Applying the Quantum Alternating Operator Ansatz to the Graph Matching Problem

TL;DR

This paper applies the Quantum Alternating Operator Ansatz (QAOA+) to graph matching problems, developing algorithms that generate superpositions over matchings and showing improved expected matching sizes on 2-regular graphs.

Contribution

It introduces quantum algorithms within the QAOA+ framework for sampling matchings, demonstrating advantages over uniform distributions and analyzing input state effectiveness.

Findings

Superpositions over matchings enable sampling from complex distributions.

Using W-states as input improves expected matching size.

Algorithm outperforms uniform distribution on 2-regular graphs.

Abstract

The Quantum Alternating Operator Ansatz (QAOA+) framework has recently gained attention due to its ability to solve discrete optimization problems on noisy intermediate-scale quantum (NISQ) devices in a manner that is amenable to derivation of worst-case guarantees. We design a technique in this framework to tackle a few problems over maximal matchings in graphs. Even though maximum matching is polynomial-time solvable, most counting and sampling versions are #P-hard. We design a few algorithms that generates superpositions over matchings allowing us to sample from them. In particular, we get a superposition over all possible matchings when given the empty state as input and a superposition over all maximal matchings when given the W -states as input. Our main result is that the expected size of the matchings corresponding to the output states of our QAOA+ algorithm when ran on a 2-regular graph is greater than the expected matching size obtained from a uniform distribution over all matchings. This algorithm uses a W -state as input and we prove that this input state is better compared to using the empty matching as the input state.