H∞ Control of Linear Quantum Stochastic Systems

Abstract

The purpose of this paper is to formulate and solve a H ∞ controller synthesis problem for a class of non-commutative linear stochastic systems which includes many examples of interest in quantum technology. The paper includes results on the class of such systems for which the quantum commutation relations are preserved (such a requirement must be satisfied in a physical quantum system). A quantum version of standard (classical) dissipativity results are presented and from this a quantum version of the Strict Bounded Real Lemma is derived. This enables a quantum version of the two Riccati solution to the H ∞ control problem to be presented. This result leads to controllers which may be realized using purely quantum, purely classical or a mixture of quantum and classical elements. This issue of physical realizability of the controller is examined in detail, and necessary and sufficient conditions are given. Our results are constructive in the sense that we provide explicit formulas for the Hamiltonian function and coupling operator corresponding to the controller.

Robust H ∞ controller design for a class of linear quantum systems with time delay

“…Quantum control engineering is a new branch of control engineering focusing on the control of physical systems whose behaviour is dominated by the laws of quantum mechanics [1,2]. Quantum technology exploiting the applications of quantum science with unique characteristics, for example, entanglement and coherence, demands novel principles of quantum control theory and provides a great impetus for research in the area of quantum feedback control systems [3][4][5][6][7]. However, in a quantum feedback control loop, it is inevitably confronted with a time delay because of, for example, measurement operation and transmission time [8,9].…”

“…However, in a quantum feedback control loop, it is inevitably confronted with a time delay because of, for example, measurement operation and transmission time [8,9]. Although the time delay appearing in feedback loops is often neglected in existing research on quantum control [3,[10][11][12], it is indeed a source of instability of quantum feedback control systems, and ignoring it may lead to incorrect analysis conclusions and design flaws. Therefore, research on quantum systems with time delays has practical significance and has received considerable attention, especially in quantum optics [13] where the problem of LQG control of linear stochastic quantum systems with time delays has been studied [14,15].…”

“…In recent years, H 1 control technique has been successfully extended to quantum feedback control systems. In Reference [3], a general framework of H 1…”

Finite horizon H∞ control for a class of sampled‐data linear quantum systems

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In this paper, a finite horizon H 1 control problem is solved for a class of linear quantum systems using a dynamic game approach for the case of sampled-data measurements. The methodology adopted involves an equivalence between the quantum problem and two auxiliary classical problems.An H 1 controller synthesis problem has been formulated for a class of linear quantum stochastic systems in [6]. In fact, H 1 methods are used in control theory to synthesize controllers in order to achieve robust performance and/or stabilization.

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“…This involved converting the equations involving complex annihilation and creation operators into a form involving the real quadratures.Another development is at the level of the H 1 control theory for a special class of quantum systems. Although the development of H 1 control theory for a more general class of linear quantum systems has been investigated in [6], the authors in [8] consider a special class of linear quantum systems (a special case of the one considered in [6]) and simplify the design of the H 1 controller for an optical cavity in [6] as the dimension of the complex Riccati equations to be solved to construct the controller is half the dimension of the corresponding real-Riccati equations required in [6] to construct the controller. Furthermore, the authors in [9] and [10] extend the theory of H 1 control to the finite horizon case for continuous time quantum systems and quantum systems with delayed measurements, respectively.In this paper, following the approach of [11], we solve a finite horizon H 1 control problem for a class of linear quantum systems for the case of sampled-data measurements.…”