论文标题
双方步行的周期性在带有条件光谱的双毛图上
Periodicity of bipartite walk on biregular graphs with conditional spectra
论文作者
论文摘要
在本文中，我们研究了一类离散的量子步行，称为双方步行。这些包括著名的格罗弗步行。任何离散的量子步行均由由ARCS或基础图的边缘索引的统一矩阵$ U $的功能给出。如果$ u^k = i $对于某些正整数$ k $，则步行是周期性的。库博塔（Kubota）在常规的两部分图上定义了步行时，对格罗弗（Grover）步行的周期性进行了描述，最多五个特征值。我们扩展了久保田的结果 - 如果双重图$ g $具有特征值，其正方形是代数整数，最多具有两个学位，我们表征了双方的周期性在其频谱上超过$ g $的周期性。我们应用了两分步道的周期性结果，以使格罗弗（Grover）在常规图上的周期性进行周期性的特征。
In this paper we study a class of discrete quantum walks, known as bipartite walks. These include the well-known Grover's walks. Any discrete quantum walk is given by the powers of a unitary matrix $U$ indexed by arcs or edges of the underlying graph. The walk is periodic if $U^k=I$ for some positive integer $k$. Kubota has given a characterization of periodicity of Grover's walk when the walk is defined on a regular bipartite graph with at most five eigenvalues. We extend Kubota's results--if a biregular graph $G$ has eigenvalues whose squares are algebraic integers with degree at most two, we characterize periodicity of the bipartite walk over $G$ in terms of its spectrum. We apply periodicity results of bipartite walks to get a characterization of periodicity of Grover's walk on regular graphs.
