  The quantum approximate optimization algorithm (QAOA) is one of the most
promising candidates for achieving quantum advantage through quantum-enhanced
combinatorial optimization. A near-optimal solution to the combinatorial
optimization problem is achieved by preparing a quantum state through the
optimization of quantum circuit parameters. Optimal QAOA parameter
concentration effects for special MaxCut problem instances have been observed,
but a rigorous study of the subject is still lacking. In this work we show
clustering of optimal QAOA parameters around specific values; consequently,
successful transferability of parameters between different QAOA instances can
be explained and predicted based on local properties of the graphs, including
the type of subgraphs (lightcones) from which graphs are composed as well as
the overall degree of nodes in the graph (parity). We apply this approach to
several instances of random graphs with a varying number of nodes as well as
parity and show that one can use optimal donor graph QAOA parameters as
near-optimal parameters for larger acceptor graphs with comparable
approximation ratios. This work presents a pathway to identifying classes of
combinatorial optimization instances for which variational quantum algorithms
such as QAOA can be substantially accelerated.
