One-dimensional Cellular Automata in Quantum and Fermionic Theories

The thesis deals with quantum cellular automata (QCAs) and Fermionic quantum cellular automata (FQCAs) on one-dimensional lattices. With the term cellular automaton, we refer to a class of algorithms that can process information distributed on a regular grid in a local fashion. Its quantum counterpart—where at each site of the grid we can find a quantum system—represents a model for massive parallel quantum computation on finitely generated grids. The model is particularly well-suited for describing and simulating a vast class of physical phenomena. The work presented in the thesis is threefold. We first introduce a new definition of QCA in terms of super maps, namely functions from quantum operations to quantum operations, that preserves locality and composition of transformations. Thereby, we define the so-called T-operator, i.e. a local operator that incorporates all the necessary information for univocally defining a QCA. The T-operator plays here the role of the Choi operator of the automaton. Secondly, we classify all nearest-neighbor FQCAs over the one-dimensional lattice where each site contains one local Fermionic mode. We observe that the Fermionic automata are divided into two classes. In the first one, we find some FQCAs that are equivalent to a subset of quantum cellular automata. On the other hand, the second class of FQCAs has been found to have no quantum counterparts. Finally, we report the experimental realization of a photonic platform to simulate the evolution of a one-dimensional quantum walk, i.e. a quantum cellular automaton whose action is linear in the field operators. Specifically, we observe the Zitterbewegung of a particle satisfying the Dirac dispersion relation. The theoretical background, numerical simulation, and optimization of the parameter space are discussed with special attention.