Quantum walks: background geometry and gauge invariance

There are many problems that cannot be solved using current classical comput- ers. One manner to approach a solution of these systems is by using quantum computers. However, building a quantum computer is really challenging from the experimental side. Quantum simulators have been capable to solve some of these problems, as they are realizable experimentally.Discrete Time Quantum Walks (DTQWs) have been proved to be an useful tool to quantum simulate physical systems. In the continuous limit, a family of differential equations can be achieved, in particular, the Dirac equation can be recovered. In this thesis we study QWs as possible schemes for quantum simula- tion. Specifically, we can summarize our results in: i) We introduce a QW-based model in which a brane theory can be simulated in the continuum, opening the possibility to study more general theories with extra dimensions; ii) Elec- tromagnetic gauge invariance in QWs is discussed, presenting some similarities and differences to previous models. This QW model also makes a connection to gauge invariance in lattice gauge theories (LGT); iii) We introduce QWs over non- rectangular lattices, such a triangular or honeycomb structures, for the purpose of simulating the Dirac equation in the continuum. We also extent these models, by introducing local coin operators, that allow us to reproduce the dynamics of quantum particles under a curved space time.