From curved spacetime to spacetime-dependent local unitaries over the honeycomb and triangular Quantum Walks

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

1 LIS - Laboratoire d'Informatique et des Systèmes (LIS) (Marseille, Toulon)
2 IXXI - Institut Rhône-Alpin des systèmes complexes

A discrete-time Quantum Walk (QW) is an operator driving the evolution of a single particle on the lattice, through local unitaries. In a previous paper, we showed that QWs over the honeycomb and triangular lattices can be used to simulate the Dirac equation. We apply a spacetime coordinate transformation upon the lattice of this QW, and show that it is equivalent to introducing spacetime-dependent local unitaries —whilst keeping the lattice fixed. By exploiting this duality between changes in geometry, and changes in local unitaries, we show that the spacetime-dependent QW simulates the Dirac equation in (2 + 1)–dimensional curved spacetime. Interestingly, the duality crucially relies on the non linear-independence of the three preferred directions of the honeycomb and triangular lattices: The same construction would fail for the square lattice. At the practical level, this result opens the possibility to simulate field theories on curved manifolds, via the quantum walk on different kinds of lattices.

Mots clés: lattice; unitarity; space-time dependence; triangulation; dimension: 3; Dirac equation; duality; continuum limit; geometry

Domaines: Physique Générale [physics.gen-ph]; Physique des Hautes Energies - Réseau [hep-lat]

Pablo Arrighi, Giuseppe Di Molfetta, Iván Márquez-Martín, Armando Pérez. From curved spacetime to spacetime-dependent local unitaries over the honeycomb and triangular Quantum Walks. Sci.Rep., 2019, 9 (1), pp.10904. ⟨10.1038/s41598-019-47535-4⟩. ⟨hal-01965370⟩