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Genetic and Evolutionary Computation Conference Companion (GECCO Companion)

Boltzmann Machine Is Useful for Enhancing Search Performance of Ising Machines

Satoru Jimbo, Kazushi Kawamura

Published Jul 14, 2025• 0 citations
Publisher

Abstract

Various methods for formulating practical combinatorial optimization problems as an Ising model have been developed, expanding real-world applications of Ising machines. Nevertheless, the usability of Ising machines remains limited since the formulation process involves complicated tasks, such as fine-tuning parameters or selecting options, which can be challenging for non-expert users. Moreover, the existing formulation technique is often insufficient to design an Ising model that facilitates efficient solution search by Ising machines. This paper proposes a novel approach in which the Boltzmann machine, an energy-based machine learning model, autonomously optimizes the Ising model based on solutions obtained from an Ising machine. The proposed combinatorial optimization flow iteratively alternates between solution search by the Ising machine and model refinement by the Boltzmann machine, gradually improving solution quality without manual adjustments of parameters or options. Experimental results demonstrate that the proposed method consistently outperforms the standalone Ising machine in several benchmark problems, achieving superior solution quality under a comparable computational budget.

DOI
10.1145/3712255.3726599

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Various methods for formulating practical combinatorial optimization problems as an Ising model have been developed, expanding real-world applications of Ising machines. Nevertheless, the usability of Ising machines remains limited since the formulation process involves complicated tasks, such as fine-tuning parameters or selecting options, which can be challenging for non-expert users. Moreover, the existing formulation technique is often insufficient to design an Ising model that facilitates efficient solution search by Ising machines. This paper proposes a novel approach in which the Boltzmann machine, an energy-based machine learning model, autonomously optimizes the Ising model based on solutions obtained from an Ising machine. The proposed combinatorial optimization flow iteratively alternates between solution search by the Ising machine and model refinement by the Boltzmann machine, gradually improving solution quality without manual adjustments of parameters or options. Experimental results demonstrate that the proposed method consistently outperforms the standalone Ising machine in several benchmark problems, achieving superior solution quality under a comparable computational budget.
Combinatorial optimization problems are central to both practical applications and the development of optimization methods. While classical and quantum algorithms have been refined over decades, machine learning-assisted approaches are comparatively recent and have not yet consistently outperformed simple, state-of-the-art classical methods. Here, we focus on a class of Quadratic Unconstrained Binary Optimization problems, specifically the challenge of finding minimum energy configurations in three-dimensional Ising spin glasses. We use a Global Annealing Monte Carlo algorithm that integrates standard local moves with global moves proposed via machine learning. We show that local moves play a crucial role in achieving optimal performance. Benchmarking against Simulated Annealing and Population Annealing, we demonstrate that Global Annealing not only surpasses the performance of Simulated Annealing but also exhibits greater robustness than Population Annealing, maintaining effectiveness across problem hardness and system size without hyperparameter tuning. These results provide clear and robust evidence that a machine learning-assisted optimization method can exceed the capabilities of classical state-of-the-art techniques in a combinatorial optimization setting.
Quantum or quantum-inspired Ising machines have recently shown promise in solving combinatorial optimization problems in a short time. Real-world and practical applications, such as time division multiple access (TDMA) scheduling for wireless multi-hop networks, financial trading, and emerging in-vehicle systems, require solving those problems sequentially where the size and characteristics change dynamically. However, using Ising machines for practical deployment involves challenges to shorten system-wide latency due to the transfer of large Ising model or the cloud access and to determine the parameters for each problem. Here we show a combinatorial optimization method using embedded Ising machines, which enables solving diverse problems at high speed without runtime parameter tuning. We customize the algorithm and circuit architecture of the simulated bifurcation-based Ising machine to compress the Ising model and accelerate computation and then build a machine learning model to estimate appropriate parameters using extensive training data. In TDMA scheduling for wireless multi-hop networks, our demonstration shows that the sophisticated system can adapt to changes in the problem and has a speed advantage over conventional methods. Quantum-inspired Ising machines can be used to address dynamically changing combinatorial problems. Here, the authors present an embedded approach that compresses models, avoids runtime parameter tuning, and uses ML for rapid adaptation, achieving faster TDMA scheduling in dynamic wireless networks than conventional methods.
