  The Quantum approximate optimization algorithm (QAOA) is one of the most
promising candidates for achieving quantum advantage through quantum-enhanced
combinatorial optimization. In a typical QAOA setup, a set of quantum circuit
parameters is optimized to prepare a quantum state used to find the optimal
solution of a combinatorial optimization problem. Several empirical
observations about optimal parameter concentration effects for special QAOA
MaxCut problem instances have been made in recent literature, however, a
rigorous study of the subject is still lacking. We show that convergence of the
optimal QAOA parameters around specific values and, consequently, successful
transferability of parameters between different QAOA instances can be explained
and predicted based on the local properties of the graphs, specifically the
types of subgraphs (lightcones) from which the graphs are composed. We apply
this approach to random regular and general random graphs. For example, we
demonstrate how optimized parameters calculated for a 6-node random graph can
be successfully used without modification as nearly optimal parameters for a
64-node random graph, with less than 1% reduction in approximation ratio as a
result. This work presents a pathway to identifying classes of combinatorial
optimization instances for which such variational quantum algorithms as QAOA
can be substantially accelerated.
