Persistence of unvisited sites in presence of a quantum random walker

TL;DR

This paper investigates the persistence dynamics of a quantum random walk, revealing how unvisited sites behave over time and highlighting fundamental differences from classical random walks.

Contribution

First analysis of persistence and first passage times in a quantum random walk, providing new insights into quantum walk dynamics and site visitation properties.

Findings

Persistence probability decays as a power law with exponent ~0.3

Fraction of unvisited sites approaches a constant value

First passage probability decays as 1/t

Abstract

A study of persistence dynamics is made for the first time in a quantum system by considering the dynamics of a quantum random walk. For a discrete walk on a line starting at $x=0$ at time $t=0$, the persistence probability $P(x,t)$ that a site at $x$ has not been visited till time $t$ has been calculated. $P(x,t)$ behaves as $(t/|x|-1)^{-\alpha}$ with $\alpha \sim 0.3$ while the global fraction ${\cal{P}}(t) = \sum_xP(x,t)/2t$ of sites remaining unvisited at time $t$ attains a constant value. $F(x,t)$, the probability that the site at $x$ is visited for the first time at $t$ behaves as $(t/|x|-1)^{-\beta}/|x|$ where $\beta = 1+ \alpha$ for $t/|x|>> 1$,and ${\cal{F}}(t) =\sum_xF(x,t)/2t \sim 1/t$. A few other properties related to the persistence and first passage times are studied and some fundamental differences between the classical and the quantum cases are observed.