Automorphism-Assisted Quantum Approximate Optimization Algorithm for efficient graph optimization

Abstract: In this article we report on the application of the Quantum Approximate Optimization Algorithm (QAOA) to solve the unweighted MaxCut problem on tree-structured graphs. Specifically, we utilize the Nauty (No Automorphisms, Yes?) package to identify graph automorphisms, focusing on determining edge equivalence classes. These equivalence classes also correspond to symmetries in the terms of the associated Ising Hamiltonian. By exploiting these symmetries, we achieve a significant reduction in the complexity of the Hamiltonian, thereby facilitating more efficient quantum simulations. We conduct benchmark experiments on graphs with up to 34 nodes on memory and CPU intensive TPU provided by google Colab, applying QAOA with a single layer ($p=1$). The approximation ratios obtained from both the full and symmetry-reduced Hamiltonians are systematically compared. Our results show that using automorphism-based symmetries to reduce the Pauli terms in the Hamiltonian can significantly decrease computational overhead without compromising the quality of the solutions obtained.