  We consider the problem of constructing a "universal set" of Markovian
processes, such that any Markovian open quantum system, described by a
one-parameter semigroup of quantum channels, can be simulated through
sequential simulations of processes from the universal set. In particular, for
quantum systems of dimension $d$, we explicitly construct a universal set of
semigroup generators, parametrized by $d^2-3$ continuous parameters, and prove
that a necessary and sufficient condition for the dynamical simulation of a $d$
dimensional Markovian quantum system is the ability to implement a) quantum
channels from the semigroups generated by elements of the universal set of
generators, and b) unitary operations on the system. Furthermore, we provide an
explicit algorithm for simulating the dynamics of a Markovian open quantum
system using this universal set of generators, and show that it is efficient,
with respect to this universal set, when the number of distinct Lindblad
operators (representing physical dissipation processes) scales polynomially
with respect to the number of subsystems.
