  The initial stages of the evolution of an open quantum system encode the key
information of its underlying dynamical correlations, which in turn can predict
the trajectory at later stages. We propose a general approach based on
non-Markovian dynamical maps to extract this information from the initial
trajectories and compress it into non-Markovian transfer tensors. Assuming
time-translational invariance, the tensors can be used to accurately and
efficiently propagate the state of the system to arbitrarily long timescales.
The non-Markovian Transfer Tensor Method (TTM) demonstrates the
coherent-to-incoherent transition as a function of the strength of quantum
dissipation and predicts the non-canonical equilibrium distribution due to the
system-bath entanglement. TTM is equivalent to solving the Nakajima-Zwanzig
equation, and therefore can be used to reconstruct the dynamical operators (the
system Hamiltonian and memory kernel) from quantum trajectories obtained in
simulations or experiments. The concept underlying the approach can be
generalized to physical observables with the goal of learning and manipulating
the trajectories of an open quantum system.
