A quantum extended Kalman filter

Abstract

A stochastic filter uses a series of measurements over time to produce estimates of unknown variables based on a dynamic model [1]. For a quantum system, such an algorithm is provided by a quantum filter [2], which is also known as a stochastic master equation (SME) [3]. For a linear quantum system subject to linear measurements and Gaussian noise, the quantum filter reduces to a quantum Kalman filter [4,5]. 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.

Filtering for a Class of Quantum Systems With Classical Stochastic Disturbances

“…The system observables, which represent the physical properties of the system, are represented by selfadjoint operators on H . The quantum state, which provides the status of a physical system, is specified by a density operator ρ ∈ S(H ), where S is the class of unity trace operators on the associated Hilbert space (Emzir et al, 2016). In this paper, the evolution of the quantum system is mostly described under the Heisenberg picture.…”

“…The fact that the computation time in simulating the filter SME scales exponentially with the dimension of the Hilbert space adds difficulty to implementing the filter in realtime. A quantum extended Kalman filter (QEKF) was introduced in (Emzir et al, 2016), aiming to reduce the computational complexity of the quantum filter. For the QEKF, the constraint that elements of observable operator vector x(t) belong to a commutative von Neumann algebra is not required and there can be noncommutating operators in the dynamic equation.…”

“…Keeping the first order term of the Taylor series, the Kalman filter gain is effectively calculated. This method was proposed to solve the filtering problem for a class of multiple output channel open quantum systems whose evolution can be described by the following QSDE (Emzir et al, 2016):…”

“…The operators L and S are the parameters from the (S, H, L) model which can be used to describe a multiple channel open quantum system (Emzir et al, 2016). The measurement dynamic equation is given by…”

Quantum mechanics and data assimilation

“…On the other hand, QMDA employs finite-rank approximations of the intrinsic evolution and measurement operators of such systems, realized through Koopman operator theory [9,10] and kernel methods for machine learning [11][12][13][14], with well-established convergence properties. It should be noted that while connections between quantum theory and data assimilation have been studied in the literature [15,16], these works have generally approached the problem of performing data assimilation for an actual physical quantum system. To our knowledge, the approach presented here, which combines the Koopman operator formalism with abstract quantum mechanical axioms to arXiv:1903.00612v2 [math-ph] 31 May 2019 construct a data assimilation algorithm for deterministic dynamical systems, as well as its approximation via machine-learning techniques, has not been studied elsewhere.…”