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arXiv:2412.07460 · Quantum Information Processing

Accuracy and performance evaluation of quantum, classical and hybrid solvers for the Max-Cut problem

Jaka Vodeb, Vid Eržen, Timotej Hrga, Janez Povh

Published Jul 3, 2026• 1 citation
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Abstract

Abstract This paper benchmarks quantum, classical, and hybrid solvers on NP-hard Max-Cut and QUBO problems, emphasizing solution quality relative to known global optima. We evaluate D-Wave’s fast annealing QPU and Hybrid solver against classical simulated annealing (SA) and Toshiba’s simulated bifurcation machine (SBM) using 139 Max-Cut instances (100 to 10,000 nodes). For small instances ( $$\le $$ ≤ 250 nodes) with known global optima, Hybrid and SA consistently achieve optimal solutions, outperforming the QPU. For larger instances, SBM and slower SA yield superior solutions, while Hybrid and faster SA perform less effectively. Computation time varies across solvers.

DOI
10.1007/s11128-026-05263-5

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Abstract This paper benchmarks quantum, classical, and hybrid solvers on NP-hard Max-Cut and QUBO problems, emphasizing solution quality relative to known global optima. We evaluate D-Wave’s fast annealing QPU and Hybrid solver against classical simulated annealing (SA) and Toshiba’s simulated bifurcation machine (SBM) using 139 Max-Cut instances (100 to 10,000 nodes). For small instances ( ≤\le ≤ ≤ 250 nodes) with known global optima, Hybrid and SA consistently achieve optimal solutions, outperforming the QPU. For larger instances, SBM and slower SA yield superior solutions, while Hybrid and faster SA perform less effectively. Computation time varies across solvers.
Abstract Quadratic unconstrained binary optimization (QUBO) formulations of combinatorial optimization problems allow for solving them using various quantum heuristics. While large-scale quantum computations are currently still out of reach, we can already numerically test such QUBO formulations on a perhaps surprisingly large scale. In this work, we benchmark Fujitsu’s Digital Annealer (DA) on the Max-Cut problem, which captures the main complexity of the QUBO problem. We make a comprehensive benchmark against leading other heuristic algorithms on graphs with up to 53,000 variables by focusing on the wall-clock time. Moreover, we compare the DA performance against published performance results of the D-Wave hybrid quantum–classical annealer and the recently proposed QIS3 heuristic. Based on performance statistics for over 2000 graphs from the MQLib, we find that the DA yields competitive results. We hope that this benchmark demonstrates the extent to which large QUBO instances can be heuristically solved today, yielding consistent results across different solvers.
QUBO formulations of combinatorial optimization problems allow for solving them using various quantum heuristics. While large-scale quantum computations are currently still out of reach, we can already numerically test such QUBO formulations on a perhaps surprisingly large scale. In this work, we benchmark Fujitsu’s Digital Annealer (DA) on the Max-Cut problem, which captures the main complexity of the QUBO problem. We make a comprehensive benchmark against leading other heuristic algorithms on graphs with up to 53,000 variables by focusing on the wall-clock time. Moreover, we compare the DA performance against published performance results of the D-Wave hybrid quantum-classical annealer and the recently proposed QIS3 heuristic. Based on performance statistics for over 2,000 graphs from the MQLib, we find that the DA yields competitive results. We hope that this benchmark demonstrates the extent to which large QUBO instances can be heuristically solved today, yielding consistent results across different solvers.
This study addresses the formidable challenge of solving large-scale Max-Cut problems (MCP). We introduce a rapid computational procedure utilizing a hybrid 1-flip/r-flip local search heuristic. This innovative strategy significantly reduces the computational time required for MCP problems while consistently generating solutions of exceptional quality. The paper presents substantial computational insights, showcasing the effectiveness of our approach on large-scale Max-Cut instances with varying densities. Our proposed heuristic is rigorously evaluated by comparing its performance against a quantum annealing solver, leveraging a multi-start Tabu Search framework. The results underscore the potency of this unique combination as an efficient and effective solution for large-scale QUBO problems. Notably, our hybrid heuristic consistently delivers high-quality solutions within the stringent CPU time limits of 600 seconds, demonstrating its efficacy across Max-Cut instances ranging from 10,000 to 40,000 variables. This research contributes to advancing the state-of-the-art in large-scale QUBO problem-solving, offering a powerful and time-efficient approach with broad applicability.
In this study, we address the formidable challenge of solving large-scale Max-Cut problems (MCP). We introduce a rapid computational procedure utilizing a hybrid 1-flip/r-flip local search heuristic. This innovative strategy significantly reduces the computational time required for MCP problems while consistently generating solutions of exceptional quality. The paper presents substantial computational insights, showcasing the effectiveness of our approach on very-large-scale Max-Cut instances with varying densities. Our proposed heuristic is rigorously evaluated by comparing its performance against a quantum annealing solver, leveraging a multi-start Tabu Search framework. The results underscore the potency of this unique combination as an efficient and effective solution for large-scale QUBO problems. Notably, our hybrid heuristic consistently delivers high-quality solutions within the stringent CPU time limits of 600 seconds, demonstrating its efficacy across Max-Cut instances ranging from 10,000 to 40,000 variables. This research contributes to advancing the state-of-the-art in large-scale QUBO problem-solving, offering a powerful and time-efficient approach with broad applicability.
Quadratic Unconstrained Binary Optimization (QUBO) provides a versatile framework for representing NP-hard combinatorial problems, yet existing solvers often face trade-offs among speed, accuracy, and scalability. In this work, we introduce a quantum-inspired solver (QIS) that unites branch-and-bound pruning, continuous gradient-descent refinement, and quantum-inspired heuristics within a fully adaptive control architecture. We benchmark QIS3 against eight state-of-the-art solvers, including genetic algorithms, coherent Ising machines, simulated bifurcation, parallel tempering, simulated annealing, our prior QIS2 version, D-Wave's simulated-annealing (Neal), and Gurobi on three canonical QUBO problem classes: Max-Cut, NAE-3SAT, and Sherrington-Kirkpatrick spin glass problems. Under a uniform runtime budget, QIS3 attains the best solution on nearly all instances, achieving optimality in 94% of max-cut instances. These results establish QIS3 as a robust, high-performance solver that bridges classical exact strategies and quantum-inspired heuristics for scalable QUBO optimization.
Recently, inspired by quantum annealing, many solvers specialized for unconstrained binary quadratic programming problems have been developed. For further improvement and application of these solvers, it is important to clarify the differences in their performance for various types of problems. In this study, the performance of four quadratic unconstrained binary optimization problem solvers, namely D-Wave Hybrid Solver Service (HSS), Toshiba Simulated Bifurcation Machine (SBM), Fujitsu Digital Annealer (DA), and simulated annealing on a personal computer, was benchmarked. The problems used for benchmarking were instances of real problems in MQLib, instances of the SAT-UNSAT phase transition point of random not-all-equal 3-SAT (NAE 3-SAT), and the Ising spin glass Sherrington-Kirkpatrick (SK) model. Concerning MQLib instances, the HSS performance ranked first; for NAE 3-SAT, DA performance ranked first; and regarding the SK model, SBM performance ranked first. These results may help understand the strengths and weaknesses of these solvers.