  Geometric phases have been shown to be feasible in implementing quantum gates
to perform quantum information processing. For all the realistic applications,
the environmental influence on the geometric phase and decoherence such as
memory effects must be properly considered in order to achieve the required
precision in geometric quantum computation. In this paper, we study the
geometric phase for a generic open quantum system based on a microscopic model.
A remarkable feature of the open system's geometric phase obtained from our
theoretical formulation is that the geometric phase can be obtained regardless
of the existence of the master equations, while the environmental noise
features such as memory effects are fully accounted for. We demonstrate that
the geometric phases for a general open quantum system can be fundamentally
modified by its non-Markovian environments.
