  Understanding the roles of the temporary and spatial structures of quantum
functional noise in open multilevel quantum molecular systems attracts a lot of
theoretical interests. I want to establish a rigorous and general framework for
functional quantum noises from the constructive and computational perspectives,
$\it{i.e.}$ how to generate the random trajectories to reproduce the kernel and
path ordering of the influence functional with effective Monte Carlo methods
for arbitrary spectral densities. This construction approach aims to unify the
existing stochastic models to rigorously describe the temporary and spatial
structure of Gaussian quantum noises. In this paper, I review the Euclidean
imaginary time influence functional and propose the stochastic matrix
multiplication scheme to calculate reduced equilibrium density matrices (REDM).
  In addition, I review and discuss the Feynman-Vernom influence functional
according to the Gaussian quadratic integral, particularly its imaginary part
which is critical to the rigorous description of the quantum detailed balance.
As a result, I establish the conditions under which the influence functional
can be interpreted as the average of exponential functional operator over
real-valued Gaussian processes for open multilevel quantum systems. I also show
the difference between the local and nonlocal phonons within this framework.
With the stochastic matrix multiplication scheme, I compare the normalized REDM
with the Boltzmann equilibrium distribution for open multilevel quantum
systems.
