Constrained mixers for the quantum approximate optimization algorithm

TL;DR

This paper introduces a framework for designing constrained quantum mixers within the QAOA, enabling the algorithm to handle optimization problems with hard constraints more efficiently and with fewer quantum gates.

Contribution

The authors generalize the XY-mixer for constrained subspaces, analyze its mathematical structure, and develop more efficient Trotterization and gate decomposition methods.

Findings

New constrained mixers for QAOA that preserve problem-specific subspaces.

Reduced CX gate count in Trotterized mixers compared to previous methods.

Algorithms for efficient basis gate decomposition of the proposed mixers.

Abstract

The quantum approximate optimization algorithm/quantum alternating operator ansatz (QAOA) is a heuristic to find approximate solutions of combinatorial optimization problems. Most literature is limited to quadratic problems without constraints. However, many practically relevant optimization problems do have (hard) constraints that need to be fulfilled. In this article, we present a framework for constructing mixing operators that restrict the evolution to a subspace of the full Hilbert space given by these constraints; We generalize the "XY"-mixer designed to preserve the subspace of "one-hot" states to the general case of subspaces given by a number of computational basis states. We expose the underlying mathematical structure which reveals more of how mixers work and how one can minimize their cost in terms of number of CX gates, particularly when Trotterization is taken into account. Our analysis also leads to valid Trotterizations for "XY"-mixer with fewer CX gates than is known to date. In view of practical implementations, we also describe algorithms for efficient decomposition into basis gates. Several examples of more general cases are presented and analyzed.