Localization of space-inhomogeneous three-state quantum walks

Abstract

Mathematical analysis on the existence of eigenvalues is essential because it is deeply related to localization, which is an exceptionally crucial property of quantum walks. We construct the method for the eigenvalue problem via the transfer matrix for space-inhomogeneous three-state quantum walks in one dimension with a self-loop, which is an extension of the technique in a previous study (Quantum Inf. Process 20(5), 2021). This method reveals the necessary and sufficient condition for the eigenvalue problem of a two-phase three-state quantum walk with one defect whose time evolution varies in the negative part, positive part, and at the origin.

Bandit Algorithm Driven by a Classical Random Walk and a Quantum Walk

“…Solving QW-based models with site-dependent coins is very difficult, in general. While some studies have addressed this matter for the case where the coin matrix only on the origin differs from the others [ 6 , 22 ] or that the coin matrices are controlled by the trigonometric function whose input is in proportion to the label position [ 12 , 14 , 21 ], the generalized case remains an open problem. To conduct a thorough analysis of this model, it is necessary to accumulate analytical results for site-dependent quantum walks over an extended period of time.…”

“…The idea of weak convergence, which is frequently used in probability theory, was introduced to show the properties of QWs [ 10 , 11 ], and since then, quantum walks have been actively studied from both fundamental and applied perspectives. In fundamental fields, there have been many attempts to analyze these evolution models mathematically [ 6 , 12 , 13 , 14 , 15 , 16 , 17 , 18 , 19 , 20 , 21 , 22 ] due to varying behavior of QWs depending on the conditions or settings of time and space. In applied fields, their unique behavior is useful for implementing quantum structures or quantum analogs of existing models; therefore, various QW-based models have been considered for subjects such as time-series analysis [ 23 ], topological insulators [ 24 , 25 ], radioactive waste reduction [ 26 , 27 ], and optics [ 28 , 29 ].…”