Quantum Fisher Information for simulation of many-body systems and new Quantum Cellular Automata protocols for computation

This work is divided into two parts. The first focuses on Quantum Fisher Information (QFI), exploring its properties and role in quantum geometry, phase estimation, and multipartite entanglement. Specifically, we study ground state QFI in one-dimensional spin-1 models, using it to witness multipartite entanglement. The models examined include the Bilinear-Biquadratic model and the XXZ spin-1 chain, all with nearest-neighbor interactions and open boundary conditions. We show that the scaling of QFI with strictly non-local observables characterizes phase diagrams, particularly in topological phases, where it exhibits maximal scaling. To conclude this part, we demonstrate how QFI can serve as a hybrid quantum-classical optimizer in variational algorithms for NISQ devices. Specifically, we evaluate the Quantum Approximate Optimization Algorithm (QAOA) using the Quantum Natural Gradient as an optimizer, showing that even with quantum noise, the QFI-based Natural Gradient improves convergence, reducing iterations compared to its classical counterpart. The second part introduces Quantum Cellular Automata (QCA) as an alternative paradigm for quantum computation, highlighting their versatility and applications. Finally, we discuss how non-unitary QCA can solve the density classification task, mapping global density information to local density. Two approaches are considered: one preserving number density and one performing majority voting. For the number-preserving case, we propose two QCAs that reach a fixed-point solution with a time complexity scaling almost quadratically with system size, both implementable via continuous-time Lindblad dynamics. Additionally, a third QCA, a hybrid rule combining discrete-time and continuous-time three-body interactions, solves the majority voting problem in linear time with system size.