The Limiting Distribution of Decoherent Quantum Random Walks

TL;DR

This paper analyzes how decoherence affects one-dimensional quantum random walks, revealing that their limiting distributions become Gaussian and providing explicit formulas for moments and distributions.

Contribution

It introduces a new analytical method to derive exact moments and Gaussian limiting distributions for decoherent quantum walks, linking decoherence levels to walk behavior.

Findings

Limiting distributions are Gaussian under decoherence.

Exact expressions for moments depend on time and noise.

Decoherence causes quantum walks to exhibit classical features.

Abstract

The behaviors of one-dimensional quantum random walks are strikingly different from those of classical ones. However, when decoherence is involved, the limiting distributions take on many classical features over time. In this paper, we study the decoherence on both position and ``coin'' spaces of the particle. We propose a new analytical approach to investigate these phenomena and obtain the generating functions which encode all the features of these walks. Specifically, from these generating functions, we find exact analytic expressions of several moments for the time and noise dependence of position. Moreover, the limiting position distributions of decoherent quantum random walks are shown to be Gaussian in an analytical manner. These results explicitly describe the relationship between the system and the level of decoherence.