  I present an explicit classical simulation of arbitrary quantum noise for
quantum models in which one qubit interacts with a quantum bath. The classical
model simulates the interaction of the bath and the qubit by random unitary
evolutions. I show that any arbitrary quantum dynamics, including quantum
dissipation, recurrence, and dephasing, can be simulated classically when one
allows the unitary operators in the classical model to depend on the initial
state of the system and bath. For initial mixed states of the system and
non-product states of the system and bath, I demonstrate that random unitary
expansion is still possible, in terms of a set of pure states.
