Time-optimal synthesis of unitary transformations in coupled fast and slow qubit system

Physical Review A (Impact Factor: 2.81). 09/2007; 77(3). DOI: 10.1103/PhysRevA.77.032332
Source: arXiv


In this paper, we study time-optimal control problems related to system of two coupled qubits where the time scales involved in performing unitary transformations on each qubit are significantly different. In particular, we address the case where unitary transformations produced by evolutions of the coupling take much longer time as compared to the time required to produce unitary transformations on the first qubit but much shorter time as compared to the time to produce unitary transformations on the second qubit. We present a canonical decomposition of SU(4) in terms of the subgroup SU(2)xSU(2)xU(1), which is natural in understanding the time-optimal control problem of such a coupled qubit system with significantly different time scales. A typical setting involves dynamics of a coupled electron-nuclear spin system in pulsed electron paramagnetic resonance experiments at high fields. Using the proposed canonical decomposition, we give time-optimal control algorithms to synthesize various unitary transformations of interest in coherent spectroscopy and quantum information processing. Comment: 8 pages, 3 figures

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    • "In the recent few years optimal control [1] [2] [3] has found its way into nuclear magnetic resonance as a powerful tool for efficient design and optimization of experiments for applications in imaging [4] [5] [6] [7], liquid-state NMR [8] [9] [10] [11] [12] [13], solid-state NMR [14] [15] [16] [17] [18] [19] [20], quantum computation [21] [22] [23] [24] [25] [26], and dynamic nuclear polarization/electronnuclear interactions [27] [28] [29] [30] [31]. This method, originally being introduced for optimizations in engineering and economy, is very attractive for optimization problems dealing with complex internal Hamiltonians and a large number of external manipulation parameters . "
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    ABSTRACT: We present the implementation of optimal control into the open source simulation package SIMPSON for development and optimization of nuclear magnetic resonance experiments for a wide range of applications, including liquid- and solid-state NMR, magnetic resonance imaging, quantum computation, and combinations between NMR and other spectroscopies. Optimal control enables efficient optimization of NMR experiments in terms of amplitudes, phases, offsets etc. for hundreds-to-thousands of pulses to fully exploit the experimentally available high degree of freedom in pulse sequences to combat variations/limitations in experimental or spin system parameters or design experiments with specific properties typically not covered as easily by standard design procedures. This facilitates straightforward optimization of experiments under consideration of rf and static field inhomogeneities, limitations in available or desired rf field strengths (e.g., for reduction of sample heating), spread in resonance offsets or coupling parameters, variations in spin systems etc. to meet the actual experimental conditions as close as possible. The paper provides a brief account on the relevant theory and in particular the computational interface relevant for optimization of state-to-state transfer (on the density operator level) and the effective Hamiltonian on the level of propagators along with several representative examples within liquid- and solid-state NMR spectroscopy.
    Preview · Article · Apr 2009 · Journal of Magnetic Resonance
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    ABSTRACT: Optimal-control-based numerical algorithms make it possible to explore the physical limits of dynamic nuclear polarization. Examples of time-optimal and relaxation-optimized electron–nuclear polarization transfer experiments are presented for simple model systems consisting of an isolated electron–nuclear spin pair in the absence and presence of relaxation. The optimized pulse sequences are compared with conventional transfer schemes, such as electron–nuclear cross polarization and selective population inversion.
    No preview · Article · Aug 2008 · Applied Magnetic Resonance
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