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arXiv:2601.09462 · Journal of the Physical Society of Japan

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

Tetsuro Abe, Kanta Hino, Shu Tanaka

Published Jun 16, 2026• 0 citations
arXivHTMLPDF

Abstract

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.

DOI
10.7566/jpsj.95.074003

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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.
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 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.
We propose a spin-variable reduction method for Ising machines to handle linear equality constraints in a combinatorial optimization problem. Ising machines including quantum-annealing machines can effectively solve combinatorial optimization problems. They are designed to find the lowest-energy solution of a quadratic unconstrained binary optimization (QUBO), which is mapped from the combinatorial optimization problem. The proposed method reduces the number of binary variables to formulate the QUBO compared to the conventional penalty method. We demonstrate a sufficient condition to obtain the optimum of the combinatorial optimization problem in the spin-variable reduction method and its general applicability. We apply it to typical combinatorial optimization problems, such as the graph κ-partitioning problem and the quadratic assignment problem. Experiments using simulated-annealing and quantum-annealing based Ising machines demonstrate that the spin-variable reduction method outperforms the penalty method. The proposed method extends the application of Ising machines to larger-size combinatorial optimization problems with linear equality constraints.
We have developed an extended Ising machine that extends the energy function, which is restricted to a binary-quadratic form in conventional Ising machines, to include a variety of penalty functions representing inequality constraints and higher-order energy terms. Such an extension opens up the possibility of dealing with problems featuring many (e.g., thousands or more) constraints. However, for problems with so many constraints, it is often necessary to adjust the value of the penalty coefficient, which represents the ratio of the penalty function to the total cost function, in order to find a solution. In this paper, we propose a penalty coefficient adjustment technique that can be combined with the parallel trial-based Exchange Monte Carlo method, a search algorithm for the extended Ising machine. The proposed algorithm enables us to solve the inequality constraint problem, which previously had the difficulty of manually determining penalty coefficients by trial and error. It also improves the solution quality. In two of the five QPLIB instances we tested, the extended Ising machine with fixed penalty coefficients obtained either a previous or a new best-known solution depending on the random seed, while the proposed coefficient adjustment method obtained a new best-known solution with all ten random seeds used.
Constrained combinatorial optimization problems, which are ubiquitous in industry, can be solved by quantum algorithms such as quantum annealing (QA) and the quantum approximate optimization algorithm (QAOA). In these quantum algorithms, constraints are typically implemented with quadratic penalty functions. This penalty method can introduce large energy scales and make interaction graphs much more dense. These effects can result in worse performance of quantum optimization, particularly on near-term devices that have sparse hardware graphs and other physical limitations. In this work, we consider linear Ising penalty functions, which are applied with local fields in the Ising model, as an alternative method for implementing constraints that makes more efficient use of physical resources. We study the behavior of the penalty method in the context of quantum optimization for customer data science problems. Our theoretical analysis and numerical simulations of QA and the QAOA indicate that this penalty method can lead to better performance in quantum optimization than the quadratic method. However, the linear Ising penalty method is not suitable for all problems as it cannot always exactly implement the desired constraint. In cases where the linear method is not successful in implementing all constraints, we propose that schemes involving both quadratic and linear Ising penalties can be effective. Published by the American Physical Society 2024
Combinatorial optimization has wide applications from industry to natural science. Ising machines bring an emerging computing paradigm for efficiently solving a combinatorial optimization problem by searching a ground state of a given Ising model. Current cutting-edge Ising machines achieve fast sampling of near-optimal solutions of the max-cut problem. However, for problems with additional constraint conditions, their advantages have been hardly shown due to difficulties in handling the constraints. In this work, we focus on benchmarks of Ising machines on the quadratic knapsack problem (QKP). To bring out their practical performance, we propose fast two-stage post-processing for Ising machines, which makes handling the constraint easier. Simulation based on simulated annealing shows that the proposed method substantially improves the solving performance of Ising machines and the improvement is robust to a choice of encoding of the constraint condition. Through evaluation using an Ising machine called Amplify Annealing Engine, the proposed method is shown to dramatically improve its solving performance on the QKP. These results are a crucial step toward showing advantages of Ising machines on practical problems involving various constraint conditions.