This manuscript gathers and subsumes a long series of works on using QW to simulate transport phenomena. Quantum Walks (QWs) consist of single and isolated quantum systems, evolving in discrete or continuous time steps according to a causal, shift-invariant unitary evolution in discrete space. We start reminding some necessary fundamentals of linear algebra, including the definitions of Hilbert space, tensor state, the definition of linear operator and then we briefly present the principles of quantum mechanics on which this thesis is grounded. After having reviewed the literature of QWs and the main historical approaches to their study, we then move on to consider a new property of QWs, the plasticity. Plastic QWs are those ones admitting both continuous time-discrete space and continuous spacetime time limit. We show that such QWs can be used to quantum simulate a large class of physical phenomena described by transport equations. We investigate this new family of QWs in one and two spatial dimensions, showing that in two dimensions, the PDEs we can simulate are more general and include dispersive terms. We show that the above results do not need to rely on the grid and we prove that such QW-based quantum simulators can be defined on 2-complex simplicia, i.e. triangular lattices. Finally, we extend the above result to any arbitrary triangulation, proving that such QWs coincide in the continuous limit to a transport equation on a general curved surface, including the curved Dirac equation in 2+1 spacetime dimensions.

翻译：此手稿收集并包含一系列关于使用 QW 模拟运输现象的长篇著作。 量子漫步( QWs) 由单项和孤立的量子系统组成, 根据离散空间的因果、 变化和变异的单一演进, 以离散空间的分解或连续时间步骤演变。 我们开始提醒线形代数的一些必要基本原理, 包括Hilbert 空间的定义、 Exor 状态、 线性操作者的定义, 然后我们简要地介绍此理论所基于的量子力学原理。 在审查了 QW 的文献及其研究的主要历史方法之后, 我们接着开始考虑 QWs 和 的三角曲线的新属性。 然后我们继续考虑 QWs 和 塑料的新的直径等值。 塑料 QW 是那些同时接受连续时间分解空间空间空间和连续时间时间限制的元素。 我们用这种QWs 来模拟运输的大型物理现象。 我们用一个和两个空基的新的量力组, 显示在两个维度上, 我们模拟的PDE是更一般的, 并且包含 QQ- 直径 直径的直径变方程式, 。 我们用直判判的直径对等的计算结果, 。 我们用在2 。 我们用直到直到直方程式 。