Quantum walking in curved spacetime

TL;DR

This paper demonstrates how a class of discrete-time quantum walks can approximate the Dirac equation in curved spacetime, providing a novel unitary model for particles in curved backgrounds using new continuum limit techniques.

Contribution

It introduces a method to derive PDEs, including the Dirac equation in curved spacetime, from quantum walks, and presents a specific QW model as a toy for particles in curved spacetime.

Findings

Quantum walks can approximate PDEs in curved spacetime.

A specific QW models a test particle in curved spacetime.

New techniques for continuum limits of quantum cellular automata.

Abstract

A discrete-time Quantum Walk (QW) is essentially a unitary operator driving the evolution of a single particle on the lattice. Some QWs admit a continuum limit, leading to familiar PDEs (e.g. the Dirac equation). In this paper, we study the continuum limit of a wide class of QWs, and show that it leads to an entire class of PDEs, encompassing the Hamiltonian form of the massive Dirac equation in $(1+1)$ curved spacetime. Therefore a certain QW, which we make explicit, provides us with a unitary discrete toy model of a test particle in curved spacetime, in spite of the fixed background lattice. Mathematically we have introduced two novel ingredients for taking the continuum limit of a QW, but which apply to any quantum cellular automata: encoding and grouping.