Exponentially decaying velocity bounds of quantum walks in periodic fields

TL;DR

This paper demonstrates that in certain quantum walks, introducing a periodic local field can exponentially suppress the walk's velocity, leading to localization effects even over many steps, especially for large periods.

Contribution

It provides a rigorous proof that periodic local fields can exponentially reduce quantum walk velocities, revealing new localization phenomena in quantum dynamics.

Findings

Velocity can be made arbitrarily small with suitable periodic fields.

Velocity bounds decay exponentially with the period length.

Localization-like effects persist over many steps for large periods.

Abstract

We consider a class of discrete-time one-dimensional quantum walks, associated with CMV unitary matrices, in the presence of a local field. This class is parametrized by a transmission parameter $t\in[0,1]$. We show that for a certain range for $t$, the corresponding asymptotic velocity can be made arbitrarily small by introducing a periodic local field with a sufficiently large period. In particular, we prove an upper bound for the velocity of the $n$-periodic quantum walk that is decaying exponentially in the period length $n$. Hence, localization-like effects are observed even after a long number of quantum walk steps when $n$ is large.