  We describe two developments of tensor network influence functionals (in
particular, influence functional matrix product states (IF-MPS)) for quantum
impurity dynamics within the fermionic setting of the Anderson impurity model.
The first provides the correct extension of the IF-MPS to continuous time by
introducing a related mathematical object, the boundary influence functional
MPS. The second connects the dynamics described by a compressed IF-MPS to that
of a quantum embedding method with a time-dependent effective bath undergoing
nonunitary dynamics. Using these concepts, we implement higher-order time
propagators for the quench dynamics of the Anderson impurity model within the
boundary IF-MPS formalism. The calculations illustrate the ability of the
current formulation to efficiently remove the time step error in standard
discrete-time IF-MPS implementations as well as to interface with state vector
propagation techniques. They also show the advantages of IF-MPS dynamics, with
its associated highly compact effective bath dynamics, over state vector
propagation with a static bath discretization.
