具有支配或固定正则初始点的Kronecker图上的最优量子行走搜索
Optimal Quantum Walk Search on Kronecker Graphs with Dominant or Fixed Regular Initiators
摘要：
在网络科学中，通过对初始图的邻接矩阵取Kronecker或张量幂而得到的图被用于构造复杂网络。在本文中，我们解析地证明了在最佳$Θ(\sqrt{N})$时间内连续时间量子行走可以搜索到这样的Kronecker图的充分条件。首先，如果发起人是正规的，它的邻接矩阵有一个主导的主特征值，这意味着它的唯一的最大特征值渐近地主导其他特征值的大小，然后由这个发起人产生的克罗内克图可以量子搜索概率为1在$π\sqrt{N}/2$时间，渐近，我们给出的临界跳跃率的行走，使这一点。其次，对于任何固定的发起人，是规则的，非二部，并连接，克罗内克图产生的量子搜索在$Θ(\sqrt{N})$时间。这极大地扩展了已知量子行走最优搜索的克罗内克图的数量。然而，如果固定的、规则的、连接的发起者是二分的，则在其克罗内克幂上的搜索不是最优的，但是如果发起者具有两个以上的顶点，则仍然优于经典计算机的$O(N)$运行时。
英文摘要：
In network science, graphs obtained by taking the Kronecker or tensor power of the adjacency matrix of an initiator graph are used to construct complex networks. In this paper, we analytically prove sufficient conditions under which such Kronecker graphs can be searched by a continuous-time quantum walk in optimal $Θ(\sqrt{N})$ time. First, if the initiator is regular and its adjacency matrix has a dominant principal eigenvalue, meaning its unique largest eigenvalue asymptotically dominates the other eigenvalues in magnitude, then the Kronecker graphs generated by this initiator can be quantum searched with probability 1 in $π\sqrt{N}/2$ time, asymptotically, and we give the critical jumping rate of the walk that enables this. Second, for any fixed initiator that is regular, non-bipartite, and connected, the Kronecker graphs generated by it are quantum searched in $Θ(\sqrt{N})$ time. This greatly extends the number of Kronecker graphs on which quantum walks are known to optimally search. If the fixed, regular, connected initiator is bipartite, however, then search on its Kronecker powers is not optimal, but is still better than classical computer's $O(N)$ runtime if the initiator has more than two vertices.
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