MAXCUT QAOA performance guarantees for p >1

TL;DR

This paper establishes performance guarantees for the QAOA algorithm on MAXCUT for 3-regular graphs at depths p=2 and 3, improving known bounds and exploring the potential for quantum advantage.

Contribution

It provides new lower bounds on QAOA's approximation ratio for p=2 and 3 on 3-regular graphs, and proposes a conjecture on worst-case graph structures for all p.

Findings

Lower bound of 0.7559 for p=2 on certain graphs

Conjecture of worst-case graphs having no cycles ≤ 2p+1

Upper bound suggesting limited quantum advantage for p<6

Abstract

We obtain worst case performance guarantees for $p=2$ and $3$ QAOA for MAXCUT on uniform 3-regular graphs. Previous work by Farhi et al obtained a lower bound on the approximation ratio of $0.692$ for $p=1$. We find a lower bound of $0.7559$ for $p=2$, where worst case graphs are those with no cycles $\leq 5$. This bound holds for any 3 regular graph evaluated at particular fixed parameters. We conjecture a hierarchy for all $p$, where worst case graphs have with no cycles $\leq 2p+1$. Under this conjecture, the approximation ratio is at least $0.7924$ for all 3 regular graphs and $p=3$. In addition, using a simple indistinguishability argument we find an upper bound on the worst case approximation ratio for all $p$, which indicates classes of graphs for which there can be no quantum advantage for at least $p<6$.