  We show explicitly that for any continuously evolving open quantum system, be
it finite ($d$-dimensional) or countably infinite dimensional, its dynamics can
be described by a time-dependent Hamiltonian and probabilistic combinations of
up to $d-1$ ($d \to \infty$ for infinite dimensional case), instead of $d^2-1$,
time-dependent unitary operators, resulting in a quadratic improvement in
simulation resources. Importantly, both types of operations must be initial
state-dependent in general, and thus the simulation is tailored to that initial
state. Such description is exact under all cases, and does not rely on any
assumptions other than the continuity and differentiability of the density
matrix. It turns out that upon generalizations, the formalism can also be used
to describe general quantum channels, which may not be complete positive or
even positive, and results in a Kraus-like representation. Experimentally, the
formalism provides a scheme to control a quantum state to evolve along designed
quantum trajectories, and can be particularly useful in quantum computing and
quantum simulation scenes since only unitary resources are needed for
implementation. Philosophically, it provides us with a new perspective to
understand the dynamics of open quantum systems and related problems such as
decoherence and quantum measurement, i.e. the non-unitary evolution of quantum
states can thereby be regarded as the combined effect of state-dependent
deterministic evolutions and probabilistic applications of unitary operators
