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arXiv:quant-ph/0503032 · Journal of Magnetic Resonance

Use of non-adiabatic geometric phase for quantum computing by NMR

Ranabir Das, Santosh Kumar, Anil Kumar

Published Sep 23, 2005• 16 citations
arXivHTMLPDF
DOI
10.1016/j.jmr.2005.07.025

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Recently, it has been proposed to perform quantum computation through the Berry phase (adiabatic cyclic geometric phase) shift with NMR. This geometric quantum gate will hopefully be fault tolerant to certain types of error because of the geometric property of the Berry phase. Here we give a scheme to realize the NMR C-NOT gate through the Aharonov-Anandan 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. Moreover, in the scheme we have chosen the zero dynamic phase time evolution loop in obtaining the geometric phase shift; we need not perform any extra operation to cancel the dynamic phase.
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. However, in quantum computation, everything must be completed 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 at the same rate as usual quantum computation.
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.