NMR C-NOT gate through Aharanov-Anandan's phase shift

Recently, it is proposed to do quantum computation through the Berry's phase(adiabatic cyclic geometric phase) shift with NMR (Jones et al, Nature, 403, 869(2000)). This geometric quantum gate is hopefully to be fault tolerant to certain types of errors because of the geometric property of the Berry phase. Here we give a scheme to realize the NMR C-NOT gate through Aharonov-Anandan's phase(non-adiabatic cyclic phase) shift on the dynamic phase free evolution loop. In our scheme, the gate is run non-adiabatically, thus it is less affected by the decoherence. And, in the scheme we have chosen the the zero dynamic phase time evolution loop in obtaining the gepmetric phase shift, we need not take any extra operation to cancel the dynamic phase.