HEOMs are developed to describe the time evolution of the density matrix ρ ρ ( t ) for an open quantum system. It is a non-perturbative, non-Markovian approach to propagating in time a quantum state. Motivated by the path integral formalism presented by Feynman and Vernon, Tanimura derive the HEOM from a combination of statistical and quantum dynamical techniques. Using a two level spin-boson system Hamiltonian