Stochastic approaches to open quantum systems and the modeling of non-standard environments

Quantum systems are inevitably influenced by the environments in which they are embedded. From molecular and nanoscale systems interacting with complex surroundings, to engineered platforms for quantum technologies subject to control imperfections and external noise, the interplay between coherent dynamics and environmental effects is unavoidable. Accurately modeling such interactions is therefore essential not only for understanding fundamental processes such as charge and energy transfer, but also for the reliable operation of quantum devices, where decoherence directly impacts performance. When environmental fluctuations deviate from idealized white-noise assumptions, capturing their effects becomes both conceptually and computationally challenging. In this thesis, we contribute to the investigation of the dynamics of open quantum systems interacting with structured and time-correlated environments, with a particular emphasis on stochastic formulations and their connection to reduced dynamics. After reviewing the main approaches to quantum master equations and their microscopic derivation, we develop and compare stochastic unraveling strategies, with a particular focus on stochastic Hamiltonian methods and numerical simulation strategies. We extend these approaches to correlated noise and process models, leading to non-Markovian master equations that require trajectory-based numerical treatments. Within this setting, we analyze the dynamical consequences of time-correlated environments, highlighting regimes exhibiting multi-timescale relaxation, long-lived coherences, and nontrivial stationary behavior. To obtain a practical closed description and physical intuition, we introduce a Redfield-inspired perturbative closure for these correlation terms, providing an effective master equation for the mean dynamics and leading to the definition of an alternative version of the Redfield tensor. To further characterize such dynamics, we introduce a trajectory-based densification measure and associated descriptors, which extract information from stochastic ensembles that is not accessible at the level of the reduced density matrix alone. Finally, we connect stochastic Hamiltonian formulations to the simulation of open quantum dynamics on digital quantum computers, analyzing a quantum trajectory-based algorithm and establishing its convergence properties and clarifying the role of quantum measurement statistics in its performance.