論文

The multi-day travel planning assists users with realistic travel itineraries by searching for the optimal travel routes through a set of candidate hotels and point-of-interests (POIs). The multi-day travel planning problem (MTPP) can be solved as an optimization problem. Although conventional methods using von Neumann computers obtain good approximate solutions to the optimization problems, large computation costs are required to solve large-scale or complex problems due to the combinatorial explosion. On the other hand, Ising machines or quantum annealing machines are non-von Neumann computers, and those machines are developed to deal with complex optimization problems. In this paper, we propose an Ising-machine-based method for the MTPP. Practical factors of the MTPP include the POI satisfaction, travel expenses, and time limits. Those factors are mapped onto quadratic unconstrained binary optimization (QUBO) forms. We evaluate the proposed method using two real-world datasets including Sapporo and Tokyo, Japan. Experimental results show that the MTPP can be effectively solved using Ising machines compared with the conventional methods in terms of the solution quality and the execution time. To the best of our knowledge, this study is the first solution of the MTPP using Ising machines.

Black-box optimization (BBO) is used in materials design, drug discovery, and hyperparameter tuning in machine learning. The world is experiencing several of these problems. In this review, a factorization machine with quantum annealing or with quadratic-optimization annealing (FMQA) algorithm to realize fast computations of BBO using Ising machines (IMs) is discussed. The FMQA algorithm uses a factorization machine (FM) as a surrogate model for BBO. The FM model can be directly transformed into a quadratic unconstrained binary optimization model that can be solved using IMs. This makes it possible to optimize the acquisition function in BBO, which is a difficult task using conventional methods without IMs. Consequently, it has the advantage of handling large BBO problems. To be able to perform BBO with the FMQA algorithm immediately, we introduce the FMQA algorithm along with Python packages to run it. In addition, we review examples of applications of the FMQA algorithm in various fields, including physics, chemistry, materials science, and social sciences. These successful examples include binary and integer optimization problems, as well as more general optimization problems involving graphs, networks, and strings, using a binary variational autoencoder. We believe that BBO using the FMQA algorithm will become a key technology in IMs, including quantum annealers.

Ising machines, including quantum annealing machines, are promising next-generation computers for combinatorial optimization problems. However, due to hardware limitations, most Ising-type hardware can only solve objective functions expressed in linear or quadratic terms of binary variables. Therefore, problems with higher-order terms require an order-reduction process, which increases the number of variables and constraints and may degrade solution quality. In this study, we evaluate the effectiveness of directly solving such problems without order reduction by using a high-performance simulated annealing-based optimization solver capable of handling polynomial unconstrained binary optimization (PUBO) formulations. We compare its performance against a conventional quadratic unconstrained binary optimization (QUBO) solver on the same hardware platform. As benchmarks, we use the low autocorrelation binary sequence (LABS) problem and the vehicle routing problem with distance balancing, 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 potential advantages of directly handling higher-order terms in practical optimization problems.

Factorization Machine with Quadratic-optimization Annealing (FMQA) is a promising technique for solving black-box optimization problems efficiently. Recently, warm-starting FMQA has been proposed to accelerate an FMQA process by initializing the Factorization Machine (FM) with a pre-computed interaction matrix. However, the mechanism underlying its effectiveness remains unclear. Therefore, in this study, we utilize random matrix theory to investigate the performance of FM initialization. We derive modified results within random matrix theory that describe the statistical properties of min-max normalized eigenvalues in random matrices. Our analysis estimates the residual coupling error that arises during FM initialization via low-rank approximation. Numerical experiments demonstrate that our theoretical predictions for the coupling error exhibit excellent agreement with empirical results. Furthermore, we show that subsequent training effectively reduces the error below the theoretical baseline defined by random matrix theory, confirming that the warm-starting approach provides a robust starting point for optimization. These insights not only reinforce the validity of warm-starting FMQA but also suggest broader applications in efficient model initialization, such as low-rank adaptation for large language models.

