Classical symmetries and the Quantum Approximate Optimization Algorithm
This work provides a formal connection between classical symmetries and QAOA dynamics, which could help in understanding and optimizing QAOA for researchers and practitioners in quantum computing.
This paper explores the relationship between the Quantum Approximate Optimization Algorithm (QAOA) and the classical symmetries of the objective function. It demonstrates that classical symmetries lead to invariant measurement outcome probabilities in QAOA and uses machine learning to predict QAOA performance based on symmetry properties, specifically predicting the minimum QAOA depth for a target approximation ratio on MaxCut with linear parameter schedules.
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.