This thesis introduces efficient approximation techniques for fermionic Gaussian influence functionals (IFs) to simulate the dynamics of quantum impurity systems coupled to non-interacting baths. The IF encapsulates the memory effects induced by the bath on the impurity, obtained by integrating out the bath degrees of freedom within the path integral formalism.

The central contribution of this work is the development of auxiliary baths that faithfully reproduce these memory effects while significantly reducing bath complexity with controlled accuracy.

Two complementary approaches are introduced: (i) A framework is developed to represent the IF as a Matrix Product State (MPS) in the temporal domain, enabling efficient computation of impurity dynamics through tensor contractions. (ii) Alternatively, a time-local description is achieved by approximating the bath hybridization function---the key determinant of the IF---as a sum of decaying exponentials. This pseudomode approach offers an intuitive mapping to an effective Lindblad equation, where the impurity interacts with a finite set of auxiliary modes that faithfully reproduce the original bath’s influence.

These approaches bridge path integral techniques with master equation formalisms and modern tensor network methods, providing both conceptual clarity and computational efficiency.

The thesis presents results on error bounds and computational complexity, alongside applications to nonequilibrium impurity problems, including quantum quenches and relaxation dynamics, demonstrating the versatility and robustness of the proposed methods.