Non-Markovian Stochastic Schr\"odinger Equation: Matrix Product State Approach to the Hierarchy of Pure States.

We derive a stochastic hierarchy of matrix product states (HOMPS) for non-Markovian dynamics in open quantum system at finite temperature, which is numerically exact and efficient. This HOMPS addresses the exponential wall issue in the recently developed stochastic hierarchy of pure states (HOPS). To this end HOPS is expressed in terms of formal creation and annihilation operators. The resulting stochastic first order differential equation can be readily adapted to a matrix product states/operators (MPS/MPO) representation and propagated with polynomial complexity. The validity and efficiency of HOMPS is demonstrated for the spin-boson model, both in the high and low temperature regime.