使用 Laplacian 和邻接矩阵量子游走搜索加权杠铃图
Searching Weighted Barbell Graphs with Laplacian and Adjacency Quantum Walks
AI总结：
本文研究了加权杠铃图上的连续时间量子搜索，发现 Laplacian 量子游走不受桥权重影响，而邻接量子游走在权重等于团大小时概率聚集，并通过两阶段算法将成功概率提升至0.996。
摘要：
由 Schrödinger 方程在离散空间中演化的量子粒子构成了在由顶点和边组成的图上的连续时间量子游走。当某个顶点被 oracle 标记时，量子游走便实现了一种量子搜索算法。先前对带有团（cliques）的图上这种量子搜索算法的研究表明，团之间的边可以被加权，以增强团之间的概率移动，从而到达标记顶点。在本文中，我们通过分析加权杠铃图（barbell graph）上的搜索来探索其最受限的形式，该图由两个相同大小的团通过一条加权边/桥连接而成。该图通常是不规则的，因此由图 Laplacian 或邻接矩阵支配的量子游走可能会有所不同。我们表明，无论桥的权重如何，Laplacian 量子游走的行为都不会改变，因此单桥的限制性太强以至于无法影响游走。类似地，对于大多数权重，邻接量子游走的行为也保持不变，但当权重等于团的大小时，概率确实会聚集在包含标记顶点的团上，并且利用每个阶段具有不同权重的两阶段算法，成功概率从 0.5 提高到了 0.996，且与杠铃图的大小无关。
英文摘要：
A quantum particle evolving by Schrödinger's equation in discrete space constitutes a continuous-time quantum walk on a graph of vertices and edges. When a vertex is marked by an oracle, the quantum walk effects a quantum search algorithm. Previous investigations of this quantum search algorithm on graphs with cliques have shown that the edges between the cliques can be weighted to enhance the movement of probability between the cliques to reach the marked vertex. In this paper, we explore the most restrictive form of this by analyzing search on a weighted barbell graph that consists of two cliques of the same size joined by a single weighted edge/bridge. This graph is generally irregular, so quantum walks governed by the graph Laplacian or by the adjacency matrix can differ. We show that the Laplacian quantum walk's behavior does not change, no matter the weight of the bridge, and so the single bridge is too restrictive to affect the walk. Similarly, the adjacency quantum walk's behavior is unchanged for most weights, but when the weight equals the size of a clique, the probability does collect at the clique containing the marked vertex, and utilizing a two-stage algorithm with different weights for each stage, the success probability is boosted from 0.5 to 0.996, independent of the size of the barbell graph.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
