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IEEE Conference on Decision and Control (CDC)

Robust observer for uncertain linear quantum systems

Naoki Yamamoto

Published Jan 1, 2006• 17 citations
Publisher

Abstract

In the theory of quantum dynamical filtering, one of the biggest issues is that the underlying system dynamics represented by a quantum stochastic differential equation must be known exactly in order that the corresponding filter provides an optimal performance; however, this assumption is in general unrealistic. Therefore, in this paper we consider a class of linear quantum systems subject to time-varying norm-bounded parametric uncertainty and then propose a robust observer such that the variance of the estimation error is guaranteed to be within a certain bound. Although the proposed observer is different from the optimal filter in the sense of the least mean square error, it is demonstrated in a typical quantum control problem that the observer is fairly robust against a parametric uncertainty even when the other estimators, the optimal Kalman filter and the risk-sensitive observer, fail in the estimation due to the uncertain perturbation

DOI
10.1109/cdc.2006.377467

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In the theory of quantum dynamical filtering, one of the biggest issues is that the underlying system dynamics represented by a quantum stochastic differential equation must be known exactly in order that the corresponding filter provides an optimal performance; however, this assumption is in general unrealistic. Therefore, in this paper we consider a class of linear quantum systems subject to time-varying norm-bounded parametric uncertainty and then propose a robust observer such that the variance of the estimation error is guaranteed to be within a certain bound. Although the proposed observer is different from the optimal filter in the sense of the least mean square error, it is demonstrated in a typical quantum control problem that the observer is fairly robust against a parametric uncertainty even when the other estimators, the optimal Kalman filter and the risk-sensitive observer, fail in the estimation due to the uncertain perturbation
In the theory of quantum dynamical filtering, one of the biggest issues is that the underlying system dynamics represented by a quantum stochastic differential equation must be known exactly in order that the corresponding filter provides an optimal performance; however, this assumption is generally unrealistic. Therefore, in this paper, we consider a class of linear quantum systems subjected to time-varying norm-bounded parametric uncertainties and then propose a robust observer such that the variance of the estimation error is guaranteed to be within a certain bound. Although in the linear case much of classical control theory can be applied to quantum systems, the quantum robust observer obtained in this paper does not have a classical analog due to the system's specific structure with respect to the uncertainties. Moreover, by considering a typical quantum control problem, we show that the proposed robust observer is fairly robust against a parametric uncertainty of the system even when the other estimators---the optimal Kalman filter and risk-sensitive observer---fail in the estimation.
Quantum parameter estimation plays a key role in many fields like quantum computation, communication, and metrology. Optimal estimation allows one to achieve the most precise parameter estimates, but requires accurate knowledge of the model. Any inevitable uncertainty in the model parameters may heavily degrade the quality of the estimate. It is therefore desired to make the estimation process robust to such uncertainties. Robust estimation was previously studied for a varying phase, where the goal was to estimate the phase at some time in the past, using the measurement results from both before and after that time within a fixed time interval up to current time. Here, we consider a robust guaranteed-cost filter yielding robust estimates of a varying phase in real time, where the current phase is estimated using only past measurements. Our filter minimizes the largest (worst-case) variance in the allowable range of the uncertain model parameter(s) and this determines its guaranteed cost. It outperforms in the worst case the optimal Kalman filter designed for the model with no uncertainty, which corresponds to the center of the possible range of the uncertain parameter(s). Moreover, unlike the Kalman filter, our filter in the worst case always performs better than the best achievable variance for heterodyne measurements, which we consider as the tolerable threshold for our system. Furthermore, we consider effective quantum efficiency and effective noise power, and show that our filter provides the best results by these measures in the worst case.
We consider classical estimators for a class of physically realizable linear quantum systems. Optimal estimation using a complex Kalman filter for this problem has been previously explored. Here, we study robust H∞ estimation for uncertain linear quantum systems. The estimation problem is solved by converting it to a suitably scaled H∞ control problem. The solution is obtained in the form of two algebraic Riccati equations. Relevant examples involving dynamic squeezers are presented to illustrate the efficacy of our method. Copyright © 2016 John Wiley & Sons, Ltd.
The aim of this note is to study the quantum robust optimal control problem for a class of linear quantum systems with uncertainties. It is shown that such problem can be converted into a mixed linear quadratic Gaussian andH∞H^{\infty }H∞quantum control problem. A physical model described by complex quantum stochastic differential equations (CQSDEs) is presented for the quantum system with uncertainties. The quantum system described by CQSDEs is called a complex quantum system in terms of annihilation and creation operators (Maalouf and Petersen, 2011). For an uncertain quantum system, we build a connection with a scaled linear system without uncertainty. The desired quantum robust optimal control results are derived based on the connection. We provide one numerical procedure for quantum robust optimal controller synthesis and then provide an example. The resulting controller presents robustness and optimal performance.
This paper considers the problem of robust stability for a class of uncertain nonlinear quantum systems subject to unknown perturbations in the system Hamiltonian. The nominal system is a linear quantum system defined by a linear vector of coupling operators and a quadratic Hamiltonian. This paper extends previous results on the robust stability of nonlinear quantum systems to allow for quantum systems with dynamic uncertainties. These dynamic uncertainties are required to satisfy a certain quantum stochastic integral quadratic constraint. The robust stability condition is given in terms of a strict bounded real condition. This result is applied to the robust stability analysis of an optical parametric amplifier.