Controlling quantum random walk with a step-dependent coin

TL;DR

This paper demonstrates that quantum random walks can be effectively controlled using a step-dependent coin, leading to diverse probability distributions and entropy behaviors, which enhances the understanding and manipulation of quantum walk dynamics.

Contribution

It introduces a novel method of controlling quantum walks through a step-dependent coin, revealing complex behaviors and entropy dynamics not seen with traditional step-independent coins.

Findings

Probability distributions include localization, Gaussian, and asymmetric types.

Entropy in position space can decrease with step-dependent coins.

Entropy in coin space shows angle-dependent oscillations.

Abstract

We report on the possibility of controlling quantum random walks with a step-dependent coin. The coin is characterized by a (single) rotation angle. Considering different rotation angles, one can find diverse probability distributions for this walk including: complete localization, Gaussian and asymmetric likes. In addition, we explore the entropy of walk in two contexts; for probability density distributions over position space and walker's internal degrees of freedom space (coin space). We show that entropy of position space can decrease for a step-dependent coin with the step-number, quite in contrast to a walk with step-independent coin. For entropy of coin space, a damped oscillation is found for walk with step-independent coin while for a step-dependent coin case, the behavior of entropy depends on rotation angle. In general, we demonstrate that quantum walks with simple initiatives may exhibit a quite complex and varying behavior if step-dependent coins are applied. This provides the possibility of controlling quantum random walk with a step-dependent coin.