  Describing a quantum impurity coupled to one or more non-interacting
fermionic reservoirs is a paradigmatic problem in quantum many-body physics.
While historically the focus has been on the equilibrium properties of the
impurity-reservoir system, recent experiments with mesoscopic and cold-atomic
systems enabled studies of highly non-equilibrium impurity models, which
require novel theoretical techniques. We propose an approach to analyze
impurity dynamics based on the matrix-product state (MPS) representation of the
Feynman-Vernon influence functional (IF). The efficiency of such a MPS
representation rests on the moderate value of the temporal entanglement (TE)
entropy of the IF, viewed as a fictitious "wave function" in the time domain.
We obtain explicit expressions of this wave function for a family of
one-dimensional reservoirs, and analyze the scaling of TE with the evolution
time for different reservoir's initial states. While for initial states with
short-range correlations we find temporal area-law scaling, Fermi-sea-type
initial states yield logarithmic scaling with time, closely related to the
real-space entanglement scaling in critical 1d systems. Furthermore, we
describe an efficient algorithm for converting the explicit form of the
reservoirs' IF to MPS form. Once the IF is encoded by a MPS, arbitrary temporal
correlation functions of the interacting impurity can be efficiently computed,
irrespective of its internal structure. The approach introduced here can be
applied to a number of experimental setups, including highly non-equilibrium
transport via quantum dots and real-time formation of impurity-reservoir
correlations.
