We propose a new family of discrete-spacetime quantum walks capable to propagate on any arbitrary triangulations. Moreover we also extend and generalise the duality principle introduced in~\cite{arrighi2019curved}, 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.

本文提出了一种新的离散时空量子行走方法，能够在任意三角形上进行传播，并且将作者中的对偶原理扩展和推广至给定三角剖分的连续局部变形和指导量子行走器的局部酉不均匀性之间。我们证明，在形式连续的时空极限下，这种新的量子行走方法收敛于曲面流形上的（1+2）D无质量Dirac方程，这一结果对于离散曲面结构上的量子传输建模/模拟以及在曲面优化方法的背景下解决快速高效的优化问题具有重要意义。

测地量子漫步