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IEEE Nanotechnology Magazine

A High-Performance Digital QUBO Solver for Quantum-Inspired Optimization

Chia‐Ho Ou, Po-Yuan Chu, Tsung-Hsuan Tsai, YiChing Chen, Chih-Yu Chen

Published Dec 30, 2025• 0 citations
Publisher

Abstract

As Moore’s Law slows, alternative computing paradigms are essential for tackling complex combinatorial optimization problems. This paper introduces DigiQUBO, a high-performance digital Quadratic Unconstrained Binary Optimization (QUBO) solver designed for quantum-inspired optimization. DigiQUBO leverages a advanced Digital Annealing (DA) approach to efficiently solve large-scale QUBO problems, offering a competitive alternative to traditional annealing methods. To evaluate its effectiveness, DigiQUBO is applied to the Traveling Salesman Problem (TSP), a well-known NP-hard problem with applications in logistics, network design, and computational biology. In solving a 40-city Traveling Salesman Problem (TSP), DigiQUBO achieved an average route distance improvement of 13% compared to Simulated Annealing (SA), and a reduction of 37% compared to D-Wave’s LeapHybridSampler, while maintaining comparable runtime to Fujitsu DA3. DigiQUBO significantly enhances solution quality and computational efficiency by integrating two key advanced techniques: two-way one-hot group encoding for improved constraint satisfaction and Bayesian Optimization for automated annealing parameter tuning. Performance comparisons against Simulated Annealing (SA), Fujitsu DA3, and D-Wave’s LeapHybridSampler demonstrate superior results in balancing accuracy and execution time. The results highlight DigiQUBO as a promising digital annealing technology that bridges the gap between classical and quantum optimization, offering a practical and efficient solution in the post-Moore’s Law era.

DOI
10.1109/mnano.2025.3638505

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As Moore’s Law slows, alternative computing paradigms are essential for tackling complex combinatorial optimization problems. This paper introduces DigiQUBO, a high-performance digital Quadratic Unconstrained Binary Optimization (QUBO) solver designed for quantum-inspired optimization. DigiQUBO leverages a advanced Digital Annealing (DA) approach to efficiently solve large-scale QUBO problems, offering a competitive alternative to traditional annealing methods. To evaluate its effectiveness, DigiQUBO is applied to the Traveling Salesman Problem (TSP), a well-known NP-hard problem with applications in logistics, network design, and computational biology. In solving a 40-city Traveling Salesman Problem (TSP), DigiQUBO achieved an average route distance improvement of 13% compared to Simulated Annealing (SA), and a reduction of 37% compared to D-Wave’s LeapHybridSampler, while maintaining comparable runtime to Fujitsu DA3. DigiQUBO significantly enhances solution quality and computational efficiency by integrating two key advanced techniques: two-way one-hot group encoding for improved constraint satisfaction and Bayesian Optimization for automated annealing parameter tuning. Performance comparisons against Simulated Annealing (SA), Fujitsu DA3, and D-Wave’s LeapHybridSampler demonstrate superior results in balancing accuracy and execution time. The results highlight DigiQUBO as a promising digital annealing technology that bridges the gap between classical and quantum optimization, offering a practical and efficient solution in the post-Moore’s Law era.
The traveling salesman problem is a well-known NP-hard problem in combinatorial optimization. This paper shows how to solve it on an Ising Hamiltonian based quantum annealer by casting it as a quadratic unconstrained binary optimization (QUBO) problem. Results of practical experiments are also presented using D-Wave’s 5,000 qubit Advantage 1.1 quantum annealer and the performance is compared to a classical solver. It is found the quantum annealer can only handle a problem size of 8 or less nodes and its performance is subpar compared to the classical solver both in terms of time and accuracy.
This paper presents a digital annealing engine capable of high-speed solving the Binary Quadratic Programming Problem (BQP) with equality and inequality constraints. While many Quadratic Unconstrained Binary Optimization (QUBO) solvers have been developed for combinatorial optimization problems, they are specialized for “unconstrained” BQP and struggle to solve “constrained” BQP at high speed. To address this issue, we developed a digital annealing engine on GPUs that leverages constraints to narrow the search space and directly evaluates inequality violations during Markov Chain Monte Carlo based searches. We evaluated the performance of the Digital Annealer (DA) system with the developed engine on benchmark problems with one-hot (a type of equality constraint) and inequality constraints. The results showed that the developed DA system achieved superior solving performance compared to the previous generation of DA. It also showed comparable performance to state-of-the-art dedicated solvers. Our results suggest that this digital annealing engine is a promising solution for high-speed solving constrained BQP.
Annealers leverage quadratic unconstrained binary optimization (QUBO) formulas to address combinatorial optimization problems (COPs) and have shown potential to outperform classical computers. This paper examines three prominent types of annealers: quantum, digital, and GPU annealers. Quantum annealers (QAs) are exemplified by the D-Wave Advantage, which relies on the quantum tunneling phenomenon to rapidly locate the minimum-energy system state corresponding to the optimal solution to a COP. Digital annealers (DAs) are typified by the Fujitsu Digital Annealing Unit (DAU), which is based on a quantum-inspired digital architecture to perform parallel and real-time optimization calculations to solve a COP. GPU annealers (GPUAs) are exemplified by the Compal Quantix solver, which harnesses graphics processing units (GPUs) to conduct the diverse adaptive bulk search for the optimal COP solution. This paper first provides an introductory overview of the QA, DA, and GPUA, and then proceeds to benchmark their performance on solving various well-known COPs such as the subset sum, maximum cut, vertex cover, 0/1 knapsack, graph coloring, Hamiltonian cycle, traveling salesperson, and job-shop scheduling problems. Their performance is also compared with that of state-of-the-art algorithms running on classical computers. Through comprehensive performance benchmarks in terms of the solution quality and the execution time, we identify the strengths and weaknesses of each annealer. In addition, we also provide recommendations to improve the performance of each annealer. Specifically, we recommend using the genetic algorithm, the improved particle swarm optimization algorithm, and the ant colony optimization algorithm in all annealers, using quantum pausing, quenching, and reverse annealing in the QA, using built-in separated penalty terms, one-way/two-way one-hot constraints, and linear inequality constraints in the DA, and implementing some parallel algorithms in the GPUA for performance improvement.
Quadratic unconstrained binary optimisation (QUBO) problems is a general class of combinatorial optimisation problems, which regained popularity after recent advances in quantum computing. Quantum-inspired technologies like Fujitsu’s Digital Annealer (DA), based on simulated annealing, can solve QUBO problems much faster than traditional computers. Penalty methods can convert constrained optimisation problems into QUBO problems. However, existing exact methods of determining penalty weights are limited to specific QUBO problems and require manual analysis by experts in QUBO. Empirical methods are general but become computationally prohibitive for large problem sizes and do not guarantee that penalty weights preserve the feasibility of global optima. We present a simple, efficient, general method applicable to any QUBO to determine exact penalty weights. Such weights are simple upper-bounds of the smallest penalty weight guaranteeing that unconstrained global optima are the same as the feasible global optima. These bounds can be iteratively improved by sequential penalty methods which we also present. Experimental results with the DA on minimum cut, travelling salesman and multi-dimensional knapsack problems show the viability of the novel methodology hybridising exact and sequential methods. This work contributes towards general, automatic and scalable penalty methods in QUBO.
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.
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.