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

Pablo Arrighi (1, 2) , Di Molfetta Giuseppe (3) , Iván Márquez-Martín , Armando Pérez

1 LMF - Laboratoire Méthodes Formelles
2 QuaCS - Quantum Computation Structures
3 LIS - Laboratoire d'Informatique et des Systèmes (LIS) (Marseille, Toulon)

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 non-rectangular 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.

Domains: Quantum Physics [quant-ph]

Pablo Arrighi, Di Molfetta Giuseppe, Iván Márquez-Martín, Armando Pérez. Dirac equation as a quantum walk over the honeycomb and triangular lattices. 2023. ⟨hal-03594743⟩