Thesis advisor: Xiao Chen / The development of highly controllable quantum coherent simulators such as superconducting qubits and Rydberg atom arrays has stimulated the study of non-equilibrium quantum dynamics, opening the door to exciting topics including dynamical phase transitions, thermalization, transport, and quantum error correction. This thesis addresses various questions from non-equilbrium quantum dynamics, with a concentration on measurement-induced phase transitions (MIPT), adaptive dynamics with feedback mechanism, and Hilbert space fragmentation. In the first part, we study the hybrid quantum automaton (QA) circuits with different symmetries subject to local composite measurements. For $\mathbb{Z}_2$-symmetric hybrid QA circuits, there exists an entanglement phase transition from a volume-law phase to a critical phase by varying the measurement rate. The special feature of QA circuits enables us to interpret the entanglement dynamics in terms of a stochastic particle model. With the help of this stochastic model, we further investigate the entanglement fluctuations and quantum error correcting property of the volume-law phase in QA circuits with no symmetry, and study the entanglement dynamics in QA circuits with U(1) symmetry. Despite being a hallmark of non-unitary quantum dynamics, MIPT is absent in the density matrix averaged over measurement outcomes. In the second part, we introduce an adaptive quantum circuit subject to measurements with feedback. The feedback is applied according to the measurement outcome and steers the system towards a unique state above certain measurement threshold. We show that there exists an absorbing phase transition in both quantum trajectories and quantum channels. In the end, we turn to the phenomenon of Hilbert space fragmentation (HSF), whereby dynamical constraints fragment Hilbert space into many disconnected sectors, providing a simple mechanism by which thermalization can be arrested. However, little is known about how thermalization occurs in situations where the constraints are not exact. To study this, we consider a situation in which a fragmented 1d chain with pair-flip constraints is coupled to a thermal bath at its boundary. For product states quenched under Hamiltonian dynamics, we numerically observe an exponentially long thermalization time, manifested in both entanglement dynamics and the relaxation of local observables. To understand this, we study an analogous model of random unitary circuit dynamics, where we rigorously prove that the thermalization time scales exponentially with system size. Slow thermalization in this model is shown to be a consequence of strong bottlenecks in configuration space, demonstrating a new way of producing anomalously slow thermalization dynamics. / Thesis (PhD) — Boston College, 2024. / Submitted to: Boston College. Graduate School of Arts and Sciences. / Discipline: Physics.