  A promising approach to the practical application of the Quantum Approximate
Optimization Algorithm (QAOA) is finding QAOA parameters classically in
simulation and sampling the solutions from QAOA with optimized parameters on a
quantum computer. Doing so requires repeated evaluations of QAOA energy in
simulation. We propose a novel approach for accelerating the evaluation of QAOA
energy by leveraging the symmetry of the problem. We show a connection between
classical symmetries of the objective function and the symmetries of the terms
of the cost Hamiltonian with respect to the QAOA energy. We show how by
considering only the terms that are not connected by symmetry, we can
significantly reduce the cost of evaluating the QAOA energy. Our approach is
general and applies to any known subgroup of symmetries and is not limited to
graph problems. Our results are directly applicable to nonlocal QAOA
generalization RQAOA. We outline how available fast graph automorphism solvers
can be leveraged for computing the symmetries of the problem in practice. We
implement the proposed approach on the MaxCut problem using a state-of-the-art
tensor network simulator and a graph automorphism solver on a benchmark of 48
graphs with up to 10,000 nodes. Our approach provides an improvement for $p=1$
on $71.7\%$ of the graphs considered, with a median speedup of $4.06$, on a
benchmark where $62.5\%$ of the graphs are known to be hard for automorphism
solvers.
