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arXiv:0903.2295 · Physical Review A

Composite pulses in NMR as nonadiabatic geometric quantum gates

Yukihiro Ota, Yasushi Kondo

Published Aug 4, 2009• 52 citations
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Abstract

We show that some composite pulses widely employed in NMR experiments are regarded as nonadiabatic geometric quantum gates with Aharanov-Anandan phases. Thus, we reveal the presence of a fundamental issue on quantum mechanics behind a traditional technique. To examine the robustness of such composite pulses against fluctuations, we present a simple noise model in a two-level system. Then, we find that the composite pulses possesses purely geometrical nature even under a certain type of fluctuations.

DOI
10.1103/physreva.80.024302

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We show that some composite pulses widely employed in NMR experiments are regarded as nonadiabatic geometric quantum gates with Aharanov-Anandan phases. Thus, we reveal the presence of a fundamental issue on quantum mechanics behind a traditional technique. To examine the robustness of such composite pulses against fluctuations, we present a simple noise model in a two-level system. Then, we find that the composite pulses possesses purely geometrical nature even under a certain type of fluctuations.
Unitary operations acting on a quantum system must be robust against systematic errors in control parameters for reliable quantum computing. Composite pulse technique in nuclear magnetic resonance realizes such a robust operation by employing a sequence of possibly poor-quality pulses. In this study, we demonstrate that two kinds of composite pulses-one compensates for a pulse length error in a one-qubit system and the other compensates for a J-coupling error in a two-qubit system-have a vanishing dynamical phase and thereby can be seen as geometric quantum gates, which implement unitary gates by the holonomy associated with dynamics of cyclic vectors defined in the text.
The nonadiabatic geometric quantum gate (NGQG) is promising for the realization of high-fidelity operation for large-scale quantum processing. Normally, conventional NGQGs can be especially robust to either the Rabi error or the detuning error, which are two typical errors in many quantum computing platforms. However, it is difficult to suppress these two types of errors at the same time. This remains a big challenge for NGQGs. Here we present a general framework to implement the optimized geometric gate, where the evolution path is performed by using a family of optimized composite pulses. These composite pulses can reduce the sensitivity to the detuning error without introducing an extra Rabi error for this path. Thus, they help fulfill the cyclic evolution condition for the geometric gate. In addition, the inserted composite pulses would not introduce unwanted dynamical phase accumulation. As a result, the designed optimized geometric gate can simultaneously mitigate both the Rabi and detuning errors even in the presence of decoherence related to the real experiments. Our work paves a way to achieve geometric quantum computation robust against multiple errors.
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.
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.
We study the performance of composite pulses in the presence of time-varying control noise on a single qubit. These protocols, originally devised only to correct for static, systematic errors, are shown to be robust to time-dependent non-Markovian noise in the control field up to frequencies as high as $\ensuremath{\sim}10%$ of the Rabi frequency. Our study combines a generalized filter-function approach with asymptotic dc-limit calculations to give a simple analytic framework for error analysis applied to a number of composite-pulse sequences relevant to nuclear magnetic resonance as well as quantum information experiments. Results include examination of recently introduced concatenated composite pulses and dynamically corrected gates, demonstrating equivalent first-order suppression of time-dependent fluctuations in amplitude and/or detuning, as appropriate for the sequence in question. Our analytic results agree well with numerical simulations for realistic 1/f1/f1/f noise spectra with a roll-off to 1/f21/{f}^{2}1/f2, providing independent validation of our theoretical insights.