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Shu Tanaka

Publications and source records attributed to Shu Tanaka.

At least 19 recordsLinked to original sources
arXiv2026

Geometric Characteristics of Subproblems in Ising-Machine-Assisted Large Neighborhood Search

Large-scale quadratic unconstrained binary optimization (QUBO) formulations of constrained combinatorial optimization problems often exceed the input-size limit of present Ising machines or suffer from degraded solution quality as the number of binary variables increases. Large neighborhood search (LNS) mitigates this difficulty by sequentially optimizing restricted subproblems, but the structural factors that distinguish subproblems beyond the number of binary variables remain insufficiently characterized. In this study, we examine vehicle routing problems and compare a construction based on the vehicle routes of the current solution, denoted by LNS-K, with a construction based on QUBO variables and constraint relations, denoted by LNS-Q, while controlling the number of binary variables in the subproblems. Under the tested conditions, LNS-K obtained shorter total distances than LNS-Q in the matched-size comparisons, and the position variance, a measure of the spatial spread of the selected customers, decreased during the iterations in LNS-K. These observations suggest that subproblem design for sequential optimization with Ising machines should consider not only subproblem size but also semantic and geometric structures inherited from the current solution.

cond-mat.stat-mech↗
arXiv2026

Ising-Machine-Assisted Large Neighborhood Search with Flexibly Tunable Subproblem Size

Ising machines are heuristic solvers for combinatorial optimization, but their solution quality can degrade when large-scale constrained problems are solved directly. Ising-machine-assisted large neighborhood search (LNS) instead repeatedly updates a feasible current solution by solving smaller subproblems. An existing feasibility-preserving method for the vehicle routing problem (VRP) re-optimizes the entire routes of selected vehicles and thus cannot adjust the subproblem size finely. We propose LNS-VT, which introduces the number of consecutive steps re-optimized per vehicle as a parameter to control the subproblem size finely while preserving feasibility. For a 300-site, 5-vehicle VRP, its best setting reduced the objective value by approximately 10\% relative to the existing method after 100 iterations, and the appropriate setting changed with the current solution quality. Applying the same principle to the quadratic multiple knapsack problem, we confirmed that an appropriate subproblem size also exists, indicating that subproblem-size control is important in Ising-machine-assisted LNS.

cond-mat.stat-mech↗
arXiv2026

Evaluating the solution performance of the augmented Lagrangian function on Ising machines

We apply the augmented Lagrangian function (ALF) as a formulation for Ising machines and evaluate its performance by time-to-epsilon (\mathrm{TT\varepsilon}). The ALF has been well studied in continuous optimization for its numerical stability and convergence, and its advantage over the penalty function formulation is demonstrated here through the following results. Using the quadratic knapsack problem as a benchmark, we examine the dependence of \mathrm{TT\varepsilon} on the hyperparameters \mu and \lambda. The augmented Lagrangian formulation reduces \mathrm{TT\varepsilon} by roughly an order of magnitude compared with the penalty function formulation, keeping \mu small while obtaining feasible solutions and, for representative parameter settings, reaching high-precision solutions earlier in the search. These findings indicate that the augmented Lagrangian function is a promising formulation for improving the solution performance of Ising machines.

cond-mat.stat-mech↗
arXiv2026

Stage-dependent integer-binary encoding in factorization-machine black-box optimization

Black-box optimization (BBO) deals with problems where objective functions lack explicit analytical forms and are expensive to evaluate. Factorization machine with quadratic-optimization annealing (FMQA) constructs a surrogate model using a factorization machine (FM) and optimizes it with an Ising machine. Conventional FMQA applies a single integer-binary encoding throughout the optimization process, although the encoding best suited to surrogate learning may differ from the one best suited to Ising-machine solution search. We propose a stage-dependent FMQA framework and derive conversion formulas between one-hot and domain-wall QUBO matrices that preserve the surrogate objective over feasible integer states up to an additive constant. We evaluate the OhDw variant, which employs one-hot encoding for learning and domain-wall encoding for search, on the Rastrigin function with input dimensions N = 2 and 5 and discretization levels q = 61 and 301. Across all conditions, the dominant factor governing optimization performance is the encoding used in the learning stage, with one-hot encoding consistently yielding lower residual errors than domain-wall or binary encoding. The additional benefit of switching to domain-wall encoding for solution search is condition-dependent. For N = 5 and q = 301, OhDw achieves a lower residual error and solutions closer to the global optimum than one-hot-only FMQA, whereas for N = 5 and q = 61 the latter achieves a lower residual error. These results indicate that one-hot encoding in the learning stage is the primary performance driver and that stage-dependent encoding can provide further improvement under finer discretization.

cs.LG↗
arXiv2026

Improving FMQA via Initial Training Data Design Considering Marginal Bit Coverage in One-Hot Encoding

