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Proceedings of the ... American Control Conference/Proceedings of the American Control Conference

Coherent control of linear quantum systems: A differential evolution approach

Hendra Gunawan Harno, Ian R. Petersen

Published Jun 1, 2010• 5 citations
Publisher

Abstract

In this paper, we propose a new method to construct an optimal coherent quantum controller, which is required to be physically realizable. This method is based on an evolutionary optimization method, namely the Differential Evolution approach. The aim is to provide a straightforward algorithm to deal with both nonlinear and nonconvex constraints arising in the quantum controller design. The solution to our problem involves the solutions of a complex algebraic Riccati equation and a Lyapunov equation. The efficacy of the proposed method is demonstrated through a case study on an entanglement control problem for an ideal quantum network comprising two cascaded optical parametric amplifiers.

DOI
10.1109/acc.2010.5531367

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In this paper, we propose a new method to construct an optimal coherent quantum controller, which is required to be physically realizable. This method is based on an evolutionary optimization method, namely the Differential Evolution approach. The aim is to provide a straightforward algorithm to deal with both nonlinear and nonconvex constraints arising in the quantum controller design. The solution to our problem involves the solutions of a complex algebraic Riccati equation and a Lyapunov equation. The efficacy of the proposed method is demonstrated through a case study on an entanglement control problem for an ideal quantum network comprising two cascaded optical parametric amplifiers.
We propose a new method to construct an optimal linear coherent quantum controller based on an evolutionary optimization method, namely a differential evolution algorithm. The aim is to provide a straightforward approach to deal with both nonlinear and nonconvex constraints arising in the coherent quantum controller synthesis. The solution to this control problem involves a complex algebraic Riccati equation, which corresponds to a physical realizability condition for the coherent quantum controller. The proposed method is demonstrated through an example of an entanglement control problem for a quantum network comprising two cascaded optical parametric amplifiers.
Coherent feedback H∞H^\inftyH∞ control of quantum linear systems
The purpose of this paper is to investigate the coherent feedback H∞H^\inftyH∞ control problem for linear quantum systems. A key contribution is a simplified design methodology that guarantees closed-loop stability and a prescribed level of disturbance attenuation. It is shown that for general linear quantum systems, a physically realizable quantum controller can be obtained by solving at most four Lyapunov equations. In the passive case, a necessary and sufficient condition is provided in terms of two uncoupled pairs of Lyapunov equations. These results represent a significant simplification over the standard approach, which requires solving two coupled algebraic Riccati equations. The effectiveness of the proposed method is demonstrated through two typical quantum optical devices: an empty optical cavity and a degenerate parametric amplifier. These results provide a computationally efficient procedure for the robust and optimal control of quantum optical and optomechanical systems.
In this article, we propose a differential evolution (DE) algorithm specifically tailored for the design of linear-quadratic-Gaussian (LQG) controllers in quantum systems. Building upon the foundational DE framework, the algorithm incorporates specialized modules, including relaxed feasibility rules, a scheduled penalty function, adaptive search range adjustment, and the "bet-and-run" initialization strategy. These enhancements improve the algorithm's exploration and exploitation capabilities while addressing the unique physical realizability requirements of quantum systems. The proposed method is applied to a quantum optical system, where three distinct controllers with varying configurations relative to the plant are designed. The resulting controllers demonstrate superior performance, achieving lower LQG performance indices compared to existing approaches. In addition, the algorithm ensures that the designs comply with physical realizability constraints, guaranteeing compatibility with practical quantum platforms. The proposed approach holds significant potential for application to other linear quantum systems in performance optimization tasks subject to physically feasible constraints.
A coherent quantum controller is itself a quantum system that is required to be physically realizable. Thus, additional non-linear and linear constraints must be imposed on the coefficients of a physically realizable quantum controller, which differs the quantum Linear Quadratic Gaussian (LQG) design from the standard LQG problem. The purpose of this paper is to propose one numerical procedure based on extended linear matrix inequality (LMI) approach and new physical realizability conditions proposed in [14] to design a coherent quantum controller. The extended synthesis linear matrix inequalities are, in addition to new analysis tools, less conservative in comparison to the conventional counterparts since the optimization variables related to the system parameters in extended LMIs are independent of the symmetric Lyapunov matrix. These features may be useful in the optimal design of quantum optical networks. For comparison, we apply our numerical procedure to the same example given in [9].
The coherent robust control problem for a class of linear quantum passive systems with model uncertainties is considered in this study, where both the plant and the controller are described by quantum stochastic differential equations (QSDEs). In the framework of language, the model uncertainties are translated into the uncertainties in the Hamiltonian, the coupling operators, and the scattering matrices. Based on the quantum bounded real lemma, the robust controller design of the original uncertain plant is transformed into the controller design of a new scaled system without model uncertainties. Further, the latter is transformed into the solving problem of a couple of Riccati equations and therefore the numerical solutions of the coefficient matrices of QSDEs associated with the controller dynamics are obtained. Finally, numerical simulations on an optical system coupled to three optical channels are performed to verify the effectiveness of the method proposed in this study.
This paper is concerned with the Coherent Quantum Linear Quadratic Gaussian (CQLQG) control problem of finding a stabilizing measurement-free quantum controller for a quantum plant so as to minimize an infinite-horizon mean square performance index for the fully quantum closed-loop system. In comparison with the observation-actuation structure of classical controllers, the coherent quantum feedback is less invasive to the quantum dynamics and quantum information. Both the plant and the controller are open quantum systems whose dynamic variables satisfy the canonical commutation relations (CCRs) of a quantum harmonic oscillator and are governed by linear quantum stochastic differential equations (QSDEs). In order to correspond to such oscillators, these QSDEs must satisfy physical realizability (PR) conditions, which are organised as quadratic constraints on the controller matrices and reflect the preservation of CCRs in time. The CQLQG problem is a constrained optimization problem for the steady-state quantum covariance matrix of the plant-controller system satisfying an algebraic Lyapunov equation. We propose a gradient descent algorithm equipped with adaptive stepsize selection for the numerical solution of the problem. The algorithm finds a local minimum of the LQG cost over the parameters of the Hamiltonian and coupling operators of a stabilizing PR quantum controller, thus taking the PR constraints into account. A convergence analysis of the proposed algorithm is presented.