A group structure arising from Grover walks on complete graphs with self-loops and its application

TL;DR

This work introduces a group-theoretic lens to the Grover walk on a complete graph with self-loops by constructing a group $K$ generated by the Grover matrix $G$ and a diagonal $S$, and a normal subgroup $H$ generated by commutators. By analyzing the quotient $K/H$, the authors show it is a finite cyclic group whose structure depends on the parity of $n$, yielding a clean algebraic description of the walk's periodicity. The main results establish that $K/H \,\cong \, Z_n \times Z_2$ when $n$ is even and $K/H \,\cong \, Z_n$ when $n$ is odd, with $H \cong Z_2^{n-1}$, and prove that the minimal $m$ with $(S^jG)^m \in H$ for all $j$ is $m = 2n$, which implies $U^{2n}=I_{n^2}$. This provides a purely algebraic explanation for the Grover walk's periodicity and introduces a modular, group-theoretic framework potentially applicable to broader quantum-walk settings.

Abstract

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.

Figures (1)

• Figure 1: The complete graph with four vertices

Theorems & Definitions (9)

• Theorem 1
• Theorem 2
• Lemma 1
• Lemma 2
• Proposition 1
• Proposition 2
• Lemma 3
• Lemma 4
• Proposition 3