Abstract
The simplicity provided by cellular automaton systems has been of great interest to the study of complex systems, particularly due to its potential to realize interesting behavior and the possibility of performing computations. They can be a valuable tool when treating such systems. One of the most interesting types of complex systems is that of quantum systems, whose complexity is unparalleled in recent times. As such, they have been fiercely studied for their possible technological breakthroughs. Part of the key to such a breakthrough might be the role of the measurement, which holds a singular place in the interaction with a quantum system. Conversely, quantum systems might necessitate a large Hilbert to be accurately described, and efficient ways of descriptions hold themselves great computational power. In this thesis, we examine these questions through the lenses of dimensional reduction, by sequentially measuring one-dimensional systems in order to derive cellular automaton-like states from it. We show that a principal component analysis performed on the correlation matrix of the system indicates properties of the underlying Hamiltonian, and we make the argument that even a coarse measure derived from such a system might hold intricate amounts of information. We then leverage this premise by presenting a possible usage of a quantum-inspired setup in the learning of probabilistic cellular automaton states. We show that our setup is able to correctly apprehend the dynamics of the probabilistic objects consisting of distributions of two and three deterministic rules of evolution. Lastly, we present results of the convergence of the learning accuracy regarding the size of training data, and also the bond dimension of the tensor networks used.