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.
翻译:量子行走动态符合量子力学定律, 并附加了位置限制, 要求进化操作员是本地化的, 也就是说行走者在努力到遥远的地方之前必须访问邻近的位置。 通常, 汉密尔顿人来自图形的相邻或斜体矩阵, 而行走者则来自垂直体到周围的脊椎。 在这项工作中, 我们定义了连续时间量子行走的版本, 允许行走者从脊椎跳到边缘, 反面跳到边缘。 作为应用程序, 我们通过修改汉密尔顿人的新版本来分析空间搜索算法, 其额外期限取决于标记的脊椎或标志边缘的位置, 类似于标准连续时间量子行走模型所做的那样。 我们显示, 找到脊椎或边缘的最佳运行时间是$O (\\ qrt{N_ e} 。 美元, 其成功概率为 $-1- o(1)美元, 作为一种应用程序, 我们分析完整的双极图的边缘数。