Quantum random walk in periodic potential on a line

TL;DR

This paper explores how quantum random walks on a line behave under periodic potentials, revealing complex probability distributions and specific behaviors of the standard deviation influenced by period and parameters.

Contribution

It introduces analysis of quantum walks with periodic potentials, highlighting how standard deviation varies with period and parameters, which was not previously detailed.

Findings

Standard deviation increases linearly with $ heta$ for certain ranges.

Standard deviation decreases with increasing period $q$ in some parameter regimes.

When $q=2$, the walk exhibits lazy behavior for specific $ heta$ intervals.

Abstract

We investigated the discrete-time quantum random walks on a line in periodic potential. The probability distribution with periodic potential is more complex compared to the normal quantum walks, and the standard deviation $\sigma$ has interesting behaviors for different period $q$ and parameter $\theta$. We studied the behavior of standard deviation with variation in walk steps, period, and $\theta$. The standard deviation increases approximately linearly with $\theta$ and decreases with $1/q$ for $\theta\in(0,\pi/4)$, and increases approximately linearly with $1/q$ for $\theta\in[\pi/4,\pi/2)$. When $q=2$, the standard deviation is lazy for $\theta\in[\pi/4+n\pi,3\pi/4+n\pi],n\in Z$.