Periodicity and absolute zeta functions of multi-state Grover walks on cycles
Jirô Akahori, Norio Konno, Iwao Sato, Yuma Tamura
TL;DR
This work analyzes Grover quantum walks on cycle graphs with an odd state count $L=2m+1$, linking their periodicity to absolute zeta functions. Using spectral factorization and cyclotomic-Polynomial reasoning, it derives exact period criteria: M-type walks have period $2L$ only when the cycle length $N$ matches $L$ (and are infinite otherwise under coprime-factor constraints), while F-type walks have period $4$ under the same conditions. It also provides explicit absolute zeta-function expressions for the M-type walk on the $L$-cycle, constructing a bridge between quantum-walk spectral properties and absolute zeta theory. The results deepen the arithmetic-underpinned understanding of quantum walks on cycles and lay groundwork for further exploration of zeta-theoretic aspects in discrete-time quantum dynamics.
Abstract
Quantum walks, the quantum counterpart of classical random walks, are extensively studied for their applications in mathematics, quantum physics, and quantum information science. This study explores the periods and absolute zeta functions of Grover walks on cycle graphs. Specifically, we investigate Grover walks with an odd number of states and determine their periods for cycles with any number of vertices greater than or equal to two. In addition, we compute the absolute zeta functions of M-type Grover walks with finite periods. These results advance the understanding of the properties of Grover walks and their connection to absolute zeta functions.