  Tensors with finite correlation afford very compact tensor network
representations. A novel tensor network-based decomposition of real-time path
integral simulations involving Feynman-Vernon influence functional is
introduced. In this tensor network path integral (TNPI) technique, the finite
temporarily non-local interactions introduced by the influence functional can
be captured very efficiently using matrix product state representation for the
path amplitude (PA) tensor. We illustrate this particular TNPI method through
various realistic examples, including a charge transfer reaction and an exciton
transfer in a dimer. We also show how it is readily applied to systems with
greater than two states by simulating a 7-site model of FMO and a molecular
wire model. The augmented propagator (AP) TNPI utilizes the symmetries of the
problem, leading to accelerated convergence and dramatic reductions of
computational effort. We also introduce an approximate method that speeds up
propagation beyond the non-local memory length. Furthermore, the structure
imposed by the tensor network representation of the PA tensor naturally
suggests other factorizations that make simulations for extended systems more
efficient. These factorizations would be the subject of future explorations.
The flexibility of the AP-TNPI framework makes it a promising new addition to
the family of path integral methods for non-equilibrium quantum dynamics.
