Quantum Alternating Operator Ansatz for Solving the Minimum Exact Cover Problem

TL;DR

This paper extends QAOA+ to solve the Minimum Exact Cover problem by transforming it into a multi-objective constrained optimization, enabling the use of non-trivial feasible solutions.

Contribution

It introduces a novel approach to apply QAOA+ to problems lacking trivial feasible solutions by decomposing the problem into manageable steps.

Findings

Successfully applied QAOA+ to MEC problem

Demonstrated the method's feasibility through numerical experiments

Expanded QAOA+'s applicability to NTFSPs

Abstract

The Quantum Alternating Operator Ansatz (QAOA+) is an extension of the Quantum Approximate Optimization Algorithm (QAOA), where the search space is smaller in solving constrained combinatorial optimization problems. However, QAOA+ requires a trivial feasible solution as the initial state, so it cannot be used for problems that are difficult to find a trivial feasible solution. For simplicity, we call them as Non-Trivial-Feasible-Solution Problems (NTFSP). In this paper, we take the Minimum Exact Cover (MEC) problem as an example, studying how to apply QAOA+ to NTFSP. As we know, exact covering (EC) is the feasible space of MEC problem, which has no trivial solutions. To overcome the above problem, the EC problem is divided into two steps to solve. First, disjoint sets are obtained, which is equivalent to solving independent sets. Second, on this basis, the sets covering all elements (i.e., EC) are solved. In other words, we transform MEC into a multi-objective constrained optimization problem, where feasible space consists of independent sets that are easy to find. Finally, we also verify the feasibility of the algorithm from numerical experiments. Our method provides a feasible way for applying QAOA+ to NTFSP, and is expected to expand its application significantly.