In this paper we extend the study of three state lively quantum walks on cycles by considering the coin operator as a linear sum of permutation matrices, which is a generalization of the Grover matrix. First we provide a complete characterization of orthogonal matrices of order $3\times 3$ which are linear sum of permutation matrices. Consequently, we determine several groups of complex, real and rational orthogonal matrices. We establish that an orthogonal matrix of order $3\times 3$ is a linear sum of permutation matrices if and only if it is permutative. Finally we determine period of lively quantum walk on cycles when the coin operator belongs to the group of orthogonal (real) linear sum of permutation matrices.

中文翻译：

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使用广义格罗弗币在周期上活跃量子行走的周期性

在本文中，我们通过将硬币算子视为置换矩阵的线性和，这是 Grover 矩阵的推广，扩展了对循环上三态活泼量子游走的研究。首先，我们提供了 $3\times 3$ 阶正交矩阵的完整表征，它们是置换矩阵的线性和。因此，我们确定了几组复数、实数和有理正交矩阵。我们确定 $3\times 3$ 阶的正交矩阵是置换矩阵的线性和当且仅当它是置换的。最后，当硬币算子属于置换矩阵的正交（实）线性和组时，我们确定活跃的量子行走周期。