Quantum Algorithms for Approximate Graph Isomorphism Testing

TL;DR

This paper introduces a quantum algorithm for approximate graph isomorphism testing that outperforms classical methods in query complexity, with extensions to spectral and weighted graphs, and demonstrates feasibility on small quantum simulators.

Contribution

The paper presents a novel quantum algorithm for approximate graph isomorphism testing with polynomial speedup and extends the framework to spectral and weighted graphs.

Findings

Quantum algorithm achieves query complexity O(n^{3/2} log n / ε)

Classical lower bound of Ω(n^2) for constant approximation

Small-scale quantum simulations support practical feasibility

Abstract

The graph isomorphism problem asks whether two graphs are identical up to vertex relabeling. While the exact problem admits quasi-polynomial-time classical algorithms, many applications in molecular comparison, noisy network analysis, and pattern recognition require a flexible notion of structural similarity. We study the quantum query complexity of approximate graph isomorphism testing, where two graphs on $n$ vertices drawn from the Erd\H{o}s--R\'enyi distribution $\mathcal{G} (n,1/2)$ are considered approximately isomorphic if they can be made isomorphic by at most $k$ edge edits. We present a quantum algorithm based on MNRS quantum walk search over the product graph $\Gamma(G,H)$ of the two input graphs. When the graphs are approximately isomorphic, the quantum walk search detects vertex pairs belonging to a dense near isomorphic matching set; candidate pairings are then reconstructed via local consistency propagation and verified via a Grover-accelerated consistency check. We prove that this approach achieves query complexity $\mathcal{O}(n^{3/2} \log n/\varepsilon)$, where $\varepsilon$ parameterizes the approximation threshold. We complement this with an $\Omega(n^2)$ classical lower bound for constant approximation, establishing a genuine polynomial quantum speedup in the query model. We extend the framework to spectral similarity measures based on graph Laplacian eigenvalues, as well as weighted and attributed graphs. Small-scale simulation results on quantum simulators for graphs with up to twenty vertices demonstrate compatibility with near-term quantum devices.