Predicting parameters for the Quantum Approximate Optimization Algorithm for MAX-CUT from the infinite-size limit - Boulebnane, Sami; Montanaro, Ashley (arXiv.org, 2021)
Combinatorial optimization is regarded as a potentially promising application of near and long-term quantum computers. The best-known heuristic quantum algorithm for combinatorial optimization on gate-based devices, the Quantum Approximate Optimization Algorithm (QAOA), has been the subject of many theoretical and empirical studies. Unfortunately, its application to specific combinatorial optimization problems poses several difficulties: among these, few performance guarantees are known, and the variational nature of the algorithm makes it necessary to classically optimize a number of parameters. In this work, we partially address these issues for a specific combinatorial optimization problem: diluted spin models, with MAX-CUT as a notable special case. Specifically, generalizing the analysis of the Sherrington-Kirkpatrick model by Farhi et al., we establish an explicit algorithm to evaluate the performance of QAOA on MAX-CUT applied to random Erdos-Renyi graphs of expected degree \(d\) for an arbitrary constant number of layers \(p\) and as the problem size tends to infinity. This analysis yields an explicit mapping between QAOA parameters for MAX-CUT on Erdos-Renyi graphs of expected degree \(d\), in the limit \(d → ∞\), and the Sherrington-Kirkpatrick model, and gives good QAOA variational parameters for MAX-CUT applied to Erdos-Renyi graphs. We then partially generalize the latter analysis to graphs with a degree distribution rather than a single degree \(d\), and finally to diluted spin-models with \(D\)-body interactions (\(D ≥ 3\)). We validate our results with numerical experiments suggesting they may have a larger reach than rigorously established; among other things, our algorithms provided good initial, if not nearly optimal, variational parameters for very small problem instances where the infinite-size limit assumption is clearly violated.