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IEEE Access

Benchmarks and Recommendations for Quantum, Digital, and GPU Annealers in Combinatorial Optimization

Jehn‐Ruey Jiang, Yu‐Chen Shu, Qiao-Yi Lin

Published Jan 1, 2024• 16 citations
Publisher

Abstract

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.

DOI
10.1109/access.2024.3455436

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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.
Quantum annealing has the potential to outperform classical transistor-based computer technologies in tackling intricate combinatorial optimization problems. However, ongoing scientific debates cast doubts on whether quantum annealing devices (or quantum annealers) can genuinely provide better problem-solving capabilities than classical computers. The question of whether quantum annealing algorithms (QAAs) running on quantum annealers have computational advantages over classical algorithms (CAs) running on classical computers still remains unclear. This paper aims to clarify the question by classifying and benchmarking QAAs that utilize quadratic unconstrained binary optimization (QUBO) formulas to solve NP-hard problems. It proposes a four-class classification of QUBO formulas and exemplifies each class by QUBO formulas used by QAAs for solving specific NP-hard problems, such as the subset sum, maximum cut, vertex cover, 0/1 knapsack, graph coloring, Hamiltonian cycle, traveling salesperson, and job shop scheduling problems. The classification is based on the following two criteria: (i) Does the number of QUBO variables scale linearly with the problem input size? (ii) Does the QUBO formula have both the constraint term and the optimization term? QAAs are implemented and run on a D-Wave quantum annealer for benchmarking. They are benchmarked against related classical algorithms (CAs) running on classical computers in terms of the quality of the solution and the time to the solution. The benchmarking results reveal which classes of QUBO formulas are likely to provide advantages to QAAs over CAs. Furthermore, based on the benchmarking results, observations and suggestions are given for each class of QUBO formulas, facilitating the adoption of appropriate actions to improve the performance of QAAs.
This paper investigates the impact of introducing disturbances into quadratic unconstrained binary optimization (QUBO) formulas to enhance the performance of both the quantum annealer (QA) and the GPU annealer (GPUA). We propose a disturbance insertion method to insert into QUBO formula coefficients some disturbances that are parameterized by the insertion ratio, Gaussian distribution mean, and standard deviation. By applying the proposed method to a QA (viz., the D-Wave Advantage annealer) and a GPUA (viz., the Compal Quantix solver) for solving two well-known combinatorial optimization problems, the traveling salesperson problem and the 0/1 knapsack problem, we identify appropriate disturbance parameters that facilitate the annealing process for better solving the two problems.
Several optimization solvers inspired by quantum annealing have been recently developed, either running on actual quantum hardware or simulating it on traditional digital computers. Industry and academics look at their potential in solving hard combinatorial optimization problems. Formally, they provide heuristic solutions for Ising models, which are equivalent to quadratic unconstrained binary optimization (QUBO). Constraints on solutions feasibility need to be properly encoded. We experiment on different ways of performing such an encoding. As benchmark we consider the cardinality constrained quadratic knapsack problem (CQKP), a minimal extension of QUBO with one inequality and one equality constraint. We consider different strategies of constraints penalization and variables encoding. We compare three QUBO solvers: quantum annealing on quantum hardware (D-Wave Advantage), probabilistic algorithms on digital hardware and mathematical programming solvers. We analyze their QUBO resolution quality and time, and the persistence values extracted in the quantum annealing sampling process. Our results show that a linear penalization of CQKP inequality improves current best practice. Furthermore, using such a linear penalization, persistence values produced by quantum hardware in a generic way allow to match a specific CQKP metric from literature. They are therefore suitable for general purpose variable fixing in core algorithms for combinatorial optimization.
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.
In more recent years, there has been increasing research interest in exploiting the use of application specific hardware for solving optimisation problems. Examples of solvers that use specialised hardware are IBM's Quantum System One and D-wave's Quantum Annealer (QA) and Fujitsu's Digital Annealer (DA). These solvers have been developed to optimise problems faster than traditional meta-heuristics implemented on general purpose machines. Previous research has shown that these solvers (can optimise many problems much quicker than exact solvers such as GUROBI and CPLEX. Such conclusions have not been made when comparing hardware solvers with classical evolutionary algorithms. Making a fair comparison between traditional evolutionary algorithms, such as Genetic Algorithm (GA), and the DA (or other similar solvers) is challenging because the later benefits from the use of application specific hardware while evolutionary algorithms are often implemented on general-purpose machines. Moreover, quantum or quantum-inspired solvers are limited to solving problems in a specific format. A common formulation used is Quadratic Unconstrained Binary Optimisation (QUBO). Many optimisation problems are however constrained and have natural representations that are non-binary. Converting such problems to QUBO can lead to more problem difficulty and/or larger search space. The question addressed in this paper is whether quantum or quantum-inspired solvers can optimise QUBO transformations of combinatorial optimisation problems faster than classical evolutionary algorithms applied to the same problems in their natural representations. We show that the DA often present better average objective function values than GA on Travelling Salesman, Quadratic Assignment and Multi-dimensional Knapsack Problem instances.
Combinatorial optimization problems are ubiquitous in various disciplines and applications. Many heuristic algorithms have been devoted to solve these types of problems. In order to increase the efficiency for finding the optimal solutions, an application-specific hardware, called digital annealer (DA) has been developed for solving combinatorial optimization problems using quadratic unconstrained binary optimization (QUBO) formulations. In this study, we formulated the number partitioning problem and the graph partitioning problem into QUBO forms and solved such problems with the DA developed by Fujitsu Ltd. The QUBO formulation of the number partitioning problem is fully connected. The DA found the overall runtime for the optimal solution to be less than 30 seconds for 6500 binary variables. For the graph partitioning problem, we adopted modularity as the metric for determining the quality of the partitions. For Zachary's Karate Club graph, the modularity obtained was 0.445, a 6% increase against D-wave Quantum Annealer and Simulated Annealing. Moreover, to explore the DA's potential applications to real-world problems, we used the search for communities or virtual microgrids in a power distribution network as an example. The problem was formulated into graph partitioning. It is shown that the DA effectively identified community structures in the IEEE 33-bus and IEEE 118-bus network.