# Directional quantum random walk induced by coherence

## TL;DR

This paper introduces a new quantum random walk model based on ensemble interpretation, showing how coherence influences directional movement and comparing it to classical and traditional quantum walks.

## Contribution

It proposes a novel ensemble-based quantum random walk model and analyzes the role of coherence in directional movement and distribution patterns.

## Key findings

- Coherence induces directional movement in the quantum random walk.
- The position distribution becomes asymmetric binomial with specific initial states.
- Decohered coins produce classical random walk distributions.

## Abstract

Quantum walk (QW), which is considered as the quantum counterpart of the classical random walk (CRW), is actually the quantum extension of CRW from the single-coin interpretation. The sequential unitary evolution engenders correlation between different steps in QW and leads to a ballistic position distribution. In this paper, we propose an alternative quantum extension of CRW from the ensemble interpretation, named quantum random walk (QRW), where the walker has many unrelated coins, modeled as two-level systems, initially prepared in the same state. We calculate the walker's position distribution in QRW for different initial coin states with the coin operator chosen as Hadamard matrix. In one-dimensional case, the walker's position is the asymmetric binomial distribution. We further demonstrate that in QRW, coherence leads the walker to perform directional movement. For an initially decoherenced coin state, the walker's position distribution is exactly the same as that of CRW. Moreover, we study QRW in 2D lattice, where the coherence plays a more diversified role in the walker's position distribution.