Geodesic quantum walks

We propose a family of discrete space-time quantum walks capable of propagating on any arbitrary triangulations. Moreover, we also extend and generalize the duality principle introduced by Arrighi et al. [Sci. Rep. 9, 10904 (2019)], 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 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 modeling and 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.