This paper introduces a group-theoretic framework to analyze the algebraic structure of the Grover walk on a complete graph with self-loops. We construct a group generated by the Grover matrix and a diagonal matrix whose entries are powers of a complex root of unity. We then characterize the resulting quotient group, which is defined using a subgroup formed by commutators involving these matrices. We show that this quotient group is isomorphic to a finite cyclic group whose structure depends on the parity of the number of vertices. This group-theoretic characterization reveals underlying symmetries in the time evolution of the Grover walk and provides an algebraic framework for understanding its periodic behavior.

翻译：本文提出了一种群论框架，用于分析具有自环的完全图上Grover游动的代数结构。我们构造了一个由Grover矩阵与一个对角矩阵生成的群，该对角矩阵的元素是单位复根的幂次。随后，我们刻画了由此得到的商群，该商群是通过由这些矩阵的换位子构成的子群来定义的。我们证明该商群同构于一个有限循环群，其结构取决于顶点数的奇偶性。这一群论刻画揭示了Grover游动时间演化中的潜在对称性，并为理解其周期性行为提供了代数框架。