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Society of Instrument and Control Engineers of Japan

Convergence analysis of adaptive tuning parameter for robust stabilizing control of $N$ -level quantum systems

Shoju Enami, Kentaro Ohki

Published Sep 8, 2021• 4 citations
OpenAlex

Abstract

This paper analyzes how the weight function affects the properties of convergence: where and how fast the estimate converges. Stabilization control of quantum systems has been getting much attention as research in quantum information technology progresses, and of course, it is necessary to consider uncertainties in practice. In order to deal with a large uncertain quantum system, we recently proposed an adaptive parameter tuning algorithm based on the study of robust stabilization of $N$ -level quantum systems by Liang, Amini, and Mason (SIAM J. Control Optim., 59(1), 669–692, 2021). In this paper, we briefly review our proposed method and analyze its convergence performance of the estimated parameter. The convergence analysis shows that certain conditions on the weight function are necessary for proper tuning, and the asymptotic convergence rate depends on only the choice of the function.

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Convergence analysis of adaptive tuning parameter for robust stabilizing control of NNN -level quantum systems
This paper analyzes how the weight function affects the properties of convergence: where and how fast the estimate converges. Stabilization control of quantum systems has been getting much attention as research in quantum information technology progresses, and of course, it is necessary to consider uncertainties in practice. In order to deal with a large uncertain quantum system, we recently proposed an adaptive parameter tuning algorithm based on the study of robust stabilization of NNN -level quantum systems by Liang, Amini, and Mason (SIAM J. Control Optim., 59(1), 669–692, 2021). In this paper, we briefly review our proposed method and analyze its convergence performance of the estimated parameter. The convergence analysis shows that certain conditions on the weight function are necessary for proper tuning, and the asymptotic convergence rate depends on only the choice of the function.
Stabilizing control synthesis is one of the central subjects in control theory and engineering, and it always has to deal with unavoidable uncertainties in practice. In this study, we propose an adaptive parameter tuning algorithm for robust stabilizing quantum feedback control of N-level quantum angular momentum systems with a robust stabilizing controller proposed by [Liang, Amini, and Mason, SIAM J. Control Optim., 59 (2021), pp. 669-692]. The proposed method ensures local convergence to the target state. Besides, numerical experiments indicate its global convergence if the learning parameters are adequately determined.
In this paper, we consider NNN-level quantum angular momentum systems interacting with electromagnetic fields undergoing quantum nondemolition measurements in continuous-time. We suppose unawareness of the initial state and physical parameters, entailing the introduction of an additional state representing the estimated quantum state. The evolution of the quantum state and its estimation is described by a coupled stochastic master equation. Here, we study the asymptotic behavior of such a system in the presence of a feedback controller. We provide sufficient conditions on the feedback controller and on the estimated parameters that guarantee exponential stabilization of the coupled stochastic system toward an eigenstate of the measurement operator. Furthermore, we estimate the corresponding rate of convergence. We also provide parametrized feedback laws satisfying such conditions. Our results show the robustness of the feedback stabilization strategy considered in [W. Liang, N. H. Amini, and P. Mason, SIAM J. Control Optim., 57 (2019), pp. 3939--3960] in the case of imprecise initialization of the estimated state and with respect to the unknown physical parameters.
This paper proposes and analyzes an adaptive quantum feedback control system designed to optimize the performance of quantum devices. The core of the system lies in a real-time monitoring and adjustment mechanism utilizing a quantum sensor to measure the state of the quantum system. This measurement data is then fed back into a control algorithm, which dynamically adjusts the system parameters to achieve precise control. Unlike traditional control systems that are sensitive to parameter uncertainties, this adaptive approach leverages the inherent properties of quantum systems to enhance stability and performance. The theoretical framework presented here establishes a foundation for future research and development in adaptive quantum control, addressing key challenges in quantum device operation and manipulation. The system aims to provide a robust and flexible solution for controlling quantum systems with varying characteristics. The primary focus is on the control loop design and the feedback algorithm implementation, with an emphasis on achieving optimal performance under conditions of parameter uncertainty. This work contributes to the growing field of quantum control by introducing a novel adaptive strategy that can significantly improve the operational capabilities of quantum devices.
Despite significant progress in theoretical and laboratory quantum control, engineering quantum systems remains principally challenging due to manifestation of noise and uncertainties associated with the field and Hamiltonian parameters. In this paper, we extend and generalize the asymptotic quantum control robustness analysis method, which provides more accurate estimates of quantum control objective moments than standard leading-order techniques, to diverse quantum observables, gates, and moments thereof and also introduce the Pontryagin maximum principle for quantum robust control. In addition, we present a Pareto optimization framework for achieving robust control via evolutionary open-loop (model-based) and closed-loop (model-free) approaches with the mechanisms of robustness and convergence described using asymptotic quantum control robustness analysis. In the open-loop approach, a multiobjective genetic algorithm is used to obtain Pareto solutions in terms of the expectation and variance of the transition probability under Hamiltonian parameter uncertainty. The set of numerically determined solutions can then be used as a starting population for model-free learning control in a feedback loop. The closed-loop approach utilizes a real-coded genetic algorithm with adaptive exploration and exploitation operators in order to preserve solution diversity and dynamically optimize the transition probability in the presence of field noise. Together, these methods provide a foundation for high-fidelity adaptive feedback control of quantum systems wherein open-loop control predictions are iteratively improved based on data from closed-loop experiments.
This paper presents a systematic method to analyze the robustness of uncertain linear stochastic quantum systems . Uncertainties are studied in the optical realization, where all the corresponding parameters are affected. First, robustly stable LSQSs are defined and then, by the use of the uncertainty decomposition algorithm, robust stability of the LSQSs is analyzed within two different approaches. In the first approach, a sufficient small-gain-like theorem is presented and in the second approach, another sufficient condition for robust stability based on Lyapunov theory is presented. The robust stability analysis is validated via a case study.
In this paper, a class of uncertain quantum systems with uncertainties in the interaction Hamiltonian is connected with a coherent controller through direct and indirect couplings. The aim of the paper is to design a robust coherent stability controller to make the closed-loop system robustly mean square stable. A sufficient and necessary condition is presented to build a connection between robust control design with uncertain parameters and scaled control design without uncertain parameters. This condition is also considered for a special class of linear passive quantum systems. A numerical procedure is proposed to obtain explicit expression for parameters of the desired coherent controller using linear matrix inequality, multi-step optimization methods and physical realizability conditions.