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arXiv:quant-ph/0305109 · Physical Review A

Proper magnetic fields for nonadiabatic geometric quantum gates in NMR

Kazuto Oshima, Koji Azuma

Published Sep 10, 2003• 6 citations
arXivHTMLPDF

Abstract

In a scheme of nonadiabatic purely geometric quantum gates in nuclear-magnetic-resonance systems we propose proper magnitudes of magnetic fields that are suitable for an experiment. We impose a natural condition and reduce the degree of freedom of the magnetic fields to the extent. By varying the magnetic fields with essentially a one-dimensional degree of freedom, any spin state can acquire an arbitrary purely geometric phase ${\ensuremath{\varphi}}_{g}=\ensuremath{-}2\ensuremath{\pi}(1\ensuremath{-}\mathrm{cos}\ensuremath{\theta}),$ $0<~\mathrm{cos}\ensuremath{\theta}<~1.$ This is an essential ingredient for constructing universal geometric quantum gates.

DOI
10.1103/physreva.68.034304

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In a scheme of nonadiabatic purely geometric quantum gates in nuclear-magnetic-resonance systems we propose proper magnitudes of magnetic fields that are suitable for an experiment. We impose a natural condition and reduce the degree of freedom of the magnetic fields to the extent. By varying the magnetic fields with essentially a one-dimensional degree of freedom, any spin state can acquire an arbitrary purely geometric phase \ensuremathφg=\ensuremath−2\ensuremathπ(1\ensuremath−cos\ensuremathθ),{\ensuremath{\varphi}}_{g}=\ensuremath{-}2\ensuremath{\pi}(1\ensuremath{-}\mathrm{cos}\ensuremath{\theta}),\ensuremathφg​=\ensuremath−2\ensuremathπ(1\ensuremath−cos\ensuremathθ), 0< cos\ensuremathθ< 1.0<~\mathrm{cos}\ensuremath{\theta}<~1.0< cos\ensuremathθ< 1. This is an essential ingredient for constructing universal geometric quantum gates.
A different way to realize nonadiabatic geometric quantum computation is proposed by varying parameters in the Hamiltonian for nuclear-magnetic resonance, where the dynamical and geometric phases are implemented separately without the usual operational process. Therefore the phase accumulated in the geometric gate is a pure geometric phase for any input state. In comparison with the conventional geometric gates by rotating operations, our approach simplifies experimental implementations making them robust to certain experimental errors. In contrast to the unconventional geometric gates, our approach distinguishes the total and geometric phases and offers a wide choice of the relations between the dynamical and geometric phases.
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.
A proposal for applying nonadiabatic geometric phases to quantum computing, called double-loop method [S.-L. Zhu and Z. D. Wang, Phys. Rev. A 67, 022319 (2003)], is demonstrated in a liquid-state nuclear magnetic-resonance quantum computer. Using a spin-echo technique, the original method is modified so that quantum gates are implemented in a standard high-precision nuclear magnetic-resonance system for chemical analysis. We show that a dynamical phase is successfully eliminated and a one-qubit quantum gate is realized although the gate fidelity is not high.
A scheme is proposed to include both cyclic and noncyclic geometric quantum computations in nuclear-magnetic-resonance system by the invariant theory. By controlling magnetic field and arbitrary parameters in the invariant operator, the phases accumulated in the entangling quantum gates for single- and two-qubit systems are pure geometric phases. Thus, fault tolerance may occur in some critical magnetic field parameters for either cyclic or noncyclic evolution by differently choosing for gate time.
A scheme to perfectly preserve an initial qubit state in geometric quantum computation is proposed for a single-qubit geometric quantum gate in a nuclear magnetic resonance system. At first, by adjusting some magnetic field parameters, one can let the dynamic phase be proportional to the geometric phase. Then, by controlling the azimuthal angle in the initial state, we may realize a geometric quantum gate whose fidelity is equal to one under cyclic evolution. This means that the quantum information is no distortion in the process of geometric quantum computation.
We propose an experimentally feasible scheme to achieve quantum computation based on nonadiabatic geometric phase shifts, in which a cyclic geometric phase is used to realize a set of universal quantum gates. Physical implementation of this set of gates is designed for Josephson junctions and for NMR systems. Interestingly, we find that the nonadiabatic phase shift may be independent of the operation time under appropriate controllable conditions. A remarkable feature of the present nonadiabatic geometric gates is that there is no intrinsic limitation on the operation time.