One-dimensional quantum walks via generating function and the CGMV method

TL;DR

This paper analyzes a one-dimensional quantum walk with a specific coin behavior, revealing localization with power-law decay and ballistic spreading, and provides a limit theorem describing its asymptotic distribution.

Contribution

It introduces a limit theorem for quantum walks with identity-tending coins, showing power-law localization and ballistic spreading, expanding understanding of quantum walk dynamics.

Findings

Localization with power-law decay around the origin

Weak convergence with delta measures at specific points

Ballistic spreading called bottom localization

Abstract

We treat a quantum walk (QW) on the line whose quantum coin at each vertex tends to be the identity as the distance goes to infinity. We obtain a limit theorem that this QW exhibits localization with not an exponential but a "power-law" decay around the origin and a "strongly" ballistic spreading called bottom localization in this paper. This limit theorem implies the weak convergence with linear scaling whose density has two delta measures at $x=0$ (the origin) and $x=1$ (the bottom) without continuous parts.