  Dephasing decoherence induced by interaction of one qubit with a quantum bath
can be simulated classically by random unitary evolution without the need for a
bath and this random unitary evolution is equivalent to the quantum case. For a
general dephasing model and a single qubit system, we explicitly construct the
noise functional and completely specify the random unitary evolution. To
demonstrate the technique, we applied our results to three paradigmatic models:
spin-boson, central spin, and quantum impurity. For multiple qubits, we
identify a class of generalized quantum dephasing models that can be simulated
classically. Finally we show that depolarizing quantum models can be simulated
classically for all dimensionalities of the principal system.
