Invariance in Quantum Walks

TL;DR

This paper explores invariance properties in quantum walks on a line, analyzing how symmetries and gauge freedoms influence their behavior, even with variable coin operators, providing explicit solutions and insights into their fundamental origins.

Contribution

It introduces the concept of invariance in quantum walks, linking it to gauge freedom, and analyzes how non-homogeneous coin operators can preserve probabilistic properties.

Findings

Invariance can be maintained despite site and time variability of the coin.

Explicit solutions for the most general homogeneous unitary operator are provided.

Invariance is connected to gauge freedom in electromagnetism.

Abstract

In this Chapter, we present some interesting properties of quantum walks on the line. We concentrate our attention in the emergence of invariance and provide some insights into the ultimate origin of the observed behavior. In the first part of the Chapter, we review the building blocks of the quantum-mechanical version of the standard random walk in one dimension. The most distinctive difference between random and quantum walks is the replacement of the random coin in the former by the action of a unitary operator upon some internal property of the later. We provide explicit expressions for the solution to the problem when the most general form for the homogeneous unitary operator is considered, and we analyze several key features of the system as the presence of symmetries or stationary limits. After that, we analyze the consequences of letting the properties of the coin operator change from site to site, and from time step to time step. In spite of this lack of homogeneity, the probabilistic properties of the motion of the walker can remain unaltered if the coin variability is chosen adequately. Finally, we show how this invariance can be connected to the gauge freedom of electromagnetism.