Classifying and Benchmarking Quantum Annealing Algorithms Based on Quadratic Unconstrained Binary Optimization for Solving NP-Hard Problems

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 CAs 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.