A ug 2 00 1 Nonadiabatic conditional geometric phase shift with NMR

A conditional geometric phase shift gate, which is fault tolerant to certain types of errors due to its geometric nature, was realized recently via nuclear magnetic resonance(NMR) under adiabatic conditions. By the adiabatic requirement, the result is inexact unless the Hamiltonian changes extremely slowly. However, in quantum computation, everything must be completed within the decoherence time. High running speed of every gate in quantum computation is demanded because the power of a quantum computer can be exponentially proportional to the maximum number of logic gate operations that can be made sequentially within the decoherence time. The adiabatic condition makes any fast conditional Berry phase(cyclic adiabatic geometric phase) shift gate impossible. Here we show that by using a newly designed sequence of simple operations with an additional vertical magnetic field, the conditional geometric phase shift gate can be run nonadiabatically. Therefore geometric quantum computation can be done