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arXiv:2108.05190 · Physical Review A

Application of the small-tip-angle approximation in the toggling frame for the design of analytic robust pulses in quantum control

Léo Van Damme, Dominique Sugny, Steffen J. Glaser

Published Oct 28, 2021• 7 citations
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Abstract

We apply the small-tip-angle approximation in the toggling frame in order to analytically design robust pulses against resonance offsets for state to state transfer in two-level quantum systems. We show that a broadband or a local robustness up to an arbitrary order can be achieved. We provide different control parametrizations to satisfy experimental constraints and limitations on the amplitude or energy of the pulse. A comparison with numerical optimal solutions is made.

DOI
10.1103/physreva.104.042226

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We apply the small-tip-angle approximation in the toggling frame in order to analytically design robust pulses against resonance offsets for state to state transfer in two-level quantum systems. We show that a broadband or a local robustness up to an arbitrary order can be achieved. We provide different control parametrizations to satisfy experimental constraints and limitations on the amplitude or energy of the pulse. A comparison with numerical optimal solutions is made.
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Control pulses that nominally optimize fidelity are sensitive to routine hardware drift and modeling errors. Robust quantum optimal control seeks error-insensitive control pulses that maintain fidelity thresholds and obey hardware constraints. Distinct numerical approximations to the first-order error susceptibility include adjoint end-point and toggling-frame approaches. Although theoretically equivalent, we provide a novel, systematic study demonstrating important numerical differences between these two approaches. We also introduce a critical discretization correction to the widely-used toggling-frame robustness estimator, measurably improving its estimate of first-order error susceptibility. We accomplish our study by positioning robustness as a first-class objective within direct, constrained optimal control. Our approach uniquely handles control and fidelity constraints while cleanly isolating robustness for dedicated optimization. In both single- and two-qubit examples under realistic constraints, our approach provides an analytic edge for obtaining precise, physics-informed robustness.
We investigate the time- and energy-minimum optimal solutions for the robust control of two-level quantum systems against offset or control-field uncertainties. Using the Pontryagin maximum principle, we derive the global optimal pulses for the first robustness orders. We show that the dimension of the control landscape is lower than or equal to 2N2N2N for a field robust to the NthN\mathrm{th}Nth order, which leads to an estimate of its complexity.
Finely manipulating a large population of interacting nuclear spins is an extremely challenging problem arising in wide-ranging applications in quantum science and technology. Prominent examples include the design of robust excitation and inversion pulses for nuclear magnetic resonance spectroscopy and imaging, coordination of spin networks for coherence transfer, and control of superposition and entanglement for quantum computation. In this paper, by integrating the technique of small angle approximation with non-harmonic Fourier analysis, we establish a systematic method to construct robust pulse sequences that neutralize the effect of coupling variations in a spin network. In addition, we explore an alternating optimization procedure for tailoring the constructed pulses to satisfy practical design criteria. We also provide numerical examples to demonstrate the efficacy of the proposed methodology.
Achieving high-fidelity control of quantum systems is essential for realization of a practical quantum computer. Composite pulse sequences which suppress different types of errors can be nested to suppress a wide variety of errors but the result is often not optimal, especially in the presence of constraints such as bandwidth limitations. Robust smooth pulse shaping provides flexibility but obtaining such analytical pulse shapes is a nontrivial problem and choosing the appropriate parameters typically requires a numerical search in a high-dimensional space. In this work, we extend a previous analytical treatment of robust smooth pulses to allow the determination of pulse parameters without numerical search. We also show that the problem can be reduced to a set of coupled ordinary differential equations which allows for a more streamlined numerical treatment.
Designing accurate and high-fidelity broadband pulses is an essential component in conducting quantum experiments across fields from protein spectroscopy to quantum optics. However, constructing exact and analytic broadband pulses remains unsolved due to the nonlinearity and complexity of the underlying spin system dynamics. Here, we present a nontrivial dynamic connection between nonlinear spin and linear spring systems and show the surprising result that such nonlinear and complex pulse design problems are equivalent to designing controls to steer linear harmonic oscillators under optimal forcing. We derive analytic broadband π/2 and π pulses that perform exact, or asymptotically exact, excitation and inversion over a defined bandwidth, and also with bounded amplitude. This development opens up avenues for pulse sequence design and lays a foundation for understanding the control of two-level systems.Coherent control of two-level systems is crucial for achieving fidelity in spectroscopy and quantum computing, but inherent nonlinearities and parameter variation have, to date, required an approximate, numerical approach. Here, Li et al. show how to map a spin ensemble to a spring model so analytic pulses can be designed using linear methods.