A combinatorial optimization problem is a problem finding an optimal combination of variables that maximizes or minimizes an objective function while satisfying given constraints. Such problems arise in various fields, including transportation and communication. Many combinatorial optimization problems are NP-hard, meaning that the number of possible combinations grows exponentially as the number of variables increases. As a result, it is often difficult to solve large-scale combinatorial optimization problems for conventional classical computers. Recently, Ising machines, including quantum annealers, have gained attention as a promising architecture for efficiently solving combinatorial optimization problems. One real-world example of a combinatorial optimization problem isuniversity course selection. Many students manually choose their courses, but this process is time consuming and labor intensive due to the large number of available options. To address this, in this paper, we formulate course selection as a combinatorial optimization problem, which we define as a Personalized Course Selection Problem (PCSP). To solve it efficiently, we use an Ising machine, a specialized computer designed for combinatorial optimization. We propose a Quadratic Unconstrained Binary Optimization (QUBO) formulation for solving PCSP and solve the problem with an Ising machine. Experimental evaluations demonstrate that, for the largest instance tested, the proposed method solves the problem 20X times faster than the conventional simulated annealing while achieving the same total course cost.
Ising machines (IMs) are hardware solvers designed to tackle computationally complex combinatorial optimization problems (COPs), harnessing physical processes such as quantum annealing to simulate the Ising model, enabling these specially designed solvers to tackle a wide range of computationally complex NP-hard problems in real-world applications, such as portfolio optimization and logistics planning. While prior works propose to fabricate dedicated integrated circuits for building IMs, we leverage off-the-shelf Graphics Processing Unit (GPU) chips for implementing Ising algorithms for quick development. In this work, we explore parallel tempering (PT), an Ising algorithm, which shows promise owing to its capability for parallel processing of multiple independent searches for the optimal solution, each with a different amount of randomness that allows escaping local minima. We propose several optimization strategies, including addressing the dependency problem in PT to enhance algorithm parallelism and integrating genetic algorithm (GA)-like operations to increase the diversity of the search process for effective solution discovery. Empirical evaluations demonstrate that our proposed Parallel Quantum-inspired Search (PQS) solver achieves a 2.66×2.66 \times2.66× speedup over the state-of-the-art GPU-based solution without sacrificing solution quality.
Ising Machines are emerging hardware architectures that efficiently solve NP-Hard combinatorial optimization problems. Generally, combinatorial problems are transformed into quadratic unconstrained binary optimization (QUBO) form, but this transformation often complicates the solution landscape, degrading performance, especially for multi-state problems. To address this challenge, we model spin interactions as generalized boolean logic function to significantly reduce the exploration space. We demonstrate the effectiveness of our approach on graph coloring problem using probabilistic Ising solvers, achieving similar accuracy compared to state-of-the-art heuristics and machine learning algorithms. It also shows significant improvement over state-of-the-art QUBO-based Ising solvers, including probabilistic Ising and simulated bifurcation machines. We also design 1024-neuron all-to-all connected probabilistic Ising accelerator on FPGA with the proposed approach that shows ~10000x performance acceleration compared to GPU-based Tabucol heuristics and reducing physical neurons by 1.5-4x over baseline Ising frameworks. Thus, this work establishes superior efficiency, scalability and solution quality for multi-state optimization problems.
Ising machines are next-generation computers expected to efficiently sample near-optimal solutions of combinatorial optimization problems. Combinatorial optimization problems are modeled as quadratic unconstrained binary optimization (QUBO) problems to apply an Ising machine. However, current state-of-the-art Ising machines still often fail to output nearoptimal solutions due to the complicated energy landscape of QUBO problems. Furthermore, the physical implementation of Ising machines severely restricts the size of QUBO problems to be input as a result of limited hardware graph structures. In this study, we take a new approach to these challenges by injecting auxiliary penalties preserving the optimum, which reduces quadratic terms in QUBO objective functions. The process simultaneously simplifies the energy landscape of QUBO problems, allowing the search for near-optimal solutions, and makes QUBO problems sparser, facilitating encoding into Ising machines with restriction on the hardware graph structure. We propose linearization of QUBO problems using variable posets as an outcome of the approach. By applying the proposed method to synthetic QUBO instances and to multi-dimensional knapsack problems, we empirically validate the effects on enhancing minorembedding of QUBO problems and the performance of Ising machines.