On Variational Perspectives To The Graph Isomorphism Problem

TL;DR

The paper investigates solving the Graph Isomorphism problem through variational quantum algorithms by encoding the problem as a QUBO and evaluating both QAOA and VQE on small graphs (4–5 nodes). QAOA provides a shallow, hardware-friendly heuristic, while VQE offers a deeper, more exact variational check; both methods exhibit energy clustering for isomorphic graph families, supporting the idea that energy landscapes can signal isomorphism. The authors compare different QUBO formulations and validate that isomorphic pairs yield zero or near-zero energies, with non-isomorphic pairs producing higher energies, a result corroborated by both QAOA and VQE. Despite promising clustering behavior, the study emphasizes limitations in qubit scalability and hardware noise, outlining directions toward compact encodings, 6–8 node scalability, and real-device experiments with error mitigation. Overall, the work highlights energy clustering as a viable variational signature for GI and outlines a practical pathway for leveraging near-term quantum devices in combinatorial problems.

Abstract

We consider a quadratic unconstrained binary optimization (QUBO) formulation of the graph isomorphism problem from a variational quantum algorithmic perspective. By treating it using the Quantum Approximate Optimization Algorithm (QAOA) and the Variational Quantum Eigensolver (VQE), this study presents results for each and highlights variations therein for graphs of four and five nodes. For isomorphic graphs having an equal number of nodes and edges, we demonstrate clustering in the energy landscape for the QAOA. This trend found in the QAOA study was further reinforced by studying the ground state energy reduction using VQEs. Furthermore, we examine the trend under which isomorphic pairs of graphs vary in the ground state energies, with varying edges and nodes.

Figures (3)

• Figure 1: The energy values of the QAOA for which the resulting values of their corresponding quadratic values yield an optimal function value of zero. (a) Shows the plot of the energy values for which the values of the quadratic program for the Graph Isomorphism Problem for a four node graph yields a function value of zero. (b) Shows the plot of the energy values for which the values of the quadratic program for the Graph Isomorphism Problem for a five node graph yields a function value of zero. The boxplot shows clearly the distribution of values, the medians and standard deviations for the absolute energy values.
• Figure 2: The illustration of variation in QAOA energies with simultaneous edge and node reduction. (a) Shows how edges were methodically removed from a five node complete graph (the graphs are reduced from (j) $\rightarrow$ (a)). A copy of each graph was used when testing, thereby yielding a pair of identical graphs, on which the graph isomorphism formulation was carried out. (b) Shows the QAOA energies of the Hamiltonian for each of these graphs for which the isomorphic pairs has the quadratic program yield a function value of zero
• Figure 3: The illustration of the energy for VQE (y-axis) for pairs of graphs with four nodes in respect with number of iterations (x-axis). The colors represent different runs on a class of graphs. It can be seen that after $400$ iterations the energies cluster.