Persistence of unvisited sites in quantum walks on a line

TL;DR

This paper investigates the long-term behavior of unvisited sites in quantum walks on a line, revealing unique scaling laws that differ from classical walks and depend on initial conditions and coin parameters.

Contribution

It introduces a detailed analysis of persistence scaling in quantum walks, highlighting the influence of initial states and coin parameters on asymptotic behavior.

Findings

Persistence in two-state quantum walks follows an inverse power-law with a coin-dependent exponent.

In three-state quantum walks, persistence exhibits both inverse power-law and exponential decay components.

Initial coin states and coherence significantly affect the persistence regimes, enabling control over decay behaviors.

Abstract

We analyze the asymptotic scaling of persistence of unvisited sites for quantum walks on a line. In contrast to the classical random walk there is no connection between the behaviour of persistence and the scaling of variance. In particular, we find that for a two-state quantum walks persistence follows an inverse power-law where the exponent is determined solely by the coin parameter. Moreover, for a one-parameter family of three-state quantum walks containing the Grover walk the scaling of persistence is given by two contributions. The first is the inverse power-law. The second contribution to the asymptotic behaviour of persistence is an exponential decay coming from the trapping nature of the studied family of quantum walks. In contrast to the two-state walks both the exponent of the inverse power-law and the decay constant of the exponential decay depend also on the initial coin state and its coherence. Hence, one can achieve various regimes of persistence by altering the initial condition, ranging from purely exponential decay to purely inverse power-law behaviour.