  Recently, geometric phases, which is fault tolerate to certain errors
intrinsically due to its geometric property, are getting considerable attention
in quantum computing theoretically. So far, only one experiment about adiabatic
geometric gate with NMR through Berry phase has been reported. However, there
are two drawbacks in it. First, the adiabatic condition of Berry phase makes
such gate very slowly. Second, the extra operation to eliminate the dynamic
phase. As we know, geometric phase can exist both adiabatic(Berry phase) and
nonadiabatic(Aharonov-Anandan phase). In this letter, we reports the first
experimental realization of nonadiabatic geometric gate with NMR through
conditional-AA phase. In our experiment the gates can be made faster and more
easily, and the two drawbacks mentioned above are removed.
