A low-rank hierarchical framework for the non-Markovian stochastic Schrödinger equation with convergence analysis

We propose and analyze a novel numerical framework for the non-Markovian stochastic Schr\"odinger equation (NMSSE) based on a low-rank approximation of the bath correlation functions. By decomposing the memory kernel into a finite-dimensional representation, we derive a truncated system of hierarchical equations that effectively balances computational tractability with physical fidelity. A rigorous convergence analysis is established for the hierarchical framework under mild assumptions. We demonstrate that our formulation serves as a mathematical generalization of the Hierarchy of Pure States (HOPS), encompassing it as a special case while offering a more flexible representation of non-Markovian effects. Numerical experiments across several benchmark models are presented to illustrate the validity and efficacy of the proposed method.