论文标题
Laplacian连续时间量子步行问题中完全连接的顶点的通用性
Universality of the fully connected vertex in Laplacian continuous-time quantum walk problems
论文作者
论文摘要
一个完全连接的顶点$ w $在简单的图形$ g $ of订单$ n $中是连接到所有其他$ n-1 $顶点的顶点。在用$ l $表示该图的拉普拉斯矩阵时，我们证明了一个沃克（Hamiltonian $ h =γl$）的连续时间量子步行（CTQW）最初以$ \ vert w \ rangle $定位的助行器并不依赖于图$ g $。我们还证明，对于任何类似Grover的CTQW-使用Hamiltonian $ h =γl +\sum_wλ_W\ vert w \ w \ rangle \ langle \ langle w \ vert $ - 在已完全连接的标记的Vertices $ W $的概率幅度不取决于$ G $。与Hamiltonian $ H =γa$（邻接矩阵）的CTQW无法得出的结果。我们将结果应用于单个和多个完全连接的标记顶点的空间搜索和量子传输，证明任何图形$ g $上的ctqws继承了已知的属性，这些属性已知相同顺序的完整图，包括空间搜索的最佳性。我们的结果为已经在文献中报道的几个部分结果提供了一个统一的框架，例如完全连接的顶点，例如CTQW的等效性和空间搜索恒星和车轮图的中心顶点，以及完整图的任何顶点。
A fully connected vertex $w$ in a simple graph $G$ of order $N$ is a vertex connected to all the other $N-1$ vertices. Upon denoting by $L$ the Laplacian matrix of the graph, we prove that the continuous-time quantum walk (CTQW) -- with Hamiltonian $H=γL$ -- of a walker initially localized at $\vert w \rangle$ does not depend on the graph $G$. We also prove that for any Grover-like CTQW -- with Hamiltonian $H=γL +\sum_w λ_w \vert w \rangle\langle w \vert$ -- the probability amplitude at the fully connected marked vertices $w$ does not depend on $G$. The result does not hold for CTQW with Hamiltonian $H=γA$ (adjacency matrix). We apply our results to spatial search and quantum transport for single and multiple fully connected marked vertices, proving that CTQWs on any graph $G$ inherit the properties already known for the complete graph of the same order, including the optimality of the spatial search. Our results provide a unified framework for several partial results already reported in literature for fully connected vertices, such as the equivalence of CTQW and of spatial search for the central vertex of the star and wheel graph, and any vertex of the complete graph.
