In discrete-variable black-box optimization, the number of candidate solutions grows combinatorially, while each evaluation is often expensive. Therefore, it is important to identify promising solutions efficiently within a limited number of trials. Bayesian Optimization of Combinatorial Structures (BOCS), an existing parametric method, works effectively when only a small amount of data is available. However, as the number of observations increases, BOCS tends to repeatedly propose points that have already been evaluated, which leads to search stagnation. A random-point addition strategy has been proposed to address this issue when an evaluated point is proposed, but it cannot sufficiently exploit information from promising data obtained so far. In this study, we propose a hybrid method that uses BOCS as the main search framework and generates alternative unevaluated points using a Gaussian process only when search stagnation is detected. In the Gaussian-process-based component, multiple Lower Confidence Bound (LCB) acquisition functions are adaptively selected to dynamically control the balance between exploitation and exploration. Numerical experiments using fully connected Quadratic Unconstrained Binary Optimization (QUBO) and Higher-order Unconstrained Binary Optimization (HUBO) as black-box functions show that the proposed method finds solutions with better objective values than the conventional random-point addition method in both settings. Additional analyses show that its effectiveness comes from selecting points that promote search progress within Hamming-distance neighborhoods, rather than simply adding low-energy points near promising solutions. Experiments with sparse surrogate models for quantum annealer applications further suggest the importance of retaining near-fully connected representational capacity.
In discrete-variable black-box optimization, the number of candidate solutions grows combinatorially, while each evaluation is often expensive. Therefore, it is important to identify promising solutions efficiently within a limited number of trials. Bayesian Optimization of Combinatorial Structures (BOCS), an existing parametric method, works effectively when only a small amount of data is available. However, as the number of observations increases, BOCS tends to repeatedly propose points that have already been evaluated, which leads to search stagnation. A random-point addition strategy has been proposed to address this issue when an evaluated point is proposed, but it cannot sufficiently exploit information from promising data obtained so far. In this study, we propose a hybrid method that uses BOCS as the main search framework and generates alternative unevaluated points using a Gaussian process only when search stagnation is detected. In the Gaussian-process-based component, multiple Lower Confidence Bound (LCB) acquisition functions are adaptively selected to dynamically control the balance between exploitation and exploration. Numerical experiments using fully connected Quadratic Unconstrained Binary Optimization (QUBO) and Higher-order Unconstrained Binary Optimization (HUBO) as black-box functions show that the proposed method finds solutions with better objective values than the conventional random-point addition method in both settings. Additional analyses show that its effectiveness comes from selecting points that promote search progress within Hamming-distance neighborhoods, rather than simply adding low-energy points near promising solutions. Experiments with sparse surrogate models for quantum annealer applications further suggest the importance of retaining near-fully connected representational capacity.
In materials informatics, searching for chemical materials with desired properties is challenging due to the vastness of the chemical space. Moreover, the high cost of evaluating properties necessitates a search with a few clues. In practice, there is also a demand for proposing compositions that are easily synthesizable. In the real world, such as in the exploration of chemical materials, it is common to encounter problems targeting black-box objective functions where formalizing the objective function in explicit form is challenging, and the evaluation cost is high. In recent research, a Bayesian optimization method has been proposed to formulate the quadratic unconstrained binary optimization (QUBO) problem as a surrogate model for black-box objective functions with discrete variables. Regarding this method, studies have been conducted using the D-Wave quantum annealer to optimize the acquisition function, which is based on the surrogate model and determines the next exploration point for the black-box objective function. In this paper, we address optimizing a black-box objective function containing discrete variables in the context of actual chemical material exploration. In this optimization problem, we demonstrate results obtaining parameters of the acquisition function by sampling from a probability distribution with variance can explore the solution space more extensively than in the case of no variance. As a result, we found combinations of substituents in compositions with the desired properties, which could only be discovered when we set an appropriate variance.
