Problem-size-independent angles for a Grover-driven quantum approximate optimization algorithm

Abstract

The quantum approximate optimization algorithm (QAOA) requires that circuit parameters are determined that allow one to sample from high-quality solutions to combinatorial optimization problems. Such parameters can be obtained using either costly outer-loop optimization procedures and repeated calls to a quantum computer or, alternatively, via analytical means. In this work, we consider a context in which one knows a probability density function describing how the objective function of a combinatorial optimization problem is distributed. We show that, if one knows this distribution, then the expected value of strings, sampled by measuring a Grover-driven, QAOA-prepared state, can be calculated independently of the size of the problem in question. By optimizing this quantity, optimal circuit parameters for average-case problems can be obtained on a classical computer. Such calculations can help deliver insights into the performance of and predictability of angles in QAOA in the limit of large problem sizes, in particular, for the number partitioning problem.