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.

翻译：图同构问题研究的是两个图是否仅在顶点重标号下相同。尽管精确版本存在拟多项式时间的经典算法，但在分子比较、噪声网络分析与模式识别等应用中，往往需要结构相似性的灵活定义。我们研究了近似图同构测试的量子查询复杂度，其中从 Erdős–Rényi 分布 $\mathcal{G} (n,1/2)$ 中抽取的两个 $n$ 顶点图，若通过至多 $k$ 条边编辑可使其同构，则视为近似同构。我们提出了一种基于 MNRS 量子游走搜索的量子算法，该算法作用于两个输入图的产品图 $Γ(G,H)$。当图近似同构时，量子游走搜索可检测属于稠密近似同构匹配集的顶点对；随后通过局部一致性传播重构候选配对，并利用 Grover 加速的一致性检验进行验证。我们证明该方法实现了查询复杂度 $\mathcal{O}(n^{3/2} \log n/\varepsilon)$，其中 $\varepsilon$ 参数化近似阈值。我们进一步针对常数近似建立了 $\Omega(n^2)$ 的经典下界，证实了查询模型下真正的多项式量子加速。该框架可扩展至基于图拉普拉斯特征值的谱相似性度量，以及加权图与属性图。针对至多二十个顶点图的小规模量子模拟器仿真结果表明，该算法与近期量子设备兼容。