  The hierarchical equations of motion (HEOM) for a generalized quantum
dissipative system is rigorously constructed in the frameworks of two different
stochastic dynamical descriptions, i.e., the non-Markovian quantum state
diffusion approach as well as the stochastic decoupling scheme. We demonstrated
that the HEOMs obtained by these two different stochastic dynamical methods are
identical. Moreover, we present some numerical examples to verify the
feasibility of our formalism.
