A crossover between open quantum random walks to quantum walks

TL;DR

This paper introduces a parameterized walk that smoothly transitions between open quantum random walks and quantum walks, revealing intermediate behaviors including localization and ballistic spreading, supported by analytical and numerical analysis.

Contribution

It defines a new intermediate walk model controlled by a parameter M, bridging open quantum random walks and quantum walks, with analytical and numerical insights into its behavior.

Findings

Intermediate behavior observed for various M values

Quantum walk characteristics appear even at small M

Three modes described by Gaussian distributions

Abstract

We propose an intermediate walk continuously connecting an open quantum random walk and a quantum walk with parameters $M\in \mathbb{N}$ controlling a decoherence effect; if $M=1$, the walk coincides with an open quantum random walk, while $M=\infty$, the walk coincides with a quantum walk. We define a measure which recovers usual probability measures on $\mathbb{Z}$ for $M=\infty$ and $M=1$ and we observe intermediate behavior through numerical simulations for varied positive values $M$. In the case for $M=2$, we analytically show that a typical behavior of quantum walks appears even in a small gap of the parameter from the open quantum random walk. More precisely, we observe both the ballistically moving towards left and right sides and localization of this walker simultaneously. The analysis is based on Kato's perturbation theory for linear operator. We futher analyze this limit theorem in more detail and show that the above three modes are described by Gaussian distributions.