Arrighi Pablo, Di Molfetta Giuseppe, Marquez-Martin Ivan, Perez Armando
Aix-Marseille Univ, Université de Toulon, CNRS, LIS, Marseille, France and IXXI, Lyon, France.
Aix-Marseille Univ, Université de Toulon, CNRS, LIS, Marseille, France and Departamento de Física Teórica and IFIC, Universidad de Valencia-CSIC, Dr. Moliner 50, 46100, Burjassot, Spain.
Sci Rep. 2019 Jul 29;9(1):10904. doi: 10.1038/s41598-019-47535-4.
A discrete-time Quantum Walk (QW) is an operator driving the evolution of a single particle on the lattice, through local unitaries. In a previous paper, we showed that QWs over the honeycomb and triangular lattices can be used to simulate the Dirac equation. We apply a spacetime coordinate transformation upon the lattice of this QW, and show that it is equivalent to introducing spacetime-dependent local unitaries -whilst keeping the lattice fixed. By exploiting this duality between changes in geometry, and changes in local unitaries, we show that the spacetime-dependent QW simulates the Dirac equation in (2 + 1)-dimensional curved spacetime. Interestingly, the duality crucially relies on the non linear-independence of the three preferred directions of the honeycomb and triangular lattices: The same construction would fail for the square lattice. At the practical level, this result opens the possibility to simulate field theories on curved manifolds, via the quantum walk on different kinds of lattices.
离散时间量子行走（QW）是一种通过局部酉变换驱动单个粒子在晶格上演化的算符。在之前的一篇论文中，我们表明蜂窝晶格和三角晶格上的量子行走可用于模拟狄拉克方程。我们对该量子行走的晶格进行时空坐标变换，并表明这等同于引入依赖于时空的局部酉变换，同时保持晶格固定。通过利用几何变化与局部酉变换变化之间的这种对偶性，我们证明依赖于时空的量子行走在（2 + 1）维弯曲时空中模拟狄拉克方程。有趣的是，这种对偶性关键依赖于蜂窝晶格和三角晶格三个优选方向的非线性独立性：对于正方形晶格，相同的构造将失败。在实际层面，这一结果开启了通过在不同类型晶格上的量子行走来模拟弯曲流形上的场论的可能性。
