A Quantum Extended Kalman Filter

TL;DR

This paper introduces a quantum extended Kalman filter (EKF) for nonlinear quantum systems, enabling approximate real-time state estimation using linearization techniques similar to classical EKF, with demonstrated effectiveness on complex quantum systems.

Contribution

It develops a quantum EKF applying commutative approximation and linearization to non-commutative QSDEs, extending classical filtering methods to quantum systems.

Findings

Effective for systems with multiple modes and nonlinear Hamiltonians

Handles jump-diffusive measurements in quantum systems

Ensures bounded estimation errors under certain conditions

Abstract

A stochastic filter uses a series of measurements over time to produce estimates of unknown variables based on a dynamic model. For a quantum system, such an algorithm is provided by a quantum filter, which is also known as a stochastic master equation (SME). For a linear quantum system subject to linear measurements and Gaussian noise, the quantum filter reduces to a 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 non-commutative quantum stochastic differential equations (QSDEs). We will show that there are conditions under which a filter similar to the classical EKF can be implemented for quantum systems. The boundedness of estimation errors and the filtering problems 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 which involve multiple modes, nonlinear Hamiltonians and simultaneous jump-diffusive measurements.