Non-Markovian Stochastic Schrodinger Equation: Matrix Product State Approach

We derive a hierarchy of matrix product states (HOMPS) method which is numerically exact and efficient for general non-Markovian dynamics in open quantum system. This HOMPS is trying to attack the exponential wall issue in the recently developed hierarchy of pure states (HOPS) scheme with two steps: a. finding an effective time-dependent Schrodinger equation which is equivalent to HOPS, b. propagating this equation within matrix product states/operators (MPS/MPO) representation. HOMPS works in linear form and takes into account finite temperature effect straightforwardly from the initial pure state. Applications of HOMPS to spin-boson model covering both high and low temperatures are provided, demonstrating the validity and efficiency of the new approach.