higher-order interactions. Results show that the PUBO solver consistently achieves superior solution quality and stability compared to its QUBO counterpart, while maintaining comparable computational time and requiring no order-reduction compilation, indicating the potential advantages of directly handling higher-order terms in practical optimization problems.

Quantum annealing is a promising algorithm for solving combinatorial optimization problems. However, various hardware restrictions significantly impede its efficient performance. Size-reduction methods provide an effective approach to addressing large-scale problems but often introduce additional difficulties. A notable hardware restriction is that quantum annealing can handle only a limited number of decision variables, compared to the size of the problem. Moreover, when employing size-reduction methods, the interactions and local magnetic fields in the Ising model — used to represent the combinatorial optimization problem — can become excessively large, making them difficult to implement on hardware. Although prior studies suggest that energy rescaling impacts the performance of quantum annealing, its interplay with size-reduction methods remains unexplored. This study examines the relationship between fixing spins, a promising size-reduction method, and the effects of energy rescaling. Numerical simulations and experiments conducted on a quantum annealer demonstrate that the fixing spins method enhances quantum annealing performance while preserving the spin-chain embedding for a homogeneous, fully connected ferromagnetic Ising model.

Ising machines, including quantum annealing machines, are promising for solving combinatorial optimization problems with high quality solutions. For constrained combinatorial optimization problems, choosing appropriate penalty coefficients is crucial for obtaining better feasible solutions. When tackling the large-scale combinatorial optimization problems, the variable reduction method is one of the most effective methods. To address these difficulties, we propose a new variable reduction method called multi-penalty sample persistence variable reduction (MP-SPVAR). The method is based on the sample persistence variable reduction method, and utilizes the solution samples across multiple penalty coefficients to fix the variables. The proposed method is evaluated on the 60-facility quadratic assignment problem tai60b from QAPLIB using an Ising machine. The experimental results show that MP-SPVAR can obtain better feasible solutions than solving the problem without any variable reduction. The proposed method is promising for solving the large-scale constrained combinatorial optimization problems with Ising machines.

Multi-stage quantum walks (MSQW) are a promising near-term quantum algorithm. Although studies on MSQW have recently been cultivated, there is still limited evidence regarding their effectiveness for constrained combinatorial optimization problems (COPs). This study applied methods proposed in previous studies, which restrict the quantum state to a feasible space and are effective for quantum annealing, to MSQW. As a result, we reveal that the proposed method has the potential to efficiently solve constrained COPs compared to penalty-based naive formulations.

In this study, we developed a method of estimating the correction terms that makes the Hamiltonian used in phase-field analysis by quantum annealing correspond to the free energy functional of the conventional phase-field analysis using the finite difference method. For the estimation of the correction terms, we employed a factorization machine. The inputs to the factorization machine were the phase-field variables in domain-wall encoding and the differences between the Gibbs free energy and Hamiltonian. We obtained the difference value in quadratic unconstrained binary optimization (QUBO) form as the output of learning using the factorization machine. The QUBO form difference was subjected to the original Hamiltonian as the correction term. The performance of this correction term was evaluated by calculating the energy for a equilibrium state of diblock copolymer. In phase-field analysis, the time evolution equation is formulated so that the total free energy decreases; hence, a lower the free energy means a more accurate result close to that of a conventional method. When we performed annealing with correction terms, the microstructure showed a Gibbs free energy that was lower than that obtained without the correction terms.

