On the Effects of Small Graph Perturbations in the MaxCut Problem by QAOA

TL;DR

This paper explores how small perturbations in graph structures affect the performance of QAOA in solving MaxCut, highlighting the role of symmetries in optimizing quantum algorithms.

Contribution

It introduces a symmetry-based heuristic approach to analyze and improve QAOA performance on perturbed graphs for MaxCut.

Findings

Symmetries significantly influence QAOA approximation ratios.

Perturbations can be managed to maintain high-quality solutions.

Insights enable reduction in quantum circuit complexity.

Abstract

We investigate the Maximum Cut (MaxCut) problem on different graph classes with the Quantum Approximate Optimization Algorithm (QAOA) using symmetries. In particular, heuristics on the relationship between graph symmetries and the approximation ratio achieved by a QAOA simulation are considered. To do so, we first solve the MaxCut problem on well-known graphs, then we consider a simple and controllable perturbation of the graph and find again the approximate MaxCut with the QAOA. Through an analysis of the spectrum of the graphs and their perturbations, as well as a careful study of the associated automorphism groups, we aim to extract valuable insights into how symmetry impacts the performance of QAOA. These insights can then be leveraged to heuristically reduce the quantum circuit complexity, the number of training steps, or the number of parameters involved, thus enhancing the efficiency and effectiveness of QAOA-based solutions.