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

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, both of interest in the study of quantum propagation on the nonrectangular grids, as in graphene-like materials. The latter, in particular, we argue, opens the door for a generalization of the Dirac equation to arbitrary discrete surfaces.

Figure 1

Left: Honeycomb QW. The particle moves first along the $u_0$ direction (blue solid line), then $u_1$ (red dot-dashed line) and finally $u_2$ (green dot line). Right: Triangular QW. Starting at the edge $k=0$, the dynamics is equivalent to the honeycomb QW, in three time steps. The circle line represents the counterclockwise rotation operator.