Eigenvalues of Two-State Quantum Walks Induced by the Hadamard Walk

Shimpei Endo; Takako Endo; Takashi Komatsu; Norio Konno

doi:10.3390/e22010127

Eigenvalues of Two-State Quantum Walks Induced by the Hadamard Walk

Entropy (Basel). 2020 Jan 20;22(1):127. doi: 10.3390/e22010127.

Authors

Shimpei Endo¹, Takako Endo², Takashi Komatsu³, Norio Konno²

Affiliations

¹ Department of Physics, Frontier Research Institute for Interdisciplinary Science, Faculty of Science, Tohoku University, 6-3, Aoba, Aramaki-aza, Aobaku, Sendai, Miyagi 980-8578, Japan.
² Department of Applied Mathematics, Faculty of Engineering, Yokohama National University, Hodogaya, Yokohama 240-8501, Japan.
³ Department of Bioengineering School of Engineering, The University of Tokyo, Bunkyo, Tokyo 113-8656, Japan.

Abstract

Existence of the eigenvalues of the discrete-time quantum walks is deeply related to localization of the walks. We revealed, for the first time, the distributions of the eigenvalues given by the splitted generating function method (the SGF method) of the space-inhomogeneous quantum walks in one dimension we had treated in our previous studies. Especially, we clarified the characteristic parameter dependence for the distributions of the eigenvalues with the aid of numerical simulation.

Keywords: classification; discrete-time quantum walks; eigenvalues; localization; stationary measure.