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IEEE International Symposium on Parallel & Distributed Processing, Workshops and Phd Forum

A Compressed QUBO Format for Traveling Salesperson Problems

Chu-Yuan Huang, Kazuhiko Komatsu, Makoto Onoda, Masahito Kumagai, Masayuki Sato, Hiroaki Kobayashi

Published Jun 3, 2025• 1 citation
Publisher

Abstract

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.

DOI
10.1109/ipdpsw66978.2025.00053

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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.
Ising machines have attracted much attention in recent years because of their potential in solving combinatorial optimization problems that are difficult to be solved using conventional computers. Ising machines solve the optimization problem that can be transformed into a quadratic unconstrained binary optimization (QUBO) problem. In the case of a problem with constraints, named a constraint problem, an objective function and constraint function with penalty coefficients are combined into a single Hamiltonian of a QUBO problem. When solving a constraint problem, the accuracy of the solution decreases compared to a QUBO problem with an objective function only. This is because penalty coefficients are set to excessively large values to prevent constraint violations. As a result, it becomes difficult to minimize the objective function. To solve this problem, this paper proposes a method that partitions constraint functions and reduces the value of the penalty coefficients. The performance evaluation using the traveling salesperson problem (TSP) with one-hot constraints demonstrates that the proposed method can improve the solution accuracy compared to the conventional method.
Abstract The traveling salesman problem (TSP) forms the foundation for many logistics and optimization tasks. At the same time, this problem is NP-hard and requires significant computational resources. Quantum computing promises to substantially accelerate computations or increase solution accuracy. The TSP can be conveniently mapped onto the quadratic unconstrained binary optimization (QUBO) formulation, which in turn can be solved using variational quantum eigensolver (VQE) and quantum approximate optimization algorithm (QAOA) methods, but this mapping requires a quadratic number of qubits. The higher-order binary optimization (HOBO) formulation requires significantly fewer qubits, but it contains higher-order terms. In our work, we thoroughly investigate both formulations and provide a practical recipe for solving the TSP. We give a comprehensive study and compare in detail QUBO and HOBO formulations.
Abstract Ising machines efficiently solve the combinatorial optimization problems described by the Ising model or the quadratic unconstrained binary optimization (QUBO) formulation. A hybrid method based on the QUBO formulation for compressed sensing is proposed. The proposed method comprises alternative steps of discrete and continuous optimization. In the discrete optimization step, the objective function is described by the QUBO formulation. Successful examples obtained via the proposed method are demonstrated. The performance of the proposed method depends highly on the initial conditions.
The QUBO framework provides a way to model, in principle, any combinatorial optimization problem and enables the use of Ising machines to solve it. Ising machines are devices designed to quickly find good solutions to QUBO problems. In previous work, Auto-QUBO was designed to automatically generate QUBO formulations from a high-level problem description. We address two shortcomings of this method. It only works on a per-instance basis, making repeated formulations for similar problems inefficient, and relies on the user to specify penalty weights. This work introduces symbolic sampling, which provides QUBO formulations for entire problem classes. We demonstrate the speedup that can be achieved with this approach using instances of the maximum clique problem. Additionally, we use proven methods to compute valid penalty weights automatically to simplify the translation process. By providing a user-friendly way to generate QUBO formulations in an efficient manner, both in terms of time and problem difficulty, we enable more people to use Ising machines for combinatorial optimization.
Abstract Over the last two decades, the travelling salesperson problem (TSP) has been cited as a benchmark problem to demonstrate the advantage of quantum computers over conventional computers. Its advantage is that it is a well-studied NP-hard optimisation problem that can be easily communicated to highlight the challenges of searching through an exponentially growing number of possible solutions to find the optimal solution. It is therefore a tempting problem to choose to explore quantum advantage. At what point, however, is a call made that quantum advantage is not likely, and efforts should be focused on other problems? This article challenges the continued use of the TSP as a benchmark for quantum optimisation methods—such as quantum annealing and gate-based quantum computing—that require the TSP to be formulated as a quadratic unconstrained binary optimisation (QUBO) problem. We offer explanations for why such quantum approaches are not well suited, nor competitive against state-of-the-art classical methods, for tackling the challenges of the TSP landscape, and we draw parallels with similar observations made almost four decades ago when QUBO-based neural networks proved to be uncompetitive for solving the TSP. After critically reviewing two decades of research effort to solve TSPs using QUBO-based quantum methods, we note a gradual shift in focus: from initial attempts to solve small sized TSPs with general-purpose QUBO-based quantum approaches, to growing evidence that competitiveness is only enhanced where TSP domain knowledge is integrated, via either modified formulations or hybridisation with TSP classical heuristics. We discuss the numerous challenges that must be overcome before QUBO-based quantum optimisers could ever be competitive with classical state-of-the-art TSP solvers. Acknowledging that there may be more promise for non-QUBO-based hybrid approaches, where quantum search accelerates components of conventional algorithms, we offer recommendations for how future studies should be conducted to compare fairly and rigorously any proposed quantum methods against state-of-the-art TSP solvers, or any classical optimisation method, when seeking to establish quantum advantage.
The Quantum Approximate Optimization Algorithm (QAOA) and its derived variants are widely in use for approximating combinatorial optimization problem instances on gate-based Noisy Intermediate Scale Quantum (NISQ) computers. Commonly, circuits required for QAOA are constructed by first reformulating a given problem as a Quadratic Unconstrained Binary Optimization (QUBO) problem. It is then straightforward to synthesize a QAOA circuit from QUBO equations. In this work, we illustrate a more qubit-efficient circuit construction for combinatorial optimization problems by the example of the Traveling Salesperson Problem (TSP). Conventionally, the qubit encoding in QAOA for the TSP describes a tour using a sequence of nodes, where each node is written as a 1-hot binary vector. We propose to encode TSP tours by selecting edges included in the tour. Removing certain redundancies, the number of required qubits can be reduced by a linear factor compared to the aforementioned conventional encoding. We examined implementations of both QAOA encoding variants in terms of their approximation quality and runtime. Our experiments show that for small instances results are just as accurate using our proposed encoding, whereas the number of required classical optimizer iterations increases only slightly.