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IEEE Access

QUBO Simplification by Singular Value Decomposition and Coefficient Elimination for Ising Machines

Shinnosuke Inaba, Takeru Ota, Nozomu Togawa

Published Jan 1, 2026• 0 citations
Publisher

Abstract

Ising machines, including quantum annealing machines, can efficiently solve combinatorial optimization problems by formulating them as Quadratic Unconstrained Binary Optimization (QUBO). However, it is known that the solving performance of Ising machines deteriorates as the energy landscape represented by the QUBObecomes more complex. Simplifying the coefficients of QUBOmatrix can improve the performance of solving combinatorial optimization problems using an Ising machine. In this paper, we propose QUBO simplification methods for Ising machines using singular value decomposition (SVD) and coefficient elimination based on their absolute value (ABS). The proposed SVD method simplifies QUBO by capturing the global structure of the QUBO matrix. QUBO is further simplified by combining it with the simple ABS method. The simplified QUBO is expected to have a less complex energy landscape, while maintaining the global structure of the original, enabling the Ising machine to efficiently find the (semi-)ground state.We demonstrate the numerical experiments using an Ising machine to evaluate the effectiveness of the SVD method alone, the ABS method alone, and the two methods combined.

DOI
10.1109/access.2026.3684571

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Ising machines, including quantum annealing machines, can efficiently solve combinatorial optimization problems by formulating them as Quadratic Unconstrained Binary Optimization (QUBO). However, it is known that the solving performance of Ising machines deteriorates as the energy landscape represented by the QUBObecomes more complex. Simplifying the coefficients of QUBOmatrix can improve the performance of solving combinatorial optimization problems using an Ising machine. In this paper, we propose QUBO simplification methods for Ising machines using singular value decomposition (SVD) and coefficient elimination based on their absolute value (ABS). The proposed SVD method simplifies QUBO by capturing the global structure of the QUBO matrix. QUBO is further simplified by combining it with the simple ABS method. The simplified QUBO is expected to have a less complex energy landscape, while maintaining the global structure of the original, enabling the Ising machine to efficiently find the (semi-)ground state.We demonstrate the numerical experiments using an Ising machine to evaluate the effectiveness of the SVD method alone, the ABS method alone, and the two methods combined.
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The performance of an Ising machine in solving combinatorial optimization problems is significantly influenced by the energy landscape of an input quadratic unconstrained binary optimization (QUBO). Previous studies have shown that the number of non-zero terms in QUBO can control the energy landscape. In this paper, we propose a QUBO conversion method, which efficiently removes non-zero coefficient terms in QUBO to improve the solving performance of Ising machines. Our pro-posed method uses the removal ratio parameter, which defines the amount of removed non-zero terms, and efficiently removes the non-zero coefficient terms in QUBO. We confirm the effectiveness of the proposed method compared to the conventional method using the capacitated vehicle routing problem (CVRP) on an Ising machine. Experimental results demonstrate that our proposed method improves the path length in CVRP up to 17% compared to the conventional method.
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.
With an increase in the complexity of society, solving multi-objective optimization problems (MOPs) has become crucial. In this study, we introduced a novel method called “quadratic unconstrained binary optimization based on the weighted normal” for solving MOPs using Ising machines, such as quantum annealing and digital annealer (DA), in the field of combinatorial optimization. The proposed method applies the penalty-based boundary intersection method to Ising machines under a setting limited to linear objective functions and maximizes the speed and performance of the DA, which is a quadratic unconstrained binary optimization-specific solver. We demonstrated the effectiveness of the proposed method by solving a real-world problem with a nonconvex shaped Pareto front (component combination problem). The results suggested that the proposed method could handle both convex- and nonconvex-shaped Pareto fronts, expanding the potential applications of Ising machines to solving complex MOPs. This development could significantly enhance decision-making processes, particularly in achieving sustainable development goals.
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.
Ising machines have recently gained a significant attention as a promising approach to combinatorial optimization problems, which are notoriously challenging for conventional computers to solve efficiently. In contrast to conventional computers, Ising machines operate based on the principles of the Ising model derived from statistical mechanics. Ising machines tackle combinatorial optimization problems by transforming them into Quadratic Unconstrained Binary Optimization (QUBO) formulations. The Traveling Salesperson Problem (TSP) is a prominent example of such optimization challenges and is widely recognized for its computational difficulty on conventional computers. Although Ising machines offer the potential for efficiently solving TSPs, the process of a QUBO formulation, namely the generation of a QUBO matrix that represents a TSP, demands substantial memory resources and considerable computation overheads. This study proposes a compressed QUBO format for TSP. This format reduces the dimensionality of the QUBO matrix for TSP by leveraging repetitive values, structural patterns, and sparsity within the matrix. The proposed format enhances the efficiency of data preprocessing, significantly reducing both the computational time required for QUBO matrix generation and the associated memory usage. These improvements contribute to the practical applicability of digital hardware-based Ising machines for addressing large-scale combinatorial optimization problems, thereby expanding their utility in solving complex real-world challenges.