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IEEE Transactions on Control Systems Technology

Filtering for a Class of Quantum Systems With Classical Stochastic Disturbances

Qi Yu, Daoyi Dong, Ian R. Petersen, Muhammad F. Emzir, Qing Zhong Gao

Published Oct 9, 2019• 10 citations
Publisher

Abstract

We present filtering equations for a class of hybrid quantum-classical systems where a quantum optical cavity system is continuously disturbed by a classical linear stochastic process. Another quantum cavity system is employed to describe the classical stochastic process as an analog. The initial hybrid quantum-classical system consisting of both a classical signal and a quantum cavity system is then modeled by a combined quantum system with two quantum cavity systems. The stochastic master equations, or filtering equations, are then derived using quantum filtering theory. A quantum extended Kalman filter method is adopted as a convenient solution to this filtering problem. The effectiveness of both methods is illustrated by numerical examples.

DOI
10.1109/tcst.2018.2868764

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We present filtering equations for a class of hybrid quantum-classical systems where a quantum optical cavity system is continuously disturbed by a classical linear stochastic process. Another quantum cavity system is employed to describe the classical stochastic process as an analog. The initial hybrid quantum-classical system consisting of both a classical signal and a quantum cavity system is then modeled by a combined quantum system with two quantum cavity systems. The stochastic master equations, or filtering equations, are then derived using quantum filtering theory. A quantum extended Kalman filter method is adopted as a convenient solution to this filtering problem. The effectiveness of both methods is illustrated by numerical examples.
A hybrid quantum-classical filtering problem, where a qubit system is disturbed by a classical stochastic process, is investigated. The strategy is to model the classical disturbance by using an optical cavity. The relations between classical disturbances and the cavity analog system are analyzed. The dynamics of the enlarged quantum network system, which includes a qubit system and a cavity system, are derived. A stochastic master equation for the qubit-cavity hybrid system is given, based on which estimates for the state of the cavity system and the classical signal are obtained. The quantum-extended Kalman filter is employed to achieve efficient computation. The numerical results are presented to illustrate the effectiveness of our methods.
A filtering problem for a class of quantum systems disturbed by a classical stochastic process is investigated in this paper. The classical disturbance process, which is assumed to be described by a linear stochastic differential equation, is modeled by a quantum cavity model. Then the hybrid quantum-classical system is described by a combined quantum system consisting of two quantum cavity subsystems. Quantum filtering theory and a quantum extended Kalman filter method are employed to estimate the states of the combined quantum system. An estimate of the classical stochastic process is derived from the estimate of the combined quantum system.
In this paper, we consider the filtering problem for a hybrid system where a quantum qubit system is disturbed by a classical signal. The quantum filtering theory, which is based on quantum probability theory, can not be directly applied to a hybrid system where a classical stochastic process is also needed in describing the system dynamics. An optical cavity system is employed to model the classical disturbance. By designing the parameters of the auxiliary cavity system, the expectation of the quadrature operator of the cavity shares the same dynamics with the classical signal. With this correspondence guaranteed, one can obtain the real time expectation of the classical signal. The quantum concatenation product is adopted to describe the quantum system which contains both the qubit subsystem and the cavity subsystem. A stochastic master equation, which provides estimates for the quantum state and the classical signal, is given. To reduce the computational complexity, the quantum extended Kalman filter is also applied to this system.
Abstract In quantum physics, a stochastic master equation (SME) estimates the state (density operator) of a quantum system in the Schrödinger picture based on a record of measurements made on the system. In the Heisenberg picture, the SME is a quantum filter. For a linear quantum system subject to linear measurements and Gaussian noise, the dynamics may be described by quantum stochastic differential equations (QSDEs), also known as quantum Langevin equations, and the quantum filter reduces to a so-called quantum Kalman filter. In this article, we introduce a quantum extended Kalman filter (quantum EKF), which applies a commutative approximation and a time-varying linearization to systems of nonlinear QSDEs. We will show that there are conditions under which a filter similar to a classical EKF can be implemented for quantum systems. The boundedness of estimation errors and the filtering problem with ‘state-dependent’ covariances for process and measurement noises are also discussed. We demonstrate the effectiveness of the quantum EKF by applying it to systems that involve multiple modes, nonlinear Hamiltonians, and simultaneous jump-diffusive measurements.
In this paper we consider an alternative formulation of a class of stochastic wave and master equations with scalar noise that are used in quantum optics for modelling open systems and continuously monitored systems. The reformulation is obtained by applying J.M.C. Clark's pathwise reformulation technique from the theory of classical nonlinear filtering. The pathwise versions of the stochastic wave and master equations are defined for all driving paths and depend continuously on them. In the case of white noise equations, we derive analogs of Clark's robust approximations. The results in this paper may be useful for implementing filters for the continuous monitoring and measurement feedback control of quantum systems, and for developing new types of numerical methods for unravelling master equations. The main ideas are illustrated by an example.
In this paper we consider an alternative formulation of a class of stochastic wave and master equations with scalar noise that are used in quantum optics for modelling open systems and continuously monitored systems. The reformulation is obtained by applying J.M.C. Clark's pathwise reformulation technique from the theory of classical nonlinear filtering. The pathwise versions of the stochastic wave and master equations are defined for all driving paths and depend continuously on them. In the case of white noise equations, we derive analogs of Clark's robust approximations. The results in this paper may be useful for implementing filters for the continuous monitoring and measurement feedback control of quantum systems, and for developing new types of numerical methods for unravelling master equations. The main ideas are illustrated by an example.