The RNA inverse folding problem aims to identify nucleotide sequences that preferentially adopt a given target secondary structure. While various heuristic and machine learning-based approaches have been proposed, many require a large number of sequence evaluations, which limits their applicability when experimental validation is costly. We propose a method to solve the problem using a factorization machine with quadratic-optimization annealing (FMQA). FMQA is a discrete black-box optimization method reported to obtain high-quality solutions with a limited number of evaluations. Applying FMQA to the problem requires converting nucleotides into binary variables. However, the influence of integer-to-nucleotide assignments and binary-integer encoding on the performance of FMQA has not been thoroughly investigated, even though such choices determine the structure of the surrogate model and the search landscape, and thus can directly affect solution quality. Therefore, this study aims both to establish a novel FMQA framework for RNA inverse folding and to analyze the effects of these assignments and encoding methods. We evaluated all 24 possible assignments of the four nucleotides to the ordered integers (0-3), in combination with four binary-integer encoding methods. Our results demonstrated that one-hot and domain-wall encodings outperform binary and unary encodings in terms of the normalized ensemble defect value. In domain-wall encoding, nucleotides assigned to the boundary integers (0 and 3) appeared with higher frequency. In the RNA inverse folding problem, assigning guanine and cytosine to these boundary integers promoted their enrichment in stem regions, which led to more thermodynamically stable secondary structures than those obtained with one-hot encoding.

In combinatorial optimization problems with degenerate ground states, fair sampling of degenerate solutions is essential. However, the quantum approximate optimization algorithm (QAOA) with a standard transverse-field mixer induces biases among degenerate states as circuit depth increases. Based on quantum–classical correspondence theory, we propose SBO-QAOA, which employs a temperature-dependent Hamiltonian encoding a Gibbs distribution as its ground state. Numerical simulations show that, unlike standard QAOA, SBO-QAOA yields ground-state probabilities converging to finite-temperature values with uniform distribution among degenerate states. These fairness and temperature-targeting properties are preserved even with only four variational parameters under a linear schedule.

Detecting high-order epistasis is a fundamental challenge in genetic association studies due to the combinatorial explosion of candidate locus combinations. Although multifactor dimensionality reduction (MDR) is a widely used method for evaluating epistasis, exhaustive MDR-based searches become computationally infeasible as the number of loci or the interaction order increases. In this paper, we define the epistasis detection problem as a black-box optimization problem and solve it with a factorization machine with quadratic-optimization annealing (FMQA). We propose an efficient epistasis detection method based on FMQA, in which the classification error rate (CER) computed by MDR is used as a black-box objective function. Experimental evaluations were conducted using simulated casecontrol datasets with predefined high-order epistasis. The results demonstrate that the proposed method successfully identified ground-truth epistasis across various interaction orders and the numbers of genetic loci within a limited number of iterations. These results indicate that the proposed method is effective and computationally efficient for high-order epistasis detection.

Error-mitigation methods for Ising machines are reexamined not merely as noise-suppression techniques but as a structural design problem of replica-coupled Ising models. Using simulated annealing as a hardware-noise-free testbed, we systematically compare the penalty-spin (PS) model, which couples replicas through a centralized auxiliary layer, with the stacked model, which couples adjacent replicas directly. Numerical experiments on the quadratic assignment problem reveal that the ferromagnetically coupled stacked model stably maintains constraint satisfaction and improves solution quality over a broad parameter range, exhibiting favorable scalability with both the number of replicas and problem size. In contrast, the PS model suffers from cooperation collapse at large parallelism: many-replica averaging in the PS layer washes out sparse solution information, preventing effective inter-replica coordination. These findings demonstrate that the topology of inter-replica couplings decisively influences search robustness, and provide practical guidelines for model selection and parameter tuning in constrained optimization.

We investigate obtaining intermediate quantum states during the quantum annealing process to address the limitation of the linear weighted sum method in multi-objective optimization, which inherently fails to reach non-convex regions of the Pareto front. We validate this approach through physical experiments utilizing quench-based readout and numerical simulations assuming ideal mid-anneal measurements. Both methods consistently demonstrate a clear tradeoff where earlier timing enhances diversity of the solutions, whereas later timing ensures convergence to non-dominated solutions. Notably, a practical compromise timing balances both metrics. The qualitative agreement between practical quench and ideal simulation indicates the potential of accessing the intermediate states for comprehensive Pareto front exploration.

