ABSTRACT

As a first step of the study of the relation between the distribution of prime, reduced cycles of a graph and the limit theorem for the probability of a quantum walk on a graph, we present a trace formula with respect to the Grover matrix that is the transition matrix of the Grover walk on a graph. We define a zeta function with respect to the Grover matrix of a graph, and differentiate the logarithm of its zeta function of a regular graph. Furthermore, we obtain a trace formula with respect to the Grover matrix by a suitable complex integral of its logarithm differential.

中文翻译：

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关于图的 Grover 矩阵的迹公式

摘要

作为研究素数分布、图的约简循环和图上量子游走概率的极限定理之间关系的第一步，我们提出了一个关于 Grover 矩阵的迹公式，即Grover 在图上行走的转移矩阵。我们针对图的 Grover 矩阵定义了一个 zeta 函数，并对其正则图的 zeta 函数的对数进行微分。此外，我们通过其对数微分的适当复积分获得了关于 Grover 矩阵的迹公式。