Quantum walk on a cylinder

TL;DR

This paper investigates a 2D quantum walk on a cylindrical lattice, analyzing how the hidden dimension influences the walk's dynamics, dispersion relations, and entanglement, with implications for simulating high-energy physics theories.

Contribution

It introduces a detailed analysis of a 2D quantum walk on a cylinder, linking the quasi-momentum in the hidden dimension to Dirac equations and exploring entanglement enhancements.

Findings

Multiple components in the walk dynamics linked to quasi-momentum values.

In the continuous limit, components correspond to Dirac equations with different masses.

Entanglement entropy increases significantly compared to single-dimensional walks.

Abstract

We consider the 2D alternate quantum walk on a cylinder. We concentrate on the study of the motion along the open dimension, in the spirit of looking at the closed coordinate as a small or "hidden" extra dimension. If one starts from localized initial conditions on the lattice, the dynamics of the quantum walk that is obtained after tracing out the small dimension shows the contribution of several components, which can be understood from the study of the dispersion relations for this problem. In fact, these components originate from the contribution of the possible values of the quasi-momentum in the closed dimension. In the continuous space-time limit, the different components manifest as a set of Dirac equations, with each quasi-momentum providing the value of the corresponding mass. We briefly discuss the possible link of these ideas to the simulation of high energy physical theories that include extra dimensions. Finally, entanglement between the coin and spatial degrees of freedom is studied, showing that the entanglement entropy clearly overcomes the value reached with only one spatial dimension.