This thesis investigates quantum many-body systems in and out of equilibrium, with a dual focus on open-system theory and advanced computational methods. Taken together, the results articulate a unifying perspective: in both classical/analog simulations and digital algorithms for many-body quantum systems, the relevant physics often concentrates in small, dynamically selected pockets of the Liouville or Hilbert space. By exposing this structure - whether as noise-protected logical manifolds in bosonic modes, low-rank sectors of the density operator during open evolution, or expressive submanifolds learned by neural networks - and by designing algorithms that operate within these reduced subspaces, the thesis advances hardware-aware encoding strategies, scalable open-system simulation, and dependable neural-network-based variational Monte Carlo methods.

The manuscript presents three original contributions: (i) a Liouville-space framework that leverages symmetries and spectral diagnostics to engineer dissipation as a resource for bosonic quantum information; (ii) a low-rank, time-dependent variational method for simulating Lindblad dynamics with adaptive error control; and (iii) methodological advances for Neural Quantum States that stabilize Monte-Carloâ driven optimization and enable reliable ground-state and dynamical calculations. These three directions structure the three main chapters of the thesis. Each chapter includes a focused, self-contained review of the necessary background, not as a textbook survey but as a coherent theoretical scaffold framing the proposed advances.