Factorization machine with quadratic-optimization annealing (FMQA) is a black-box optimization method that combines a factorization machine (FM) surrogate with QUBO-based search by an Ising machine. When FMQA is applied to integer or discretized continuous variables via one-hot encoding, uniform random initial sampling can leave many binary variables never active in the initial training data, and the corresponding FM parameters receive no direct gradient updates from the observed responses. We address this by designing the initial training data to achieve complete marginal bit coverage, namely, ensuring that every binary variable obtained by one-hot encoding takes the value one at least once. We use two space-filling sampling methods, Latin hypercube sampling (LHS) and the Sobol' sequence, yielding LHS-FMQA and Sobol'-FMQA. On the human-powered aircraft wing-shape optimization benchmark with 17 and 32 design variables, both proposed methods achieved numerically higher mean final cruising speeds than the baseline FMQA, with the advantage more pronounced on the 32-variable problem.

cs.LG↗
arXiv2026

Unfair Sampling of Quantum Annealing in Weighted Graph Bipartitioning Problems

Quantum annealing (QA) is a promising approach for solving combinatorial optimization problems; however, it is known to exhibit unfair sampling, in which degenerate ground states are not sampled with equal probability even for sufficiently long annealing times. Fair sampling is important in applications such as solution diversity assessment and combinatorial counting, yet the mechanism of unfair sampling remains poorly understood, particularly in constrained combinatorial optimization problems. In this work, we investigate unfair sampling of QA in weighted graph bipartitioning problems (GBP), a representative constrained optimization problem. We study how the penalty coefficient in the penalty method affects sampling fairness. Through numerical simulations and experiments on the D-Wave Advantage2 system, we show that increasing the penalty coefficient reduces unfair sampling in a representative single instance, and that this qualitative behavior is also observed on actual hardware. A scaling analysis over randomly generated instances with up to 12 spins reveals that, while this trend does not hold universally, more than 70% of instances exhibit monotonically increasing sampling fairness as the penalty coefficient increases, even at the largest system size studied. These results show that increasing the penalty coefficient improves sampling fairness, though at the cost of ground-state probability under practical annealing conditions, and call for a deeper theoretical understanding of unfair sampling in constrained optimization problems.

quant-ph↗
arXiv2026

Parallelizable Search-Space Decomposition for Large-Scale Combinatorial Optimization Problems Using Ising Machines

Combinatorial optimization problems are crucial in industry. However, many COPs are NP-hard, causing the search space to grow exponentially with problem size and rendering large-scale instances computationally intractable. Conventional solvers typically treat problems as monolithic entities, leading to significant efficiency degradation as structural complexity increases. To address this issue, we propose a novel search-space decomposition method that leverages the inherent structure of variables to systematically reduce the size of the master problem. We formulate interaction costs between variables and individual variable costs as a constrained maximum cut problem and convert it into a quadratic unconstrained binary optimization formulation using penalty terms. An Ising-model solver is used to rapidly decompose the problem into independent small-scale subproblems, which are subsequently solved in parallel using mathematical optimization solvers. We validated this method on the capacitated vehicle routing problem. Results demonstrate three significant benefits: a substantial enhancement in feasible solution rates, accelerated convergence, achieving in 1 min the accuracy that the naive method required 30 min to reach, and a variable reduction of up to 95.32\%. These findings suggest that search-space decomposition is a promising strategy for efficiently solving large-scale combinatorial optimization problems.

cs.ET↗
arXiv2026

Factorization Machine with Quadratic-Optimization Annealing for RNA Inverse Folding and Evaluation of Binary-Integer Encoding and Nucleotide Assignment

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.

cs.LG↗
arXiv2026

Effectiveness of Binary Autoencoders for QUBO-Based Optimization Problems

In black-box combinatorial optimization, objective evaluations are often expensive, so high quality solutions must be found under a limited budget. Factorization machine with quantum annealing (FMQA) builds a quadratic surrogate model from evaluated samples and optimizes it on an Ising machine. However, FMQA requires binary decision variables, and for nonbinary structures such as integer permutations, the choice of binary encoding strongly affects search efficiency. If the encoding fails to reflect the original neighborhood structure, small Hamming moves may not correspond to meaningful modifications in the original solution space, and constrained problems can yield many infeasible candidates that waste evaluations. Recent work combines FMQA with a binary autoencoder (bAE) that learns a compact binary latent code from feasible solutions, yet the mechanism behind its performance gains is unclear. Using a small traveling salesman problem as an interpretable testbed, we show that the bAE reconstructs feasible tours accurately and, compared with manually designed encodings at similar compression, better aligns tour distances with latent Hamming distances, yields smoother neighborhoods under small bit flips, and produces fewer local optima. These geometric properties explain why bAE+FMQA improves the approximation ratio faster while maintaining feasibility throughout optimization, and they provide guidance for designing latent representations for black-box optimization.

cs.LG↗
arXiv2026

Fair sampling with temperature-targeted QAOA based on quantum-classical correspondence theory

