The Dirac equation as a quantum walk over the honeycomb and triangular lattices

TL;DR

This paper demonstrates that the Dirac equation in (2+1) dimensions can be simulated using discrete-time quantum walks on honeycomb and triangular lattices, expanding the potential for modeling physics on various discrete surfaces.

Contribution

It introduces a method to simulate the Dirac equation on non-square lattices, specifically honeycomb and triangular, using local unitaries in quantum walks.

Findings

Dirac equation simulated on honeycomb lattice relevant to graphene.

Triangular lattice simulation suggests generalization to arbitrary discrete surfaces.

Quantum walks can replicate continuum physics on non-rectangular grids.

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 well-known physics partial differential equations, such as the Dirac equation. We show that these simulation results need not rely on the grid: the Dirac equation in $(2+1)$--dimensions can also be simulated, through local unitaries, on the honeycomb or the triangular lattice. The former is of interest in the study of graphene-like materials. The latter, we argue, opens the door for a generalization of the Dirac equation to arbitrary discrete surfaces.