Trapping and spreading properties of quantum walk in homological structure
Trapping and spreading properties of quantum walk in homological structure
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同调结构中量子行走的捕获和扩散特性
DOI：
10.1007/s11128-014-0819-6
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发表时间：
2014
影响因子：
2.5
通讯作者：
Etsuo Segawa
中科院分区：
文献类型：
--
作者：
Takuya Machida;Etsuo Segawa
我们试图通过Grover行走来提取两类图的同调结构。第一个是由一个圈和两条半无限直线组成的，第二个是由圈的周期嵌入组装而成的。我们证明了这两个图具有本质上相同的特征值，这些特征值是由无限图中圈的存在引起的。在Grover游动中，同调结构的特征空间表现为所谓的局部化，其中游动部分地被同调结构捕获。另一方面，它们之间的谱的绝对连续部分的差异提供了不同的行为。我们用弱收敛定理中的密度函数刻画了这两种行为：第一种是底部的δ测度，第二种是两种连续函数，它们分别具有不同的有限支集和.
We attempt to extract a homological structure of two kinds of graphs by the Grover walk. The first one consists of a cycle and two semi-infinite lines, and the second one is assembled by a periodic embedding of the cycles in. We show that both of them have essentially the same eigenvalues induced by the existence of cycles in the infinite graphs. The eigenspace of the homological structure appears as so calledlocalizationin the Grover walks, in which the walk is partially trapped by the homological structure. On the other hand, the difference of the absolutely continuous part of spectrum between them provides different behaviors. We characterize the behaviors by the density functions in the weak convergence theorem: The first one is the delta measure at the bottom, while the second one is expressed by two kinds of continuous functions, which have different finite supportsand, respectively.