In an amusement park, an attraction-visiting route considering the waiting time and traveling time improves visitors' satisfaction and experience. We focus on Ising machines to solve the problem, which are recently expected to solve combinatorial optimization problems at high speed by mapping the problems to Ising models or quadratic unconstrained binary optimization (QUBO) models. We propose a mapping of the visiting-route recommendation problem in amusement parks to a QUBO model for solving it using Ising machines. By using an actual Ising machine, we could obtain feasible solutions 15 times faster with almost the same accuracy as the simulated annealing method for the visiting-route recommendation problem.

In the introductory article of this theme issue, we provide an overview of quantum annealing and computation with a very brief summary of the individual contributions to this issue made by experts as well as a few young researchers. We hope the readers will get the touch of the excitement as well as the perspectives in this unusually active field and important developments there. This article is part of the theme issue 'Quantum annealing and computation: challenges and perspectives'.

Quantum annealing has been used to predict molecular adsorption on solid surfaces. Evaluation of adsorption, which takes place in all solid surface reactions, is a crucially important subject for study in various fields. However, predicting the most stable coordination by theoretical calculations is challenging for multimolecular adsorption because there are numerous candidates. This report presents a novel method for quick adsorption coordination searches using the quantum annealing principle without combinatorial explosion. This method exhibited much faster search and more stable molecular arrangement findings than conventional methods did, particularly in a high coverage region. We were able to complete a configurational prediction of the adsorption of 16 molecules in 2286 s (including 2154 s for preparation, only required once), whereas previously it has taken 38 601 s. This approach accelerates the tuning of adsorption behavior, especially in composite materials and large-scale modeling, which possess more combinations of molecular configurations.

This paper describes a hybrid quantum-classical computational approach to designing synthesizable deuterated tris(8-hydroxyquinolinato) aluminum (Alq3) emitters with desirable emission quantum efficiency (QE). This multi-pronged approach first uses classical quantum chemistry to create a machine learning dataset, which is then used to construct an Ising Hamiltonian by a factorization-machine-based model to predict the QEs of Alq3 emitters. Finally, the Ising Hamiltonian is applied to perform simulations using the variational quantum eigensolver (VQE) and quantum approximate optimization algorithm (QAOA) on a quantum device to discover molecules with optimal QE. Moreover, to improve the simulations on the noisy quantum device, we developed the recursive probabilistic variable elimination method, which recursively eliminates qubits depending on the probability that each qubit has a binary value. We demonstrated that the accuracy of VQE and QAOA optimized for a noisy device can be improved from a probability of 0.075 to 0.97.

Ising machines are expected to be dedicated computers that are capable of efficiently searching for good solutions to combinatorial optimization problems. When solving combinatorial optimization problems with an Ising machine, the objective function and constraints of the combinatorial optimization problems are expressed using the Ising model. Then, the low-energy states of the Ising model are searched by the internal algorithms of Ising machines. Typical internal algorithms are Simulated Annealing (SA) and Quantum Annealing (QA). Both algorithms exhibit stochastic behavior, so the final solution may contain errors. Therefore, it is an essential issue in the field of Ising machines to reduce errors as much as possible. A previous study proposed an error reduction method for QA machines, namely the Quantum Annealing Correction (QAC) model. This method reduces error by introducing penalty interaction and performing a specific decoding method. Building on these valuable insights, we proposed an alternative error reduction method, the stacked model. To verify the effectiveness, we examined on the relationship between the penalty interaction and the success probability of the QAC model and the stacked model on SA. The obtained results differ from the facts observed in QA. In summary, a decoding method different from the one considered effective in QA was found to be effective in SA. This result suggests that when considering an appropriate error reduction method, it is necessary to consider the internal algorithm of the Ising machine.

Ising machines are promising next-generation computers that aim to efficiently obtain good solutions to combinatorial optimization problems, which are converted to the Ising model. To obtain the ground state of the Ising model with high accuracy, tuning the hyperparameters in the Ising model is of great importance. Recently, a new method called Energy Landscape Transformation of Ising Problems (ELTIP) has been proposed which does not change any of the original eigenenergies at all. The previous study demonstrated that the Time-to-Solution (TTS) of Quantum Annealing (QA) decreases for the specific problems after ELTIP. We reveal the scope of the problems where ELTIP can be sufficiently beneficial. To this end, we analyze the correlation between the mean Hamming distance and the ground state probability in both the static and dynamic analysis.

