Quantum walking in curved spacetime: $(3+1)$ dimensions, and beyond

TL;DR

This paper extends the framework of discrete-time Quantum Walks to arbitrary dimensions, demonstrating their ability to simulate the Dirac equation in curved spacetime, thus linking quantum computation models with general relativity.

Contribution

It generalizes previous results to higher dimensions and internal degrees of freedom, enabling the simulation of the Dirac equation in $(3+1)$ curved spacetime using local unitaries.

Findings

Quantum Walks can simulate the Dirac equation in curved spacetime.

The metric field is represented by local unitaries on a lattice.

Extension to arbitrary space dimensions and internal degrees of freedom.

Abstract

A discrete-time Quantum Walk (QW) is essentially an operator driving the evolution of a single particle on the lattice, through local unitaries. Some QWs admit a continuum limit, leading to familiar PDEs (e.g. the Dirac equation). Recently it was discovered that prior grouping and encoding allows for more general continuum limit equations (e.g. the Dirac equation in $(1+1)$ curved spacetime). In this paper, we extend these results to arbitrary space dimension and internal degree of freedom. We recover an entire class of PDEs encompassing the massive Dirac equation in $(3+1)$ curved spacetime. This means that the metric field can be represented by a field of local unitaries over a lattice.