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Acta Physica Sinica

The nonadiabatic conditional geometric phase shift in a coiled fiber system

Shen Jian-Qi, Fei Zhuang, (1)杭州师范学院物理系,杭州310012; (2)浙江大学光及电磁波研究中心，瑞典皇家工学院浙江大学联合光子学研究中心,杭州310027

Published Jan 1, 2005• 1 citation
Publisher

Abstract

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.

DOI
10.7498/aps.54.1048

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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 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).
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 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.
A single-loop scenario is proposed to realize nonadiabatic geometric quantum computation. Conventionally, a so-called multiloop approach is used to remove the dynamical phase accumulated in the operation process for geometric quantum gates. More intriguingly, we here illustrate in detail how to use a special single-loop method to remove the dynamical phase and thus to construct a set of universal quantum gates based on the nonadiabatic geometric phase shift. The present scheme is applicable to NMR systems and may be feasible in other physical systems.
Nonadiabatic geometric quantum computation in decoherence-free subspaces has received increasing attention due to the merits of its high-speed implementation and robustness against both control errors and decoherence. However, all the previous schemes in this direction have been based on the conventional geometric phases, of which the dynamical phases need to be removed. In this paper, we put forward a scheme of nonadiabatic geometric quantum computation in decoherence-free subspaces based on unconventional geometric phases, of which the dynamical phases do not need to be removed. Specifically, by using three physical qubits undergoing collective dephasing to encode one logical qubit, we realize a universal set of geometric gates nonadiabatically and unconventionally. Our scheme not only maintains all the merits of nonadiabatic geometric quantum computation in decoherence-free subspaces, but also avoids the additional operations required in the conventional schemes to cancel the dynamical phases.