  We describe an iterative formalism to compute influence functionals that
describe the general quantum dynamics of a subsystem beyond the assumption of
linear coupling to a quadratic bath. We use a space-time tensor network
representation of the influence functional and investigate its approximability
in terms of the bond dimensions and time-like entanglement in the tensor
network description. We study two numerical models, the spin-boson model and a
model of interacting hard-core bosons in a 1D harmonic trap. We find that the
influence functional and the intermediates involved in its construction can be
efficiently approximated by low bond dimension tensor networks in certain
dynamical regimes, which allows the quantum dynamics to be accurately computed
for longer times than with direct time evolution methods. However, as one
iteratively integrates out the bath, the correlations in the influence
functional can first increase before decreasing, indicating that the final
compressibility of the influence functional is achieved via non-trivial
cancellation.
