Discrete-time quantum walks: continuous limit and symmetries

TL;DR

This paper investigates the conditions under which one-dimensional discrete-time quantum walks with variable coefficients have a continuous limit, revealing that certain families lead to Dirac-like or Klein-Gordon equations and exploring their symmetries.

Contribution

It identifies which families of quantum walks admit a continuous limit and derives the corresponding Dirac-like and Klein-Gordon equations, including their variational principles and symmetries.

Findings

Certain families of quantum walks have a well-defined continuous limit.

The continuous limit is described by Dirac-like or Klein-Gordon equations.

Variational principles and invariance properties are established for these equations.

Abstract

The continuous limit of one dimensional discrete-time quantum walks with time- and space-dependent coefficients is investigated. A given quantum walk does not generally admit a continuous limit but some families (1-jets) of quantum walks do. All families (1-jets) admitting a continuous limit are identified. The continuous limit is described by a Dirac-like equation or, alternately, a couple of Klein-Gordon equations. Variational principles leading to these equations are also discussed, together with local invariance properties.