Quantum walking in curved spacetime: discrete metric

TL;DR

This paper characterizes and constructs discrete-time quantum walks that simulate scalar transport with tunable speeds in curved spacetime, facilitating experimental implementation and insights into quantizing the metric field.

Contribution

It provides a method to tune the speed of quantum walks via local coin operators, enabling discrete models of curved spacetime with practical and theoretical applications.

Findings

Constructed QWs with tunable scalar transport speeds

Provided techniques to control propagation speed using finite coin operators

Facilitated potential experimental implementations and metric quantization studies

Abstract

A discrete-time quantum walk (QW) is essentially a unitary operator driving the evolution of a single particle on the lattice. Some QWs have familiar physics PDEs as their continuum limit. Some slight generalization of them (allowing for prior encoding and larger neighbourhoods) even have the curved spacetime Dirac equation, as their continuum limit. In the $(1+1)-$dimensional massless case, this equation decouples as scalar transport equations with tunable speeds. We characterise and construct all those QWs that lead to scalar transport with tunable speeds. The local coin operator dictates that speed; we provide concrete techniques to tune the speed of propagation, by making use only of a finite number of coin operators---differently from previous models, in which the speed of propagation depends upon a continuous parameter of the quantum coin. The interest of such a discretization is twofold : to allow for easier experimental implementations on the one hand, and to evaluate ways of quantizing the metric field, on the other.