The Quantum Approximate Optimization Algorithm (QAOA) is a general purpose
quantum algorithm designed for combinatorial optimization. We analyze its
expected performance and prove concentration properties at any constant level
(number of layers) on ensembles of random combinatorial optimization problems
in the infinite size limit. These ensembles include mixed spin models and
Max-q-XORSAT on sparse random hypergraphs. To enable our analysis, we prove a
generalization of the multinomial theorem which is a technical result of
independent interest. We then show that the performance of the QAOA at constant
levels for the pure q-spin model matches asymptotically the ones for
Max-q-XORSAT on random sparse Erd\H{o}s-R\'{e}nyi hypergraphs and every
large-girth regular hypergraph. Through this correspondence, we establish that
the average-case value produced by the QAOA at constant levels is bounded away
from optimality for pure q-spin models when q≥4 is even. This limitation
gives a hardness of approximation result for quantum algorithms in a new regime
where the whole graph is seen.Comment: 12+46 page