Geodesic Quantum Walks
This work provides a tool for simulating quantum transport on curved discrete structures and optimizing in curved spaces, though it is incremental as it extends existing duality principles.
The authors introduced a new family of quantum walks that can propagate on arbitrary triangulations and converge to the massless Dirac equation on curved manifolds in the continuous limit, enabling modeling of quantum transport on discrete curved structures like fullerene molecules.
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.