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Physica Scripta

Automatic generation of nonadiabatic conditional geometric phase shift with a noncoplanar fibre system

Jian Qi Shen

Published Dec 22, 2005• 2 citations
Publisher

Abstract

A new scheme for realizing a nonadiabatic conditional geometric phase shift via a noncoplanar (coiled) fibre system is presented. It is shown that the effective Hamiltonian that describes the interaction of polarized photons with a fibre medium is just the Wang–Matsumoto type of Hamiltonian. This, therefore, means that the coiled fibre system may be an ideal implementation for realizing the nonadiabatic geometric phase gates involved in the topological quantum computation. A remarkable feature of the present method is that it can automatically meet the conditions and requirements proposed in the Wang–Matsumoto nuclear magnetic resonance (NMR) scheme: specifically, (i) in the coiled fibre system, the dynamical phase of photon wavefunction caused by the interaction Hamiltonian automatically vanishes; (ii) the Wang–Matsumoto requirement for the parameters in the Wang–Matsumoto NMR Hamiltonian can be exactly satisfied automatically in such a fibre system; and (iii) the conditional initial state can be easily achieved by manipulating the initial wave vector of polarized photons.

DOI
10.1088/0031-8949/73/1/011

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A new scheme for realizing a nonadiabatic conditional geometric phase shift via a noncoplanar (coiled) fibre system is presented. It is shown that the effective Hamiltonian that describes the interaction of polarized photons with a fibre medium is just the Wang–Matsumoto type of Hamiltonian. This, therefore, means that the coiled fibre system may be an ideal implementation for realizing the nonadiabatic geometric phase gates involved in the topological quantum computation. A remarkable feature of the present method is that it can automatically meet the conditions and requirements proposed in the Wang–Matsumoto nuclear magnetic resonance (NMR) scheme: specifically, (i) in the coiled fibre system, the dynamical phase of photon wavefunction caused by the interaction Hamiltonian automatically vanishes; (ii) the Wang–Matsumoto requirement for the parameters in the Wang–Matsumoto NMR Hamiltonian can be exactly satisfied automatically in such a fibre system; and (iii) the conditional initial state can be easily achieved by manipulating the initial wave vector of polarized photons.
A new scheme of realizing the nonadiabatic conditional geometric phase shift via a noncoplanar (and coiled) fiber system is presented in this Letter. It is shown that the effective Hamiltonian that describes the interaction of polarized photons with the fiber medium is just the Wang-Keiji type of Hamiltonian. This, therefore, means that the coiled fiber system may be an ideal implementation of realizing the nonadiabatic geometric phase gates for the topological quantum computation. The remarkable feature of the present method is that it can automatically meet the conditions and requirements proposed in the Wang-Keiji scheme: (i) in the coiled fiber system, the dynamical phase of photon wavefunction caused by the interaction Hamiltonian automatically vanishes; (ii) the Wang-Keiji requirement for the parameters in the Wang-Keiji Hamiltonian can be exactly satisfied automatically in the fiber system; (iii) the conditional initial state can be easily achieved by manipulating the initial wave vector of polarized photons. Due to these three advantages, the coiled fiber system may be a potentially practical way of achieving the nonadiabatic conditional geometric phase shift (and hence the nonadiabatic geometric quantum gates).
The nonadiabatic conditional geometric phase gate is required in the topological quantum computation in order to overcome the conflict between the requirement of adiabatic condition(to avoid the severe distortion from the nonadiabaticity to the results) and the removal of decoherence effects. It was demonstrated that the effective Hamiltonian that describes the propagation of photon fields inside the coiled fiber is just the WangMatsumoto type of Hamiltonian. Thus, the coiled fiber system will automatically generate the nonadiabatic conditional geometric phase shift. In addition, it was shown that the dynamical phase(resulting from the effective Hamiltonian) acquired by the polarized photons vanishes, and the conditional initial state can be easily prepared only by controlling the initial wave vector of photons. In a word, the coiled fiber system can automatically satisfy the requirements and conditions, which were proposed by Wang and Matsumoto in order to create the nonadiabatic conditional geometric phase shift in their NMR scheme.
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 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.
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.
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.