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Recently, it is proposed to do quantum computation through the Berry'sphase(adiabatic cyclic geometric phase) shift with NMR (Jones et al, Nature,403, 869(2000)). This geometric quantum gate is hopefully to be fault tolerantto certain types of errors because of the geometric property of the Berryphase. Here we give a scheme to realize the NMR C-NOT gate throughAharonov-Anandan's phase(non-adiabatic cyclic phase) shift on the dynamic phasefree evolution loop.  In our scheme, the gate is run non-adiabatically, thus it is less affected bythe decoherence. And, in the scheme we have chosen the the zero dynamic phasetime evolution loop in obtaining the gepmetric phase shift, we need not takeany extra operation to cancel the dynamic phase.Comment: 5 pages, 1 figur

NMR C-NOT gate through the Aharanov-Anandan phase shift