Skeleton structure inherent in discrete-time quantum walks

TL;DR

This paper reveals a fundamental skeleton structure underlying discrete-time quantum walks on a one-dimensional lattice, which simplifies understanding their transition probabilities and relates to quantum-walk-replicating random walks.

Contribution

It identifies a universal skeleton structure in quantum walks that is independent of initial states and partially of the coin matrix, providing a simplified formula for transition probabilities.

Findings

Skeleton structure is present in quantum walks regardless of initial state.

Constructed random walk with transition probabilities based on the skeleton mimics quantum walk properties.

Skeleton structure offers a simplified understanding of quantum walk transition dynamics.

Abstract

In this paper, we claim that a common underlying structure--a skeleton structure--is present behind discrete-time quantum walks (QWs) on a one-dimensional lattice with a homogeneous coin matrix. This skeleton structure is independent of the initial state, and partially, even of the coin matrix. This structure is best interpreted in the context of quantum-walk-replicating random walks (QWRWs), i.e., random walks that replicate the probability distribution of quantum walks, where this newly found structure acts as a simplified formula for the transition probability. Additionally, we construct a random walk whose transition probabilities are defined by the skeleton structure and demonstrate that the resultant properties of the walkers are similar to both the original QWs and QWRWs.