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arXiv:2501.13209 · IEEE Control Systems Letters

Geometric Interpretation of Sensitivity to Structured Uncertainties in Spintronic Networks

Sean P. O’Neil, EDMOND A. JONCKHEERE, Sophie Shermer

Published Jan 1, 2025• 0 citations
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Abstract

We present a geometric model of the differential sensitivity of the fidelity error for state transfer in a spintronic network based on the relationship between a set of matrix operators. We show an explicit dependence of the differential sensitivity on the fidelity (error), and we further demonstrate that this dependence does not require a trade-off between the fidelity and sensitivity. We prove that for closed systems, ideal performance in the sense of perfect state transfer is both necessary and sufficient for optimal robustness in terms of vanishing sensitivity. We demonstrate the utility of this geometric interpretation of the sensitivity by applying the model to explain the sensitivity versus fidelity error data in two examples.

DOI
10.1109/lcsys.2025.3567866

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We present a geometric model of the differential sensitivity of the fidelity error for state transfer in a spintronic network based on the relationship between a set of matrix operators. We show an explicit dependence of the differential sensitivity on the fidelity (error), and we further demonstrate that this dependence does not require a trade-off between the fidelity and sensitivity. We prove that for closed systems, ideal performance in the sense of perfect state transfer is both necessary and sufficient for optimal robustness in terms of vanishing sensitivity. We demonstrate the utility of this geometric interpretation of the sensitivity by applying the model to explain the sensitivity versus fidelity error data in two examples.
A novel quantum landscape optimization with respect to bias field control inputs is developed with the goal of achieving optimal transfer fidelity subject to robustness against bias field, spin couplings and other uncertainties. This objective is achieved by minimization of a convex combination of fidelity error and worst-case perturbation of fidelity error under directional perturbation of uncertain parameters. The novelty is that the end-point perturbations of the parameters are points of a random uniform sampling of the sphere centered at the nominal values of the parameters. This reveals that the previously developed perfect state transfer with zero sensitivity solution keeps high fidelity and robustness under large rather than differential perturbations.
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.
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.
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 laws for selective transfer of information encoded in excitations of a quantum network, based on shaping the energy landscape using time-invariant, spatially-varying bias fields, can be successfully designed by using numerical optimization. Such control laws, already departing from classicality by replacing closed-loop asymptotic stability with alternative notions of localization, have the intriguing property that for all practical purposes they achieve the upper bound on the fidelity, yet the (logarithmic) sensitivity of the fidelity to such structured perturbation as spin coupling errors and bias field leakages is nearly vanishing. Here, these differential sensitivity results are extended to large structured variations using μ-design tools to reveal a crossover region in the space of controllers where objectives usually thought to be conflicting are actually concordant.