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AVS Quantum Science

Robustness of dynamic quantum control: Differential sensitivity bounds

Sean P. O’Neil, Carrie Ann Weidner, EDMOND A. JONCKHEERE, Frank C. Langbein, S. G. Schirmer

Published Jul 1, 2024• 3 citations
Publisher

Abstract

Dynamic control via optimized, piecewise-constant pulses is a common paradigm for open-loop control to implement quantum gates. While numerous methods exist for the synthesis of such controls, there are many open questions regarding the robustness of the resulting control schemes in the presence of model uncertainty; unlike in classical control, there are generally no analytical guarantees on the control performance with respect to inexact modeling of the system. In this paper, a new robustness measure based on the differential sensitivity of the gate fidelity error to parametric (structured) uncertainties is introduced, and bounds on the differential sensitivity to parametric uncertainties are used to establish performance guarantees for optimal controllers for a variety of quantum gate types, system sizes, and control implementations. Specifically, it is shown how a maximum allowable perturbation over a set of Hamiltonian uncertainties that guarantees a given fidelity error can be reliably computed. This measure of robustness is inversely proportional to the upper bound on the differential sensitivity of the fidelity error evaluated under nominal operating conditions. Finally, the results show that the nominal fidelity error and differential sensitivity upper bound are positively correlated across a wide range of problems and control implementations, suggesting that in the high-fidelity control regime, rather than there being a trade-off between fidelity and robustness, higher nominal gate fidelities are positively correlated with increased robustness of the controls in the presence of parametric uncertainties.

DOI
10.1116/5.0196110

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Dynamic control via optimized, piecewise-constant pulses is a common paradigm for open-loop control to implement quantum gates. While numerous methods exist for the synthesis of such controls, there are many open questions regarding the robustness of the resulting control schemes in the presence of model uncertainty; unlike in classical control, there are generally no analytical guarantees on the control performance with respect to inexact modeling of the system. In this paper, a new robustness measure based on the differential sensitivity of the gate fidelity error to parametric (structured) uncertainties is introduced, and bounds on the differential sensitivity to parametric uncertainties are used to establish performance guarantees for optimal controllers for a variety of quantum gate types, system sizes, and control implementations. Specifically, it is shown how a maximum allowable perturbation over a set of Hamiltonian uncertainties that guarantees a given fidelity error can be reliably computed. This measure of robustness is inversely proportional to the upper bound on the differential sensitivity of the fidelity error evaluated under nominal operating conditions. Finally, the results show that the nominal fidelity error and differential sensitivity upper bound are positively correlated across a wide range of problems and control implementations, suggesting that in the high-fidelity control regime, rather than there being a trade-off between fidelity and robustness, higher nominal gate fidelities are positively correlated with increased robustness of the controls in the presence of parametric uncertainties.
Dynamic control via optimized, piecewise-constant pulses is a common paradigm for open-loop control to implement quantum gates. While numerous methods exist for the synthesis of such controls, there are many open questions regarding the robustness of the resulting control schemes in the presence of model uncertainty; unlike in classical control, there are generally no analytical guarantees on the control performance with respect to inexact modeling of the system. In this paper a new robustness measure based on the differential sensitivity of the gate fidelity error to parametric (structured) uncertainties is introduced, and bounds on the differential sensitivity to parametric uncertainties are used to establish performance guarantees for optimal controllers for a variety of quantum gate types, system sizes, and control implementations. Specifically, it is shown how a maximum allowable perturbation over a set of Hamiltonian uncertainties that guarantees a given fidelity error, can be reliably computed. This measure of robustness is inversely proportional to the upper bound on the differential sensitivity of the fidelity error evaluated under nominal operating conditions. Finally, the results show that the nominal fidelity error and differential sensitivity upper bound are positively correlated across a wide range of problems and control implementations, suggesting that in the high-fidelity control regime, rather than there being a trade-off between fidelity and robustness, higher nominal gate fidelities are positively correlated with increased robustness of the controls in the presence of parametric uncertainties.
Control of quantum systems via time-varying external fields optimized to maximize a fidelity measure at a given time is a mainstay in modern quantum control. However, save for specific systems, current analysis techniques for such quantum controllers provide no analytical robustness guarantees. In this letter we provide analytical bounds on the differential sensitivity of the gate fidelity error to structured uncertainties for a closed quantum system controlled by piecewise-constant, optimal control fields. We additionally determine those uncertainty structures that result in this worst-case maximal sensitivity. We then use these differential sensitivity bounds to provide conditions that guarantee performance, quantified by the fidelity error, in the face of parameter uncertainty.
We develop a structure-specific fidelity-threshold robustness margin for finite-dimensional closed quantum systems under piecewise-constant coherent control. A scalar physical parameter may perturb the drift, a control Hamiltonian, or another declared Hamiltonian component across the control horizon. A differential sensitivity bound for trace-amplitude gate fidelity yields a threshold-dependent Lipschitz constant on the connected safe parameter component and hence a certified finite perturbation radius. Recentering this certificate produces an iterative one-dimensional method that takes certified safe steps toward the first fidelity-threshold boundary in either parameter direction. A three-qubit gate-control example shows that these finite margins vary by up to a factor of three across controllers of comparable nominal fidelity and contain structure-dependent information not captured by nominal differential sensitivity alone.
Control of quantum systems via time-varying external fields optimized to maximize a fidelity measure at a given time is a mainstay in modern quantum control. However, save for specific systems, current analysis techniques for such quantum controllers provide no analytical robustness guarantees. In this letter we provide analytical bounds on the differential sensitivity of the gate fidelity error to structured uncertainties for a closed quantum system controlled by piecewise-constant, optimal control fields. We additionally determine those uncertainty structures that result in this worst-case maximal sensitivity. We then use these differential sensitivity bounds to provide conditions that guarantee performance, quantified by the fidelity error, in the face of parameter uncertainty.
This dataset contains the code and data for investigation of the differential sensitivity of the gate infidelity to parametric (structured) uncertainty. The provided controllers are piecewise constant, dynamic controls optimized to maximize gate fidelity across nine problems of varying system complexity and control paradigms. The analysis code computes bounds on the differential sensitivity and statistically analyses the relationship between the infidelity and sensitivity bounds. The analysis code also provides plots to aid in visualization of the statical data.
We derive a universal performance limit for coherent quantum control in the presence of modeled and unmodeled uncertainties. For any target unitary WWW that is implementable in the absence of error, we prove that the worst-case (and hence the average) gate fidelity obeys the lower bound $F \ge \Flb\bigl(\tf \Omeff\bigr)$, where $\tf$ is the gate duration and $\Omeff$ is a single frequency-like measure that aggregates \emph{all} bounded uncertainty sources, e.g., coherent control imperfections, unknown couplings, and residual environment interactions, without assuming an initially factorizable system-bath state or a completely positive map. The bound is obtained by combining an interaction-picture averaging method with a Bellman-Gronwall inequality and holds for any finite-norm Hamiltonian decomposition. Hence it applies equally to qubits, multi-level qudits, and ancilla-assisted operations. Because $\Flb$ depends only on the dimensionless product $\tf\Omeff$, it yields a device-independent metric that certifies whether a given hardware platform can, in principle, reach a specified fault-tolerance threshold, and also sets a quantitative target for robust-control synthesis and system identification.