  We introduce an efficient method to simulate dynamics of an interacting
quantum impurity coupled to non-interacting fermionic reservoirs. Viewing the
impurity as an open quantum system, we describe the reservoirs by their
Feynman-Vernon influence functionals (IF). The IF are represented as
matrix-product states in the temporal domain, which enables an efficient
computation of dynamics for arbitrary interactions. We apply our method to
study quantum quenches and transport in an Anderson impurity model, including
highly non-equilibrium setups, and find favorable performance compared to
state-of-the-art methods. The computational resources required for an accurate
computation of dynamics scale polynomially with evolution time, indicating that
a broad class of out-of-equilibrium quantum impurity problems are efficiently
solvable. This approach will provide new insights into dynamical properties of
mesoscopic devices and correlated materials.
