  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.
