  The difficulty to simulate the dynamics of open quantum systems resides in
their coupling to many-body reservoirs with exponentially large Hilbert space.
Applying a tensor network approach in the time domain, we demonstrate that
effective small reservoirs can be defined and used for modeling open quantum
dynamics. The key element of our technique is the timeline reservoir network
(TRN), which contains all the information on the reservoir's characteristics,
in particular, the memory effects timescale. The TRN has a one-dimensional
tensor network structure, which can be effectively approximated in full analogy
with the matrix product approximation of spin-chain states. We derive the
sufficient bond dimension in the approximated TRN with a reduced set of
physical parameters: coupling strength, reservoir correlation time, minimal
timescale, and the system's number of degrees of freedom interacting with the
environment. The bond dimension can be viewed as a measure of the open dynamics
complexity. Simulation is based on the semigroup dynamics of the system and
effective reservoir of finite dimension. We provide an illustrative example
showing scope for new numerical and machine learning-based methods for open
quantum systems.
