Quantum cellular automata and quantum field theory in two spatial dimensions

TL;DR

This paper constructs a quantum cellular automaton in two spatial dimensions that converges to the 2D Dirac quantum field theory, overcoming previous obstacles and extending the one-dimensional QCA/QFT correspondence.

Contribution

It introduces a novel construction method for 2D QCA that yields the Dirac QFT, addressing prior limitations in higher dimensions.

Findings

Constructed a 2D QCA that reproduces the Dirac QFT in the long-wavelength limit.

Demonstrated how to evade previous no-go theorems for higher-dimensional QCA.

Outlined potential extensions to three spatial dimensions.

Abstract

Quantum walks on lattices can give rise to one-particle relativistic wave equations in the long-wavelength limit. In going to multiple particles, quantum cellular automata (QCA) are natural generalizations of quantum walks. In one spatial dimension, the quantum walk can be "promoted" to a QCA that, in the long-wavelength limit, gives rise to the Dirac quantum field theory (QFT) for noninteracting fermions. This QCA/QFT correspondence has both theoretical and practical applications, but there are obstacles to similar constructions in two or more spatial dimensions. Here we show that a method of construction employing distinguishable particles confined to the completely antisymmetric subspace yields a QCA in two spatial dimensions that gives rise to the 2D Dirac QFT. Generalizing to 3D will entail some additional complications, but no conceptual barriers. We examine how this construction evades the "no go" results in earlier work.