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arXiv:1706.03247 · IEEE Transactions on Automatic Control

Structured Singular Value Analysis for Spintronics Network Information Transfer Control

EDMOND A. JONCKHEERE, S. G. Schirmer, Frank C. Langbein

Published Jun 12, 2017• 11 citations
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Abstract

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.

DOI
10.1109/tac.2017.2714623

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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.
Information encoded in networks of stationary, interacting spin-1/2 particles is central for many applications ranging from quantum spintronics to quantum information processing. Without control, however, information transfer through such networks is generally inefficient. Currently available control methods to maximize the transfer fidelities and speeds mainly rely on dynamic control using time-varying fields and often assume instantaneous readout. We present an alternative approach to achieving efficient, high-fidelity transfer of excitations by shaping the energy landscape via the design of time-invariant feedback control laws without recourse to dynamic control. Both instantaneous readout and the more realistic case of finite readout windows are considered. The technique can also be used to freeze information by designing energy landscapes that achieve Anderson localization. Perfect state or superoptimal transfer and localization are enabled by conditions on the eigenstructure of the system and signature properties for the eigenvectors. Given the eigenstructure enabled by superoptimality, it is shown that feedback controllers that achieve perfect state transfer are, surprisingly, also the most robust with regard to uncertainties in the system and control parameters.
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.
Selective transfer of information between spin-1/2 particles arranged in a ring is achieved by optimizing the transfer fidelity over a readout time window via shaping, externally applied, static bias fields. Such static control fields have properties that clash with the expectations of classical control theory. Previous work has shown that there are cases in which the logarithmic differential sensitivity of the transfer fidelity to uncertainty in coupling strength or spillage of the bias field to adjacent spins is minimized by controllers that produce the best fidelity. Here we expand upon these examples and examine cases of both classical and non-classical behavior of logarithmic sensitivity to parameter uncertainty and robustness as measured by the p function for quantum systems. In particular we examine these properties in an 11-spin ring with a single uncertainty in coupling strength or a single bias spillage.
Propagation of information encoded in spin degrees of freedom through networks of coupled spins enables important applications in spintronics and quantum information processing. We study control of information propagation in networks of spin-1/2 particles with uniform nearest neighbor couplings forming a ring with a single excitation in the network as simple prototype of a router for spin-based information. Specifically optimizing spatially distributed potentials, which remain constant during information transfer, simplifies the implementation of the routing scheme. However, the limited degrees of freedom makes finding a control that maximizes the transfer probability in a short time difficult. We show that the structure of the eigenvalues and eigenvectors must fulfill a specific condition to be able to maximize the transfer fidelity, and demonstrate that a specific choice among the many potential structures that fulfill this condition significantly improves the solutions found by optimal control.
Abstract The design and analysis of controllers to regulate excitation transport in quantum spin rings presents challenges in the application of classical feedback control techniques to synthesize effective control and generates results in contradiction to the expectations of classical control theory. This paper examines the robustness of controllers designed to optimize the fidelity of an excitation transfer to uncertainty in system and control parameters. We use the logarithmic sensitivity of the fidelity error as the robustness measure, drawing on the classical control analog of the sensitivity of the tracking error. Our analysis shows that quantum systems optimized for coherent transport demonstrate significantly different correlation between error and the log-sensitivity depending on whether the controller is optimized for readout at an exact time T or over a time-window T ± Δ /2.
We study the problem of robust performance of quantum systems under structured uncertainties. A specific feature of closed (Hamiltonian) quantum systems is that their poles lie on the imaginary axis and that neither a coherent controller nor physically relevant structured uncertainties can alter this situation. This changes for open systems where decoherence ensures asymptotic stability and creates a unique landscape of pure performance robustness, with the distinctive feature that closed-loop stability is secured by the underlying physics and needs not be enforced. This stability, however, is often detrimental to quantum-enhanced performance, and additive perturbations of the Hamiltonian give rise to dynamic generators that are nonlinear in the perturbed parameters, invalidating classical paradigms to assess robustness to structured perturbations such as singular value analysis. This problem is addressed using a fixed-point iteration approach to determine a maximum perturbation strength δmaxthat ensures that the transfer function remains bounded, ║Tδ║−1for δmax.