通过连续时间量子游走在顶点与边上行走
Walking on Vertices and Edges by Continuous-Time Quantum Walk
AI总结：
本文定义了一种允许在顶点与边间跳跃的连续时间量子游走新版本，并将其应用于完全二部图上的空间搜索，证明了寻找顶点或边的最优运行时间为 $O(\sqrt{N_e})$ 且成功概率为 $1-o(1)$。
摘要：
量子游走动力学遵循量子力学定律并带有额外的局域性约束，该约束要求演化算子具有局域性，即行走者在前往远处之前必须先访问相邻位置。通常，哈密顿量由图的邻接矩阵或拉普拉斯矩阵得出，行走者从顶点跳跃到相邻顶点。在本研究中，我们定义了一种连续时间量子游走的版本，允许行走者在顶点与边之间跳跃，反之亦然。作为应用，我们通过在哈密顿量的新版本中添加一个依赖于标记顶点或标记边位置的额外项，分析了完全二部图上的空间搜索算法，这与标准连续时间量子游走模型中的做法类似。我们证明，找到顶点或边的最优运行时间为 $O(\sqrt{N_e})$，成功概率为 $1-o(1)$，其中 $N_e$ 是完全二部图的边数。
英文摘要：
The quantum walk dynamics obey the laws of quantum mechanics with an extra locality constraint, which demands that the evolution operator is local in the sense that the walker must visit the neighboring locations before endeavoring to distant places. Usually, the Hamiltonian is obtained from either the adjacency or the laplacian matrix of the graph and the walker hops from vertices to neighboring vertices. In this work, we define a version of the continuous-time quantum walk that allows the walker to hop from vertices to edges and vice versa. As an application, we analyze the spatial search algorithm on the complete bipartite graph by modifying the new version of the Hamiltonian with an extra term that depends on the location of the marked vertex or marked edge, similar to what is done in the standard continuous-time quantum walk model. We show that the optimal running time to find either a vertex or an edge is $O(\sqrt{N_e})$ with success probability $1-o(1)$, where $N_e$ is the number of edges of the complete bipartite 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。
