  We study quantum decoherence of single-qubit and two-qubit Aharonov-Anandan
(AA) geometric phase gates realized in a multistep scheme. Each AA gate is also
compared with the dynamical phase gate performing the same unitary
transformation within the same time period and coupled with the same
environment, which is modeled as harmonic oscillators. It is found that the
fidelities and the entanglement protection of the AA phase gates are enhanced
by the states being superpositions of different eigenstates of the
environmental coupling, and the noncommutativity between the qubit interaction
and the environmental coupling.
