Mapping graph coloring to quantum annealing

Abstract

Quantum annealing provides a method to solve combinatorial optimization problems in complex energy landscapes by exploiting thermal fluctuations that exist in a physical system. This work introduces the mapping of a graph coloring problem based on pseudo-Boolean constraints to a working graph of the D-Wave Systems Inc. We start from the problem formulated as a set of constraints represented in propositional logic. We use the SATyrus approach to transform this set of constraints to an energy minimization problem. We convert the formulation to a quadratic unconstrained binary optimization problem (QUBO), applying polynomial reduction when needed, and solve the problem using different approaches: (a) classical QUBO using simulated annealing in a von Neumann machine; (b) QUBO in a simulated quantum environment; (c) actual quantum 1, QUBO using the D-Wave quantum machine and reducing polynomial degree using a D-Wave library; and (d) actual quantum 2, QUBO using the D-Wave quantum machine and reducing polynomial degree using our own implementation. We study how the implementations using these approaches vary in terms of the impact on the number of solutions found (a) when varying the penalties associated with the constraints and (b) when varying the annealing approach, simulated (SA) versus quantum (QA). Results show that both SA and QA produce good heuristics for this specific problem, although we found more solutions through the QA approach.