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arXiv:2309.14437 · Physical Review Letters

Universally Robust Quantum Control

Pablo M. Poggi, Gabriele De Chiara, Steve Campbell, Anthony Kiely

Published May 7, 2024• 36 citations
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Abstract

We study the robustness of the evolution of a quantum system against small uncontrolled variations in parameters in the Hamiltonian. We show that the fidelity susceptibility, which quantifies the perturbative error to leading order, can be expressed in superoperator form and use this to derive control pulses that are robust to any class of systematic unknown errors. The proposed optimal control protocol is equivalent to searching for a sequence of unitaries that mimics the first-order moments of the Haar distribution, i.e., it constitutes a 1-design. We highlight the power of our results for error-resistant single- and two-qubit gates.

DOI
10.1103/physrevlett.132.193801

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We study the robustness of the evolution of a quantum system against small uncontrolled variations in parameters in the Hamiltonian. We show that the fidelity susceptibility, which quantifies the perturbative error to leading order, can be expressed in superoperator form and use this to derive control pulses that are robust to any class of systematic unknown errors. The proposed optimal control protocol is equivalent to searching for a sequence of unitaries that mimics the first-order moments of the Haar distribution, i.e., it constitutes a 1-design. We highlight the power of our results for error-resistant single- and two-qubit gates.
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.
Adiabatic pulses are used extensively to enable robust control of quantum operations. We introduce an approach to adiabatic control that uses the superadiabatic quality factor as a performance metric to design robust, high-fidelity pulses. This approach permits the systematic design of quantum control schemes to maximize the adiabaticity of a unitary operation in a particular time interval given the available control resources. The interplay between adiabaticity, fidelity, and robustness of the resulting pulses is examined for the case of single-qubit inversion, and superadiabatic pulses are demonstrated to have improved robustness to control errors. A numerical search strategy is developed to find a broader class of adiabatic operations, including multiqubit adiabatic unitaries. We illustrate the utility of this search strategy by designing control waveforms that adiabatically implement a two-qubit entangling gate for a model NMR system.
Constructing high-fidelity control fields that are robust to control, system, and/or surrounding environment uncertainties is a crucial objective for quantum information processing. Using the two-state Landau-Zener model for illustrative simulations of a controlled qubit, we generate optimal controls for $\ensuremath{\pi}/2$ and $\ensuremath{\pi}$ pulses and investigate their inherent robustness to uncertainty in the magnitude of the drift Hamiltonian. Next, we construct a quantum-control protocol to improve system-drift robustness by combining environment-decoupling pulse criteria and optimal control theory for unitary operations. By perturbatively expanding the unitary time-evolution operator for an open quantum system, previous analysis of environment-decoupling control pulses has calculated explicit control-field criteria to suppress environment-induced errors up to (but not including) third order from $\ensuremath{\pi}/2$ and $\ensuremath{\pi}$ pulses. We systematically integrate this criteria with optimal control theory, incorporating an estimate of the uncertain parameter to produce improvements in gate fidelity and robustness, demonstrated via a numerical example based on double quantum dot qubits. For the qubit model used in this work, postfacto analysis of the resulting controls suggests that realistic control-field fluctuations and noise may contribute just as significantly to gate errors as system and environment fluctuations.
Robust quantum control usually assumes one of two extremes: the error is of a known form and only its strength is unknown, or nothing at all is known about it. In practice one often sits in between. The size of a static error is unknown, but its class is known, for example, it only shifts energy levels, or it acts on each subsystem separately. We ask how fast a control can cancel, to first order, every error in such a class. A time-independent Hamiltonian can do this if and only if all errors in the class have zero diagonal in a common basis. In that case an equally spaced energy ladder in this basis does it after one period. A lower bound that holds for any time-dependent control with bounded drive power shows that this simple constant control is optimal for several natural classes, including unknown level shifts and unknown local errors on two subsystems. Completely unknown errors need a time-dependent control, and there, knowing the error class shortens the minimum time by a factor d2+1\sqrt{d^2+1}d2+1​. So partial knowledge of the error changes not only how long an optimal robust control takes, but also its form.
This thesis details work performed on the improvement of experimental control of quantum systems through the use of optimal control theory. We pursue this objective through both conceptual and practical routes. We first devise improved methods of optimal control for quantum systems. We then apply them, as well as more established methods, to experimental situations where robustness to inaccuracy and error is sought. We develop a method for improving the computational efficiency of optimal control theory in the context of quantum control. This approach makes use of the iterative nature of optimisation algorithms, by using perturbation theory to minimise the effort needed to calculate the time propagator associated with a controlled Hamiltonian. We also create a new way of defining a figure of merit when optimising for robustness against error in an experiment. This method specifically applies to errors that have a relatively small effect on the controlled system's Hamiltonian. We compare this with an ensemble method based around randomly sampling several points on the control landscape, and find these methods to be largely equivalent for small errors, with the ensemble method being superior for larger error. On the applied side, we design radio frequency magnetic pulses that create robustness in spin ensembles to inhomogeneous broadening and variations of coupling strength to the control field for two applications. In the first, we create reliable gates in a situation where limitations are imposed on the control that can be applied. In the second application, we design controls for radio frequency magnetic pulses to create a superradiant state in a spin ensemble. Finally, we collaborate with the experiment to measure the electrons electric dipole moment that is being performed at Imperial College London. For this collaboration, we shape the envelope of the radio frequency control pulses used in the experiment to create superpositions of Zeeman and Stark-shifted excited states in YbF molecules. This shaping allows us to impart pre-determined phases to these superpositions, in a manner that is robust to errors in the control pulse envelope. We obtain sets of robust and sensitive pulses that have been sent to the experiment for testing purposes.
The optimization of robust quantum control is often tailored to specific tasks and suffers from inefficiencies due to the complexity of cost functions. Our recent findings indicate a highly effective methodology for the engineering of quantum gates by initiating the process with a robust control configuration of any arbitrary gate. We first introduce the Quantum Control Robustness Landscape (QCRL), a conceptual framework that maps control parameters to noise susceptibility. This framework facilitates a systematic investigation of equally robust controls for diverse quantum operations. By navigating through the level sets of the QCRL, our Robustness-Invariant Pulse Variation (RIPV) algorithm allows for the variation of control pulses while preserving robustness. Numerical simulations demonstrate that our single- and two-qubit gates exceed the quantum error correction threshold even with substantial noise. This methodology opens up a new paradigm for quantum gate engineering capable of effectively suppressing generic noise.