A black-box optimization algorithm such as Bayesian optimization finds the extremum of an unknown function by alternating the inference of the underlying function and optimization of an acquisition function. In a high-dimensional space, such algorithms perform poorly due to the difficulty of acquisition function optimization. Herein, we apply Ising machines to overcome the difficulty in the continuous black-box optimization. As an Ising machine specializes in optimization of binary problems, a continuous vector has to be encoded to binary, and the solution by Ising machines has to be translated back. Our method has the following three parts: (1) random subspace coding based on axis-parallel hyperrectangles from continuous vector to binary vector, (2) a quadratic unconstrained binary optimization (QUBO) defined by the acquisition function based on the nonnegative-weighted linear regression model, which is solved by Ising machines, and (3) a penalization scheme to ensure that the solution can be translated back. It is shown with benchmark tests that its performance using the D-Wave Advantage quantum annealer and simulated annealing is competitive with a state-of-the-art method based on the Gaussian process in high-dimensional problems. Our method may open up the possibility of Ising machines and other QUBO solvers, including a quantum approximate optimization algorithm using gated-quantum computers, and may expand its range of application to continuous-valued problems.
A black-box optimization algorithm such as Bayesian optimization finds extremum of an unknown function by alternating inference of the underlying function and optimization of an acquisition function. In a high-dimensional space, such algorithms perform poorly due to the difficulty of acquisition function optimization. Herein, we apply quantum annealing (QA) to overcome the difficulty in the continuous black-box optimization. As QA specializes in optimization of binary problems, a continuous vector has to be encoded to binary, and the solution of QA has to be translated back. Our method has the following three parts: 1) Random subspace coding based on axis-parallel hyperrectangles from continuous vector to binary vector. 2) A quadratic unconstrained binary optimization (QUBO) defined by acquisition function based on nonnegative-weighted linear regression model which is solved by QA. 3) A penalization scheme to ensure that the QA solution can be translated back. It is shown in benchmark tests that its performance using D-Wave AdvantageTM quantum annealer is competitive with a state-of-the-art method based on the Gaussian process in high-dimensional problems. Our method may open up a new possibility of quantum annealing and other QUBO solvers including quantum approximate optimization algorithm (QAOA) using a gated-quantum computers, and expand its range of application to continuous-valued problems.
Bayesian optimization has emerged as a powerful technique for tackling high-dimensional black-box optimization problems. However, its performance often degrades significantly when the dimensionality of the search space increases, a phenomenon known as the "curse of dimensionality." This paper proposes a novel approach – Quantum-Inspired Bayesian Optimization – that addresses this limitation by incorporating principles inspired by quantum computing, specifically quantum annealing. The core idea is to utilize a modified version of quantum annealing to guide the acquisition of function evaluations, exploiting superposition and interference to enhance exploration in high-dimensional spaces. We formulate the optimization problem as a quantum-inspired search, where candidate solutions exist in a superposition of states and are refined through interference effects. The algorithm iteratively updates the probability distribution over the search space, favoring regions with high potential based on the quantum-inspired exploration strategy. We demonstrate the effectiveness of this approach through theoretical analysis and illustrate its potential through conceptual examples, highlighting its ability to mitigate the curse of dimensionality and improve optimization performance compared to traditional Bayesian optimization methods. The key contributions of this work include the formalization of the quantum-inspired Bayesian optimization framework, the design of the modified quantum annealing algorithm for Bayesian optimization, and the theoretical justification for its improved exploration capabilities.
Bayesian Optimization (BO) has emerged as a powerful technique for optimizing black-box functions, particularly in scenarios where evaluations are expensive. However, traditional BO often suffers from inefficient exploration due to the reliance on a fixed, uniform prior distribution. This paper proposes a novel Bayesian Optimization framework, termed Quantum-Inspired Bayesian Optimization with Adaptive Entanglement (QIBOA), which draws inspiration from quantum mechanics, specifically the concept of adaptive entanglement, to dynamically adjust the prior distribution. The core idea is to represent the prior distribution using entangled states, where the entanglement strength is adaptively modified based on the observed function values. Higher entanglement indicates greater uncertainty and encourages exploration, while lower entanglement signifies higher confidence and promotes exploitation. The adaptive nature of this entanglement allows QIBOA to effectively balance exploration and exploitation, leading to improved optimization performance. We demonstrate the effectiveness of QIBOA through theoretical analysis and illustrate its advantages with simulations against established BO algorithms. The key contributions of this work include a new exploration strategy rooted in quantum-inspired concepts and a framework for dynamically adapting the prior distribution in Bayesian Optimization.
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.