Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:1603.01890 · Journal of Physics A Mathematical and Theoretical

A quantum extended Kalman filter

Muhammad F. Emzir, Matthew J. Woolley, Ian R. Petersen

Published Apr 21, 2017• 30 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1088/1751-8121/aa6e5e

Related papers

State Estimation of a Quantum System Using Extended Kalman Filter

Nidhi Agarwal, Akanksha Sondhi, Ghanapriya Singh

In this paper we propose to estimate the state of a quantum system dynamically evolving in the presence of white Gaussian noise (WGN) under Schrödinger dynamics using Extended Kalman Filter (EKF) algorithm. In the proposed measurement model the EKF is applied to obtain the conditional expectation of the state at a given observation time up to that time. First the complex Schrödinger equation of dimension N is transformed into a real Schrödinger equation of dimension 2N. MATLAB programs were implemented to plot the wave function, error and noise to signal ratio (NSR). NSR within minimum bounds is obtained.

International Conference on Cutting-edge Technologies in Engineering (ICon-CuTE)•Nov 1, 2019•1 citation

State Estimation and Control for Stochastic Quantum Dynamics With Homodyne Measurement: Stabilizing Qubits Under Uncertainty

Nahid Binandeh Dehaghani, A. Pedro Aguiar, Rafał Wiśniewski

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.

arXiv:2403.07021•IEEE Access•Jan 1, 2024•2 citations

Review of the Applications of Kalman Filtering in Quantum Systems

Kezhao Ma, Jia Kong, Yihan Wang +1 more

State variable and parameter estimations are important for signal sensing and feedback control in both traditional engineering systems and quantum systems. The Kalman filter, which is one of the most popular signal recovery techniques in classical systems for decades, has now been connected to the stochastic master equations of linear quantum mechanical systems. Various studies have invested effort on mapping the state evolution of a quantum system into a set of classical filtering equations. However, establishing proper evolution models with symmetry to classical filter equation for quantum systems is not easy. Here, we review works that have successfully built a Kalman filter model for quantum systems and provide an improved method for optimal estimations. We also discuss a practical scenario involving magnetic field estimations in quantum systems, where non-linear Kalman filters could be considered an estimation solution.

Symmetry•Nov 22, 2022•13 citations

Quantum-Inspired Stochastic Differential Equations (QSDEs)

Jincheng Zhang

This paper introduces Quantum-Inspired Stochastic Differential Equations (QSDEs), a novel approach to modeling stochastic systems by leveraging core concepts from quantum mechanics. Traditional Stochastic Differential Equations (SDEs) often rely on Gaussian noise, which can be a significant limitation when accurately representing complex, real-world phenomena. QSDEs propose a fundamental shift by incorporating superposition and entanglement principles to model noise, moving beyond Gaussian distributions. The core mechanism involves introducing a "quantum noise" term, denoted as *η(t, x, P)*, where *η* is a superposition of probability distributions dynamically adjusted based on the system's state *x*. This allows for a more nuanced representation of uncertainty, potentially leading to vastly improved simulations across diverse fields. We outline the theoretical framework, explore the key mathematical formulations, and discuss the potential impact of QSDEs on areas such as financial modeling, physics simulations, and control systems. The central claim of this work is that classical SDEs can be significantly enhanced by incorporating quantum mechanical principles.

•Sep 7, 2026•0 citations

Hybrid Filtering for a Class of Quantum Systems with Classical Disturbances

Qi Yu, Daoyi Dong, Ian R. Petersen +1 more

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.

arXiv:1703.08976•IFAC-PapersOnLine•Jul 1, 2017•5 citations

A Quantum Kalman Filter-Based PID Controller

John Edward Gough

We give a concrete description of a controlled quantum stochastic dynamical model corresponding to a quantum system (a cavity mode) undergoing continual quadrature measurements, with a PID controller acting on the filtered estimate for the mode operator. The control has three feedback paths: P (proportional), I (integral) and D (derivative) of the error signal. Central use is made of the input and output pictures when constructing the model: these unitarily equivalent pictures are presented in the paper, and used to transfer concepts relating to the controlled internal dynamics to those relating to measurement output, and vice versa. The approach shows the general principle for investigating mathematically and physically consistent models in which standard control theoretic methods are to be extended to the quantum setting.

arXiv:1701.06578•Open Systems & Information Dynamics•Sep 1, 2020•2 citations
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 propose to estimate the state of a quantum system dynamically evolving in the presence of white Gaussian noise (WGN) under Schrödinger dynamics using Extended Kalman Filter (EKF) algorithm. In the proposed measurement model the EKF is applied to obtain the conditional expectation of the state at a given observation time up to that time. First the complex Schrödinger equation of dimension N is transformed into a real Schrödinger equation of dimension 2N. MATLAB programs were implemented to plot the wave function, error and noise to signal ratio (NSR). NSR within minimum bounds is obtained.
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.
State variable and parameter estimations are important for signal sensing and feedback control in both traditional engineering systems and quantum systems. The Kalman filter, which is one of the most popular signal recovery techniques in classical systems for decades, has now been connected to the stochastic master equations of linear quantum mechanical systems. Various studies have invested effort on mapping the state evolution of a quantum system into a set of classical filtering equations. However, establishing proper evolution models with symmetry to classical filter equation for quantum systems is not easy. Here, we review works that have successfully built a Kalman filter model for quantum systems and provide an improved method for optimal estimations. We also discuss a practical scenario involving magnetic field estimations in quantum systems, where non-linear Kalman filters could be considered an estimation solution.
This paper introduces Quantum-Inspired Stochastic Differential Equations (QSDEs), a novel approach to modeling stochastic systems by leveraging core concepts from quantum mechanics. Traditional Stochastic Differential Equations (SDEs) often rely on Gaussian noise, which can be a significant limitation when accurately representing complex, real-world phenomena. QSDEs propose a fundamental shift by incorporating superposition and entanglement principles to model noise, moving beyond Gaussian distributions. The core mechanism involves introducing a "quantum noise" term, denoted as *η(t, x, P)*, where *η* is a superposition of probability distributions dynamically adjusted based on the system's state *x*. This allows for a more nuanced representation of uncertainty, potentially leading to vastly improved simulations across diverse fields. We outline the theoretical framework, explore the key mathematical formulations, and discuss the potential impact of QSDEs on areas such as financial modeling, physics simulations, and control systems. The central claim of this work is that classical SDEs can be significantly enhanced by incorporating quantum mechanical principles.
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.
We give a concrete description of a controlled quantum stochastic dynamical model corresponding to a quantum system (a cavity mode) undergoing continual quadrature measurements, with a PID controller acting on the filtered estimate for the mode operator. The control has three feedback paths: P (proportional), I (integral) and D (derivative) of the error signal. Central use is made of the input and output pictures when constructing the model: these unitarily equivalent pictures are presented in the paper, and used to transfer concepts relating to the controlled internal dynamics to those relating to measurement output, and vice versa. The approach shows the general principle for investigating mathematically and physically consistent models in which standard control theoretic methods are to be extended to the quantum setting.