塞格迪量子行走中的单链路移除扰动：从图完整性测试到完整性监测
Single Link Removal Perturbation in Szegedy Quantum Walk: from Graph Completeness Testing to Integrity Monitoring
中文总结 AI 辅助
研究塞格迪量子行走在完全图有标记节点且存在单链路移除异常时的情况，基于图完整性测试框架量化单链路移除对相关谱量的影响，证明扰动谱范数等结论，为量子行走拓扑完整性监测提供理论基础和缩放极限。
中文摘要 AI 辅助
我们对具有标记节点的完全图上的塞格迪量子行走搜索算法进行了严格的微扰分析，此时图中存在特定异常。这是由用量子辅助程序监测密集可信通信网络完整性的问题所推动的。这些网络的拓扑结构被建模为完全图，感兴趣的异常是单个通信链路的消失，这代表了最难检测的最小结构缺陷。基于图完整性测试算法框架，我们量化了移除单个未标记 - 未标记边如何通过塞格迪量子行走的相关谱量传播。用\(n\)表示图的节点总数，\(m\)表示标记节点数，我们证明对转移矩阵的扰动具有谱范数\(\Theta(1/n)\)，并且对于每个\(n\)和每个标记节点数\(m\)，间隙特征值经历严格负的一阶偏移，为我们在完整性测试算法工作中的一个猜想提供了形式证明；在与搜索算法相关的\(m = \Theta(n)\)的情况下，这种偏移的大小为\(\Theta(1/n^2)\)。相应的本征相移在相同情况下满足\(\Delta\theta_\star = \Theta(1/n^2)\)。我们确定在这种扰动下有效子空间的旋转角度对于\(m = \Theta(n)\)为\(O(1/n)\)。最后，我们将成功概率的变化限制在相同标记节点情况下为\(O(1/\sqrt{n})\)，并表明这个界限主要由有效子空间的几何失准而非本征相的谱移主导。这些结果为基于量子行走的拓扑完整性监测在最小结构扰动下提供了理论基础和基本缩放极限。
英文摘要
We present a rigorous perturbative analysis of the Szegedy quantum walk search algorithm on the complete graph with marked nodes, when a specific anomaly is present in the graph. This is motivated by the problem of monitoring the integrity of dense trusted communication networks with a quantum-assisted procedure. The topology of these networks is modeled as a complete graph, and the anomaly of interest is the disappearance of a single communication link which represents the minimal and spectrally hardest structural defect to detect. Building on the graph-completeness testing algorithm framework, we quantify how the removal of a single unmarked-unmarked edge propagates through the relevant spectral quantities of the Szegedy quantum walk. Denoting by $n$ the total number of nodes of the graph and by $m$ the number of marked nodes, we prove that the perturbation to the transition matrix has spectral norm $Θ(1/n)$, and that the gap eigenvalue undergoes a strictly negative first-order shift for every $n$ and every number of marked nodes $m$, providing a formal proof of a conjecture from our completeness testing algorithm work; in the regime $m = Θ(n)$ relevant for the search algorithm, this shift has magnitude $Θ(1/n^2)$. The corresponding eigenphase shift satisfies $Δθ_\star = Θ(1/n^2)$ in the same regime. We establish that the rotation angle of the effective subspace under this perturbation is $O(1/n)$ for $m = Θ(n)$. Finally, we bound the change in success probability to $O(1/\sqrt{n})$ in this same regime of marked nodes, and show that this bound is dominated by the geometric misalignment of the effective subspace rather than by the spectral shift of the eigenphase. These results provide both the theoretical foundations and the fundamental scaling limits of quantum walk-based topology integrity monitoring under minimal structural perturbations.
