Quantum Algorithms for Approximate Graph Isomorphism Testing

TL;DR

This work presents a quantum algorithm based on MNRS quantum walk search over the product graph $\Gamma(G,H)$ of the two input graphs, and complements this with an $\Omega(n^2)$ classical lower bound for constant approximation, establishing a genuine polynomial quantum speedup in the query model.

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 Erdős--Rényi 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 $Γ(G,H)$ 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 $Ω(n^2)$ 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.

Figures (2)

• Figure 1: Empirical query count to achieve $90\%$ discrimination accuracy versus graph size $n$. Blue circles: quantum algorithm (MNRS walk search + reconstruction + verification). Red squares: classical random sampling baseline with the same query budget. Dashed/dotted curves show $n^{3/2}$ and $n^{2}$ reference scalings, confirming the predicted polynomial separation.
• Figure 2: Discrimination accuracy versus approximation parameter $\varepsilon$ at fixed $n=14$. The full algorithm degrades gracefully, maintaining $\ge 83\%$ accuracy even at $\varepsilon=0.20$, while the classical baseline drops to near-random.

Theorems & Definitions (88)

• Definition 2.1: Adjacency matrix
• Definition 2.2: Graph Laplacian
• Definition 2.3: Graph isomorphism
• Definition 2.4: Edit distance
• Definition 2.5: Quantum query complexity
• Theorem 2.6: Grover search, Grover1996
• Theorem 2.7: Amplitude estimation, BrassardHMT2002
• Definition 2.8: Szegedy walk
• Theorem 2.9: Szegedy spectral theorem, Szegedy2004
• Theorem 2.10: Quantum walk search, MNRS2011