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International Conference on Cutting-edge Technologies in Engineering (ICon-CuTE)

State Estimation of a Quantum System Using Extended Kalman Filter

Nidhi Agarwal, Akanksha Sondhi, Ghanapriya Singh

Published Nov 1, 2019• 1 citation
Publisher

Abstract

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.

DOI
10.1109/icon-cute47290.2019.8991472

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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.
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 extraction of weak signals plays a crucial role in quantum precision measurement, where the estimation results are often limited by low signal-to-noise ratios. Here, we demonstrate a parameter-estimation framework based on the adaptive extended Kalman filter for dynamic magnetic-field estimation in quantum systems using spin-noise measurements -- a challenging regime characterized by weak signals. By modeling the magnetic field as an unknown parameter, the proposed approach alleviates model dependence in state estimation. Furthermore, by introducing an adaptive algorithm with real-time noise estimation, our method overcomes the measurement noise intensity constraints of conventional extended Kalman filtering and enhances its practical applicability. Numerical simulations covering three representative magnetic-field dynamics validate the capability of the proposed framework, while experimental results demonstrate successful tracking of a seismo-magnetic-like signal beyond the intrinsic sensitivity of conventional spin-noise spectroscopy.
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.
The Quantum Kalman Filter (QKF) emerges as an innovative approach to address the challenges of state estimation in quantum systems, adapting classical Kalman filtering to the complexities of quantum mechanics. This paper examines the theoretical framework, practical implementation hurdles, and potential applications of QKF in quantum information processing and sensing. By incorporating quantum measurement principles and stochastic dynamics, QKF offers a solution for optimal state estimation in noisy quantum environments. The study presents simulation results and comparative analyses with classical Kalman filtering for tracking single targets, underscoring the advantages and limitations of QKF in advancing quantum technology applications.
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.
The measurement precision of the static atomic gravimetry is limited by white Gaussian noise in short term, which costs previous works an inevitable integration to reach the precision demanded. Here, we propose a statistical model based on the quantum projection noise and apply the associated Kalman filter with the waveform estimation in static atomic gravimetry. With the white Gaussian noise significantly removed by the Kalman-filter formalism, the measurement noise of the gravimetry is reshaped in short term and shows τ1/2 feature that corresponds to a random walk. During 200 h of static measurement of gravity, the atomic gravimeter using Kalman filter demonstrates a sensitivity as good as 0.6 nms2/s and highlights a precision of 1.7 nms2 at the measuring time of a single sample. The measurement noise achieved is also lower than the quantum projection limit below ∼30 s.