Predicting parameters for the Quantum Approximate Optimization Algorithm for MAXCUT from the infinitesize limit
Abstract
Combinatorial optimization is regarded as a potentially promising application of near and longterm quantum computers. The bestknown heuristic quantum algorithm for combinatorial optimization on gatebased 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 MAXCUT as a notable special case. Specifically, generalizing the analysis of the SherringtonKirkpatrick model by Farhi et al., we establish an explicit algorithm to evaluate the performance of QAOA on MAXCUT applied to random ErdosRenyi 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 MAXCUT on ErdosRenyi graphs of expected degree $d$, in the limit $d \to \infty$, and the SherringtonKirkpatrick model, and gives good QAOA variational parameters for MAXCUT applied to ErdosRenyi graphs. We then partially generalize the latter analysis to graphs with a degree distribution rather than a single degree $d$, and finally to diluted spinmodels with $D$body interactions ($D \geq 3$). 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 infinitesize limit assumption is clearly violated.
 Publication:

arXiv eprints
 Pub Date:
 October 2021
 arXiv:
 arXiv:2110.10685
 Bibcode:
 2021arXiv211010685B
 Keywords:

 Quantum Physics;
 Computer Science  Data Structures and Algorithms
 EPrint:
 59 pages, 8 figures