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arXiv:2004.07050 · IEEE Transactions on Cybernetics

Hybrid Filtering for a Class of Nonlinear Quantum Systems Subject to Classical Stochastic Disturbances

Qi Yu, Daoyi Dong, Ian R. Petersen

Published May 4, 2020• 9 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1109/tcyb.2020.2985016

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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.
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.
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.
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.
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.
The cavity is a fundamental ingredient of quantum optical systems. This paper concerns the behavior of a quantum cavity driven by non-classical field in single-photon state. To this end, the number operator has been opted to reveal the number of photons inside the cavity. Then, the quantum filtering equations have been employed to derive a stochastic master equation for the cavity which is driven by single-photon and is observed by either Homodyne or photon-counting detector. Finally, the state of the cavity has been estimated by the derived equations, and the results have been compared with the conventional master equation.
This paper introduces a Lyapunov-based control strategy alongside two filtering methods for controlling and estimating the evolution of coherence vector elements from sequential homodyne measurements. The methods include traditional quantum filtering and a novel extended Kalman filter, which explicitly addresses the dynamics of a stochastic master equation with correlated noise, thereby ensuring the quantum properties of the estimated state variable by design. We also explore scenarios where the system’s Hamiltonian is unknown, demonstrating that both filters exhibit reduced performance with increased estimation errors. To address this, we propose a multiple model estimation scheme applicable to either filter. The estimated density operator is then controlled using the proposed switching-based Lyapunov scheme, which guarantees noise-to-state practical stability in probability. We demonstrate the effectiveness of our approach in stabilizing a qubit coupled to a leaky cavity under homodyne detection, even with uncertainty in the resonance frequency.