The Overlap Gap Property limits limit swapping in QAOA

The Quantum Approximate Optimization Algorithm (QAOA) is a quantum algorithm designed for Combinatorial Optimization Problem (COP). We show that if a COP with an underlying Erd\"os--R\'enyi hypergraph exhibits the Overlap Gap Property (OGP), then a random regular hypergraph exhibits it as well. Given that Max-$q$-XORSAT on an Erd\"os--R\'enyi hypergraph is known to exhibit the OGP, and since the performance of QAOA for the pure $q$-spin model matches asymptotically for Max-$q$-XORSAT on large-girth regular hypergraph, we show that the average-case value obtained by QAOA for the pure $q$-spin model for even $q\ge 4$ is bounded away from optimality even when the algorithm runs indefinitely. This suggests that a necessary condition for the validity of limit swapping in QAOA is the absence of OGP in a given combinatorial optimization problem. Furthermore, 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.