In this study, we propose a new method for constrained combinatorial optimization using variational quantum circuits. Quantum computers are considered to have the potential to solve large combinatorial optimization problems faster than classical computers. Variational quantum algorithms, such as Variational Quantum Eigensolver (VQE), have been studied extensively because they are expected to work on noisy intermediate scale devices. Unfortunately, many optimization problems have constraints, which induces infeasible solutions during VQE process. Recently, several methods for efficiently solving constrained combinatorial optimization problems have been proposed by designing a quantum circuit so as to output only the states that satisfy the constraints. However, the types of available constraints are still limited. Therefore, we have started to develop variational quantum circuits that can handle a wider range of constraints. The proposed method utilizes a forwarding operation that maps from feasible states for subproblems to those for larger subproblems. As long as appropriate forwarding operations can be defined, iteration of this process can inductively construct variational circuits outputting feasible states even in the case of multiple and complex constraints. In this paper, the proposed method was applied to facility location problem. As a result, feasible solutions were obtained with 100%, and the probability of obtaining optimal solutions was over 22 times higher than that of conventional VQEs. Nevertheless, the cost of the obtained circuit was comparable to that of conventional circuits.

This study investigates the application of Factorization Machines with Quantum Annealing (FMQA) to address the crystal structure problem (CSP) in materials science. FMQA is a black-box optimization algorithm that combines machine learning with annealing machines to find samples to a black-box function that minimize a given loss. The CSP involves determining the optimal arrangement of atoms in a material based on its chemical composition, a critical challenge in materials science. We explore FMQA’s ability to efficiently sample optimal crystal configurations by setting the loss function to the energy of the crystal configuration as given by a predefined interatomic potential. Further, we investigate how well the energies of the various metastable configurations, or local minima of the potential, are learned by the algorithm. Our investigation reveals FMQA’s potential in quick ground state sampling and in recovering relational order between local minima.

Quantum annealing can efficiently obtain solutions to combinatorial optimization problems. Size-reduction methods are used to treat large-scale combinatorial optimization problems that cannot be input directly into a quantum annealer because of its size limitation. Various size-reduction methods using fixing spins have been proposed as quantum-classical hybrid methods to obtain solutions. However, the high performance of these hybrid methods is yet to be clearly elucidated. In this study, we adopted a parameterized fixing spins method to verify the effects of fixing spins. The results revealed that setting the appropriate number of spins of the subproblem is crucial for obtaining a satisfactory solution, and the energy gap expansion is confirmed after fixing spins.

Ising machines are expected to solve combinatorial optimization problems faster than the existing integer programming solvers. These problems, particularly those encountered in practical situations, typically involve inequality constraints. However, owing to the hardware limitations of the current Ising machines, solving combinatorial optimization problems with inequality constraints remains challenging. The Capacitated Vehicle Routing Problem (CVRP) is a typical example of a problem with inequality constraints. The CVRP is classified as NP-hard and, thus, is commonly solved using heuristic algorithms, such as column generation. Column generation attempts to iteratively generate only the promising routes, as the number of feasible routes increases exponentially. Within this framework, the CVRP is formulated as a set cover problem. The corresponding dual solutions are used to define the pricing subproblem, which is intended to create a new route. By applying Ising machines to this pricing subproblem, the overall computation time can be reduced. This study aims to solve combinatorial optimization problems with inequality constraints using a hybrid algorithm that combines column generation and Ising machines, thereby extending the applications of the latter. Additionally, we introduced limited column generation, wherein we fixed the decision variables to eliminate the overlap between the generated columns. By reducing unnecessary variables, our proposed method can overcome the hardware limitations where a small number of variables is available on real devices. We parameterize the difficulty of the inequality constraints and demonstrate that our annealing-assisted column generation can converge to a better lower bound.

In this study, we developed a new method of topology optimization for truss structures by quantum annealing. To perform quantum annealing analysis with real variables, representation of real numbers as a sum of random number combinations is employed. The nodal displacement is expressed with binary variables. The Hamiltonian H is formulated on the basis of the elastic strain energy and position energy of a truss structure. It is confirmed that truss deformation analysis is possible by quantum annealing. For the analysis of the optimization method for the truss structure, the cross-sectional area of the truss is expressed with binary variables. The iterative calculation for the changes in displacement and cross-sectional area leads to the optimal structure under the prescribed boundary conditions.