  This paper provides a detailed account of the numerical implementation of the
stochastic equation of motion (SEOM) method for the dissipative dynamics of
fermionic open quantum systems. To enable direct stochastic calculations, a
minimal auxiliary space (MAS) mapping scheme is adopted, with which the
time-dependent Grassmann fields are represented by c-numbers noises and a set
of pseudo-operators. We elaborate on the construction of the system operators
and pseudo-operators involved in the MAS-SEOM, along with the analytic
expression for the particle current. The MASSEOM is applied to study the
relaxation and voltage-driven dynamics of quantum impurity systems described by
the single-level Anderson impurity model, and the numerical results are
benchmarked against those of the highly accurate hierarchical equations of
motion (HEOM) method. The advantages and limitations of the present MAS-SEOM
approach are discussed extensively.
