Classical symmetries and the Quantum Approximate Optimization Algorithm

TL;DR

This paper explores how classical symmetries of objective functions influence the Quantum Approximate Optimization Algorithm (QAOA), revealing that symmetries lead to invariant measurement outcomes and can predict QAOA performance.

Contribution

It formalizes the connection between classical symmetries and QAOA dynamics, and demonstrates how symmetry properties can predict algorithm performance and required depth.

Findings

Classical symmetries lead to invariant measurement probabilities in QAOA.

Symmetry properties can predict the minimum QAOA depth for MaxCut.

Machine learning can leverage symmetry features to estimate QAOA performance.

Abstract

We study the relationship between the Quantum Approximate Optimization Algorithm (QAOA) and the underlying symmetries of the objective function to be optimized. Our approach formalizes the connection between quantum symmetry properties of the QAOA dynamics and the group of classical symmetries of the objective function. The connection is general and includes but is not limited to problems defined on graphs. We show a series of results exploring the connection and highlight examples of hard problem classes where a nontrivial symmetry subgroup can be obtained efficiently. In particular we show how classical objective function symmetries lead to invariant measurement outcome probabilities across states connected by such symmetries, independent of the choice of algorithm parameters or number of layers. To illustrate the power of the developed connection, we apply machine learning techniques towards predicting QAOA performance based on symmetry considerations. We provide numerical evidence that a small set of graph symmetry properties suffices to predict the minimum QAOA depth required to achieve a target approximation ratio on the MaxCut problem, in a practically important setting where QAOA parameter schedules are constrained to be linear and hence easier to optimize.