The Overlap Gap Property limits limit swapping in the QAOA

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 Erdös--Rényi hypergraph, then a random regular hypergraph is similarly limited in performance as well. As such, we re-derived the fact that the average-case value obtained by the QAOA for even $q\ge 4$, Max-$q$-XORSAT is bounded away from optimality when optimised using asymptotic analysis due to 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 $p \le ε\log n$ providing an improvement over limitation at constant depth. Furthermore, the extension to logarithmic depth leads to a tightening of the upper bound that the QAOA outputs at logarithmic depth for MaxCUT and Max-$q$-XORSAT problems. We also provide some numerical evidence the limitation should be extended to odd $q$ by showing that the OGP exists for the Max-$3$-XORSAT on random regular graphs.