Periodicity for the 3-state quantum walk on cycles
Quantum Physics
2019-07-04 v1
Abstract
Dukes (2014) and Konno, Shimizu, and Takei (2017) studied the periodicity for 2-state quantum walks whose coin operator is the Hadamard matrix on cycle graph C_N with N vertices. The present paper treats the periodicity for 3-state quantum walks on C_N. Our results follow from a new method based on cyclotomic field. This method shows a necessary condition for the coin operator of quantum walks to have the finite period. Moreover, we reveal the period T_N of two kinds of typical quantum walks, the Grover and Fourier walks. We prove that both walks do not have any finite period except for N=3, in which case T_3=6 (Grover), =12 (Fourier).
Cite
@article{arxiv.1907.01725,
title = {Periodicity for the 3-state quantum walk on cycles},
author = {Takeshi Kajiwara and Norio Konno and Shohei Koyama and Kei Saito},
journal= {arXiv preprint arXiv:1907.01725},
year = {2019}
}
Comments
6 pages