  We examine the adiabatic dynamics of a quantum system coupled to a noisy
classical control field. A stochastic phase shift is shown to arise in the
off-diagonal elements of the system's density matrix which can cause
decoherence. We derive the condition for onset of decoherence, and identify the
noise properties that drive decoherence. We show how this decoherence mechansim
causes: (1) a dephasing of the observable consequences of the adiabatic
geometric phase; and (2) the loss of computational efficiency of the Shor
algorithm when run on a sufficiently noisy geometric quantum computer.
