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Robustness of Energy Landscape Controllers for XX Spin Rings

Sean P. O’Neil, Frank C. Langbein, Sophie Shermer

Published Mar 22, 2023• 0 citations
Publisher

Abstract

This project consists of code and data to facilitate investigation of the robustness of energy landscape controllers for transporting a single excitation in XX-coupled spin-1/2 rings of size 5 and 6 under decoherence. The dataset provides controllers optimized for three different performance objectives, and the analysis code is designed to investigate robustness through both statistical estimation and analytic calculation of the log-sensitivity of the fidelity error.

DOI
10.33774/coe-2023-x52n7

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This project consists of code and data to facilitate investigation of the robustness of energy landscape controllers for transporting a single excitation in XX-coupled spin-1/2 rings of size 5 and 6 under decoherence. The dataset provides controllers optimized for three different performance objectives, and the analysis code is designed to investigate robustness through both statistical estimation and analytic calculation of the log-sensitivity of the fidelity error.
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.
This data set contains energy landscape controllers for information propagation in networks of spin-1/2 particles (qubits) with a ring topology and uniform nearest neighbor couplings of XXZ type. The controllers are optimized to maximize the fidelity of excitation transfer between nodes of ring in the single excitation subspace. The network can be regarded as simple prototype of a router for spin-based information. The dataset contains controllers for rings ranging in size from 3 to 20 spins with excitation transfer from node 1 to nodes 2, 3, ... ceil(N/2) considered. Controllers optimized for both instantaneous readout and time-windowed (finite bandwidth) readout are included.
This data set contains energy landscape controllers for information propagation in networks of spin-1/2 particles (qubits) with a ring topology and uniform nearest neighbor couplings of XXZ type. The controllers are optimized to maximize the fidelity of excitation transfer between nodes of ring in the single excitation subspace. The network can be regarded as simple prototype of a router for spin-based information. The dataset contains controllers for rings ranging in size from 3 to 20 spins with excitation transfer from node 1 to nodes 2, 3, ... ceil(N/2) considered. Controllers optimized for both instantaneous readout and time-windowed (finite bandwidth) readout are included.
As shown in previous work, in some cases closed quantum systems exhibit a non-conventional absence of trade-off between performance and robustness in the sense that controllers with the highest fidelity can also provide the best robustness to parameter uncertainty. As the dephasing induced by the interaction of the system with the environment guides the evolution to a more classically mixed state, it is worth investigating what effect the introduction of dephasing has on the relationship between performance and robustness. In this paper we analyze the robustness of the fidelity error, as measured by the logarithmic sensitivity function, to dephasing processes. We show that introduction of dephasing as a perturbation to the nominal unitary dynamics requires a modification of the log-sensitivity formulation used to measure robustness about an uncertain parameter with non-zero nominal value used in previous work. We consider controllers optimized for a number of target objectives ranging from fidelity under coherent evolution to fidelity under dephasing dynamics to determine the extent to which optimizing for a specific regime has desirable effects in terms of robustness. Our analysis is based on two independent computations of the log-sensitivity: a statistical Monte Carlo approach and an analytic calculation. We show that despite the different log-sensitivity calculations employed in this study, both demonstrate that the log-sensitivity of the fidelity error to dephasing results in a conventional trade-off between performance and robustness.
Abstract As shown in previous work, in some cases closed quantum systems exhibit a non-conventional absence of trade-off between performance and robustness in the sense that controllers with the highest fidelity can also provide the best robustness to parameter uncertainty. As the dephasing induced by the interaction of the system with the environment guides the evolution to a more classically mixed state, it is worth investigating what effect the introduction of dephasing has on the relationship between performance and robustness. In this paper we analyze the robustness of the fidelity error, as measured by the logarithmic sensitivity function, to dephasing processes. We show that introduction of dephasing as a perturbation to the nominal unitary dynamics requires a modification of the log-sensitivity formulation used to measure robustness about an uncertain parameter with nonzero nominal value used in previous work. We consider controllers optimized for a number of target objectives ranging from fidelity under coherent evolution to fidelity under dephasing dynamics to determine the extent to which optimizing for a specific regime has desirable effects in terms of robustness. Our analysis is based on two independent computations of the log-sensitivity: a statistical Monte Carlo approach and an analytic calculation. We show that despite the different log-sensitivity calculations employed in this study, both demonstrate that the log-sensitivity of the fidelity error to dephasing results in a conventional trade-off between performance and robustness.
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.