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.

quant-ph↗
arXiv2026

Structural Comparison of Error Mitigation Methods for Ising Machines: Penalty-Spin Model versus Stacked Model

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.

cond-mat.stat-mech↗
arXiv2026

High-Order Epistasis Detection Using Factorization Machine with Quadratic Optimization Annealing and MDR-Based Evaluation

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 case-control 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.

cs.LG↗
arXiv2026

Utilizing intermediate states in quantum annealing for multi-objective 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 trade-off 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.

quant-ph↗
arXiv2025

Statistics of Min-max Normalized Eigenvalues in Random Matrices

Random matrix theory has played an important role in various areas of pure mathematics, mathematical physics, and machine learning. From a practical perspective of data science, input data are usually normalized prior to processing. Thus, this study investigates the statistical properties of min-max normalized eigenvalues in random matrices. Previously, the effective distribution for such normalized eigenvalues has been proposed. In this study, we apply it to evaluate a scaling law of the cumulative distribution. Furthermore, we derive the residual error that arises during matrix factorization of random matrices. We conducted numerical experiments to verify these theoretical predictions.

cs.LG↗
arXiv2025

Evaluating the Performance of Direct Higher-Order Formulations in Combinatorial Optimization Problems

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.

cond-mat.stat-mech↗
arXiv2025

Effectiveness of cardinality-return weighted maximum independent set approach for financial portfolio optimization

The portfolio optimization problem is a critical issue in asset management and has long been studied. Markowitz's mean-variance model has fundamental limitations, such as the assumption of a normal distribution for returns and sensitivity to estimation errors in input parameters. In this research, we propose a novel graph theory-based approach, the cardinality-return weighted maximum independent set (CR-WMIS) model, to overcome these limitations. The CR-WMIS model pursues the optimization of both return and risk characteristics. It integrates the risk diversification effect by selecting the largest number of weakly correlated stocks, a feature of the maximum independent set (MIS) model, with the weighting effect based on expected returns from the weighted maximum independent set (WMIS) model. We validated the effectiveness of the proposed method through a five-year backtesting simulation (April 2019 - March 2024) using real market data from the S&P 500. For this task, we employed a simulated-bifurcation-based solver for finding high-quality solutions to large-scale combinatorial optimization problems. In our evaluation, we conducted a comprehensive risk assessment, which has not been sufficiently explored in previous MIS and WMIS studies. The results demonstrate that the CR-WMIS model exhibits superiority in both return and risk characteristics compared to the conventional MIS and WMIS models, as well as the market index (S&P 500). This study provides a practical portfolio optimization method that overcomes the theoretical limitations of the mean-variance model, contributing to both the advancement of academic theory and the support of practical investment decision-making.

cond-mat.stat-mech↗
arXiv2025

Extending Sample Persistence Variable Reduction for Constrained Combinatorial Optimization Problems

Constrained combinatorial optimization problems (CCOPs) are challenging to solve due to the exponential growth of the solution space. When tackled with Ising machines, constraints are typically enforced by the penalty function method, whose coefficients must be carefully tuned to balance feasibility and objective quality. Variable-reduction techniques such as sample persistence variable reduction (SPVAR) can mitigate hardware limitations of Ising machines, yet their behavior on CCOPs remains insufficiently understood. Building on our prior proposal, we extend and comprehensively evaluate multi-penalty SPVAR (MP-SPVAR), which fixes variables using solution persistence aggregated across multiple penalty coefficients. Experiments on benchmark problems, including the quadratic assignment problem and the quadratic knapsack problem, demonstrate that MP-SPVAR attains higher feasible-solution ratios while matching or improving approximation ratios relative to the conventional SPVAR algorithm. An examination of low-energy states under small penalties clarifies when feasibility degrades and how encoding choices affect the trade-off between solution quality and feasibility. These results position MP-SPVAR as a practical variable-reduction strategy for CCOPs and lay a foundation for systematic penalty tuning, broader problem classes, and integration with quantum-inspired optimization hardware as well as quantum algorithms.

cond-mat.stat-mech↗
arXiv2025

Frustration-enhanced quantum annealing correction models with additional inter-replica interactions

Quantum annealing correction (QAC) models provide a promising approach for mitigating errors in quantum annealers. Previous studies have established that QAC models are crucial for ensuring the robustness of the ground state of the Ising model on hardware. In this work, the effects of QAC models incorporating replicas with additional interactions, specifically, the penalty spin model and the stacked model, are investigated for problems characterized by a small energy gap between the ground and first excited states during quantum annealing, a well-known bottleneck to reaching the ground state. The results demonstrate that these QAC models can obtain the optimal solution within short annealing times by exploiting diabatic transitions, even for problems with a small energy gap. These findings highlight the potential of QAC models as practical near-term algorithms for hardware subject to runtime limitations and control noise.

cond-mat.stat-mech↗