Exact Model Reduction For Quantum Systems

This thesis focuses on the problem of finding reduced quantum dynamical models that exactly reproduce the dynamics of certain observable quantities when starting from given initial conditions. The main results of the thesis revolve around an algebraic framework that guarantees the reduced model obtained from the procedure to satisfy the preservation of positivity and total probability. The problem of finding smaller mathematical descriptions of dynamical quantum systems is instrumental to a variety of tasks, ranging from efficient simulations on classical and quantum devices to simplified implementation of quantum filters, as well as to the analysis of the minimal resources needed to realize a certain quantum process. The tools used in this work are borrowed from many fields, including operator algebras and non-commutative probability theory, system theory and quantum mechanics. New theoretical results on finite-dimensional distorted algebras allow us to devise algorithms for systematic model reduction, for both continuous and discrete dynamics. The proposed methods are tested on prototypical examples which include the quantum walk realizing Grover's algorithm and open quantum spin systems with both purely Hamiltonian or dissipative dynamics.