On The Variational Perspectives To The Graph Isomorphism Problem

Abstract: This paper studies the Graph Isomorphism Problem from a variational algorithmic perspective, specifically studying the Quadratic Unconstrained Binary Optimization (QUBO) formulation of the Graph Isomorphism Problem and subsequent execution using the Quantum Approximate Optimization Algorithm (QAOA) and the Variational Quantum Eigensolver (VQE). This study presents the results of these algorithms and the variations that occur therein for graphs of four and five nodes. The main findings of this paper include the clustering in the energy landscape for the QAOA in isomorphic graphs having an equal number of nodes and edges. This trend found in the QAOA study was further reinforced by studying the ground state energy reduction using VQEs. Furthermore, this paper examines the trend under which isomorphic pairs of graphs vary in the ground state energies, with varying edges and nodes.