Different flavors of collision unravelling: analysis of finite-sample convergence in quantum molecular dynamics

The interaction between a quantum system and its surrounding environment is a defining feature of molecular and condensed-phase dynamics, giving rise to unavoidable decoherence and relaxation processes. To properly account for these environmental effects, the framework of Open Quantum Systems must be adopted. This requires shifting from the unitary evolution of a wavefunction to a density matrix description, capable of tracking the loss of quantum coherence. In the Markovian regime, the density matrix dynamics is described by the Lindblad-Gorini-Kossakowski-Sudarshan (LGKS) Quantum Master Equation. Since direct propagation of the density matrix scales quadratically with the Hilbert-space dimension, it becomes quickly computationally demanding. An alternative but equivalent representation of the dynamics can be generated through the so-called unraveling of the master equation. This approach reconstructs the density matrix as an ensemble average over stochastic wavefunction trajectories. A powerful framework to systematically construct these unravelings is provided by Collision Models (CMs). The environment is modeled as a collection of auxiliary quantum systems, that sequentially interact with the main system, and are subsequently measured, generating stochastic wavefunction trajectories that reproduce the LGKS dynamics on average. This thesis investigates the formal structure and finite-sample convergence of CM-generated unravelings. We show that a continuous family of collisional unravelings can be generated by rotating the auxiliary systems measurement basis, while leaving the unconditional master-equation dynamics unchanged. At finite time steps, these schemes produce distinct stochastic trajectories, whereas in the continuous-time limit all fixed non-quantum jump configurations converge to the same diffusive process. Our results demonstrate that no unraveling scheme is universally optimal: its finite-sample efficiency depends on the target observable and on the physical structure of the problem. In rare-event regimes, quantum-jump trajectories may exhibit a pronounced pre-asymptotic sampling barrier, whereas for other observables they can outperform diffusive trajectories. By decomposing the total observable variance into quantum and trajectory-to-trajectory statistical contributions, we provide a physical criterion for anticipating the sampling efficiency of different unravelings. This criterion connects convergence to the interplay among the observable, jump operators, system Hamiltonian, and initial state. To explore these concepts, we analyze three paradigmatic case studies: site-dephasing in an excitonic dimer, internal conversion in the thymine molecule, and exciton transport within the Fenna-Matthews-Olson (FMO) complex. Overall, these findings establish the choice of unraveling as an observable- and system-dependent optimization problem and provide a physically motivated strategy for reducing the sampling cost of stochastic open-system simulations.