Quantum Approximate Optimization Algorithm and Symmetry

 Our new paper QAOA symmetry and its predictability using classical symmetry is out! We analyze the connections between the symmetries of the objective function and the symmetries of QAOA dynamics, and applications to performance prediction, simulation and more https://scirate.com/arxiv/2012.04713

Ruslan Shaydulin, Stuart Hadfield, Tad Hogg, Ilya Safro "Classical symmetries and QAOA", 2020

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.