The Overlap Gap Property limits limit swapping in QAOA (2404.06087v4)

Abstract: The Quantum Approximate Optimization Algorithm (QAOA) is a quantum algorithm designed for Combinatorial Optimization Problem (COP). We show that if a local algorithm is limited in performance at logarithmic depth for a spin glass type COP with an underlying Erd\"os--R\'enyi hypergraph, then a random regular hypergraph exhibits it as well. As such, we re-derived the fact that the average-case value obtained by QAOA for the Max-$q$-XORSAT for even $q\ge 4$ is bounded away from optimality even when the algorithm runs indefinitely if optimised using the so-called tree parameters due to the presence of the Overlap Gap Property (OGP). While this result was proven before, the proof is rather technical compared to ours. In addition, we show that the earlier result implicitly also implies limitation at logarithmic depth $p \le \epsilon \log n$ providing an improvement over limitation at superconstant depth. Lastly, the results suggests that even when sub-optimised, the performance of QAOA on spin glass is equal in performance to classical algorithms in solving the mean field spin glass problem providing further evidence that the conjecture of getting the exact solution under limit swapping for the Sherrington--Kirkpatrick model to be true.