Simulation of non-equilibrium properties of quantum systems is one of the central challenges in quantum physics. In my talk, I will concentrate on the quantum impurity models (QIM) describing the small interacting degree of freedom coupled to a free and large bath. These models exhibit rich phenomena such as phase transitions and non-Markovian behavior but pose significant computational challenges. In this talk, I will review the influence matrix (IM) approach, which allows to approximate an infinite bath by a finite set of effective modes while preserving accuracy guarantees. I will demonstrate that this approach enables simulations with computational complexity scaling polynomially with both the evolution time and target error. Additionally, I will introduce a numerical routine for extracting optimal pseudomodes, enhancing practical applicability. The talk will draw on results from arXiv:2307.15592 and arXiv:2409.08816.