A discrete-time quantum walk (QW) is essentially a unitary operator driving the evolution of a single particle on the lattice. Some QWs have familiar physics PDEs as their continuum limit. Some slight generalization of them (allowing for prior encoding and larger neighbourhoods) even have the curved spacetime Dirac equation, as their continuum limit. In the $(1+1)-$dimensional massless case, this equation decouples as scalar transport equations with tunable speeds. We characterise and construct all those QWs that lead to scalar transport with tunable speeds. The local coin operator dictates that speed; we provide concrete techniques to tune the speed of propagation, by making use only of a finite number of coin operators---differently from previous models, in which the speed of propagation depends upon a continuous parameter of the quantum coin. The interest of such a discretization is twofold : to allow for easier experimental implementations on the one hand, and to evaluate ways of quantizing the metric field, on the other.
From curved spacetime to spacetime-dependent local unitaries over the honeycomb and triangular Quantum Walks
离散时间量子行走（英语：Discrete time Quantum Walk，QW）是一种通过局域幺正驱动晶格上单个粒子演化的算子。一些量子阱承认，作为它们的连续极限，一个著名的物理方程。在arXiv：1803.01015中，量子阱位于蜂窝和三角晶格上，并模拟狄拉克方程。我们在这个量子阱的晶格上应用时空坐标变换，并证明它等价于引入时空依赖的局域幺正-同时保持晶格固定。通过利用这种对偶性的变化之间的几何，并在本地幺正的变化，我们表明，时空依赖量子阱模拟狄拉克方程在$(2+1)$维弯曲时空。有趣的是，这种对偶性主要依赖于蜂窝晶格和三角晶格的三个优选方向的非线性独立性：同样的构造对于正方形晶格会失败。在实践层面上，这一结果打开了通过不同种类的晶格上的量子行走来模拟弯曲流形上的场论的可能性。
