Experimental Analysis of Quantum Annealers and Hybrid Solvers Using Benchmark Optimization Problems

Abstract

This paper studies the Hamiltonian cycle problem (HCP) and the traveling salesman problem (TSP) on D-Wave quantum systems. Motivated by the fact that most libraries present their benchmark instances in terms of adjacency matrices, we develop a novel matrix formulation for the HCP and TSP Hamiltonians, which enables the seamless and automatic integration of benchmark instances in quantum platforms. We also present a thorough mathematical analysis of the precise number of constraints required to express the HCP and TSP Hamiltonians. This analysis explains quantitatively why, almost always, running incomplete graph instances requires more qubits than complete instances. It turns out that QUBO models for incomplete graphs require more quadratic constraints than complete graphs, a fact that has been corroborated by a series of experiments. Moreover, we introduce a technique for the min-max normalization for the coefficients of the TSP Hamiltonian to address the problem of invalid solutions produced by the quantum annealer, a trend often observed. Our extensive experimental tests have demonstrated that the D-Wave Advantage_system4.1 is more efficient than the Advantage_system1.1, both in terms of qubit utilization and the quality of solutions. Finally, we experimentally establish that the D-Wave hybrid solvers always provide valid solutions, without violating the given constraints, even for arbitrarily big problems up to 120 nodes.

Comparison among Classical, Probabilistic and Quantum Algorithms for Hamiltonian Cycle Problem

“…In addition, many heuristics and strategic approaches exist for TSP, such as Ant Colony Optimization Algorithm, Particle Swarm Optimization Algorithm, Artificial Bee Colony Algorithm and recently Genetic Algorithm [15,16], but they do not find the exact solution efficiently; they only search for one solution near to the optimal. Alternatively, also for TSP, as for HCP, quantum annealing algorithms have been proposed [17][18][19], but they still have too many limitations regarding noise and number of nodes of the graph that have to be elaborated [17].…”