# 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.

## Key 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.