Quantum annealing is a promising metaheuristic for solving constrained combinatorial optimization problems. However, coefficient tuning difficulties and hardware noise often prevent optimal solutions from being properly encoded as the ground states of the problem Hamiltonian. This study investigates mid-anneal measurement as a mitigation approach for such situations, analyzing its effectiveness and underlying physical mechanisms. We introduce a quantitative metric to evaluate the effectiveness of mid-anneal measurement and apply it to the graph bipartitioning problem and the quadratic knapsack problem. Our findings reveal that mid-anneal measurement is most effective when the energy difference between desired solutions and ground states is small, with effectiveness strongly governed by the energy structure. Moreover, analysis of fully-connected Ising models demonstrates that the effectiveness of mid-anneal measurement persists with increasing system size, indicating its scalability and practical applicability to large-scale quantum annealing.

Quantum computers are expected to enable fast solving of large-scale combinatorial optimization problems. However, their limitations in fidelity and the number of qubits prevent them from handling real-world problems. Recently, a quantum-inspired solver using tensor networks has been proposed, which works on classical computers. Particularly, tensor networks have been applied to constrained combinatorial optimization problems for practical applications. By preparing a specific tensor network to sample states that satisfy constraints, feasible solutions can be searched for without the method of penalty functions. Previous studies have been based on profound physics, such as U(1) gauge schemes and high-dimensional lattice models. In this study, we devise to design feasible tensor networks using elementary mathematics without such a specific knowledge. One approach is to construct tensor networks with nilpotent-matrix manipulation. The second is to algebraically determine tensor parameters. We showed mathematically that such feasible tensor networks can be constructed to accommodate various types of constraints. For the principle verification, we numerically constructed a feasible tensor network for facility location problem, to find much faster construction than conventional methods. Then, by performing imaginary time evolution, feasible solutions were always obtained, ultimately leading to the optimal solution.

In this study, we proposed a method using an Ising machine to support a mathematical programming solver in order to partition the search space for large-scale combinatorial optimization problems, especially for the capacitated vehicle routing problem. As a method for partitioning the search space, we use the constrained maximum cut problem (CMC), which allows for efficient embedding and is easy to solve on the Ising machine. By using CMC, it is possible to partition the search space for solutions as subproblems within the feasible space of the master problem. In order to evaluate the effectiveness of the proposed method, experiments were carried out using multiple instances. The results indicated that feasible solutions to the master problem can be obtained quickly using the proposed method.

Derivative-free (DF) optimization problems aim to identify an input that maximizes or minimizes the output of an objective function whose input-output relationship is unknown. Factorization machine with quadratic-optimization annealing (FMQA) is a promising approach to this task, employing a factorization machine (FM) as a surrogate model to iteratively guide the solution search via an Ising machine. Although FMQA has demonstrated strong optimization performance across various applications, its performance often stagnates as the number of optimization iterations increases. One contributing factor to this stagnation is the growing number of data points in the dataset used to train FM. As more data are accumulated, the contribution of newly added data points tends to become diluted within the entire dataset. Based on this observation, we hypothesize that such dilution reduces the impact of new data on improving the prediction accuracy of FM. To address this issue, we propose a novel method named sliding window for iterative factorization training combined with FMQA (SWIFT-FMQA). This method improves upon FMQA by utilizing a sliding-window strategy to sequentially construct a dataset that retains at most a specified number of the most recently added data points. SWIFT-FMQA is designed to enhance the influence of newly added data points on the surrogate model. Numerical experiments demonstrate that SWIFT-FMQA obtains lower-cost solutions with fewer objective function evaluations compared to FMQA.

Quantum Annealing (QA) can efficiently solve combinatorial optimization problems whose objective functions are represented by Quadratic Unconstrained Binary Optimization (QUBO) formulations. For broader applicability of QA, quadratization methods are used to transform higher-order problems into QUBOs. However, quadratization methods for complex problems involving Machine Learning (ML) remain largely unknown. In these problems, strong nonlinearity and dense interactions prevent conventional methods from being applied. Therefore, we model target functions by the sum of rectified linear unit bases, which not only have the ability of universal approximation, but also have an equivalent quadratic-polynomial representation. In this study, the proof of concept is verified both numerically and analytically. In addition, by combining QA with the proposed quadratization, we design a new black-box optimization scheme, in which ML surrogate regressors are inputted to QA after the quadratization process.