Geodesic Quantum Walks

TL;DR

This paper introduces a new class of discrete-spacetime quantum walks on arbitrary triangulations, which converge to the (1+2)D massless Dirac equation on curved manifolds, with applications in quantum transport and optimization.

Contribution

It develops a generalized family of quantum walks capable of propagating on any triangulation and links local deformations to inhomogeneous unitaries, extending the duality principle.

Findings

Quantum walks on triangulations converge to Dirac equation on curved spaces.

The framework models quantum transport on discrete curved structures.

Potential applications in optimization and quantum simulation.

Abstract

We propose a new family of discrete-spacetime quantum walks capable to propagate on any arbitrary triangulations. Moreover we also extend and generalize the duality principle introduced by one of the authors, linking continuous local deformations of a given triangulation and the inhomogeneity of the local unitaries that guide the quantum walker. We proved that in the formal continuous limit, in both space and time, this new family of quantum walks converges to the (1+2)D massless Dirac equation on curved manifolds. We believe that this result has relevance in both modelling/simulating quantum transport on discrete curved structures, such as fullerene molecules or dynamical causal triangulation, and in addressing fast and efficient optimization problems in the context of the curved space optimization methods.