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Quantum projection filtering

ZT Wu, Qing Zhong Gao, GF Zhang

Published Jan 1, 2017• 3 citations
Publisher

Abstract

A quantum filter extracts the optimal estimate of quantum system states from the past history of weak continuous measurements on a quantum system by using the quantum Bayesian theory. However, the online computation burden of a quantum filter increases quadratically along with the increase of the quantum system dimension. We propose a quantum projection filtering method, aiming to reduce the online computation burden of the quantum filter. By using a differential geometric approach, the trajectory of the resulting quantum filter is constrained to be evolving within a finite-dimensional differentiable submanifold. In other words, the quantum state can be calculated through online calculation of the coordinate system of this submanifold. Simulation results from a two-level quantum system demonstrate the effectiveness of the proposed method.

DOI
10.7641/cta.2017.70717

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A quantum filter extracts the optimal estimate of quantum system states from the past history of weak continuous measurements on a quantum system by using the quantum Bayesian theory. However, the online computation burden of a quantum filter increases quadratically along with the increase of the quantum system dimension. We propose a quantum projection filtering method, aiming to reduce the online computation burden of the quantum filter. By using a differential geometric approach, the trajectory of the resulting quantum filter is constrained to be evolving within a finite-dimensional differentiable submanifold. In other words, the quantum state can be calculated through online calculation of the coordinate system of this submanifold. Simulation results from a two-level quantum system demonstrate the effectiveness of the proposed method.
This paper presents an approximation quantum projection filtering strategy, aiming to reduce the computational cost in calculating the standard quantum filter equation in time. By using a differential geometric approach, the trajectory of the resulting quantum projection filter is constrained to evolve in a finite-dimensional differentiable manifold consisting of an exponential family of quantum density operators. A convenient design of the differentiable manifold is then developed through reduction of the local approximation errors, which allows simplification of the quantum projection filter equations. Finally, simulation results from a two-level quantum system example illustrate the approximation performance of the proposed filtering scheme. The proposed approach is expected to be of practical use in developing more efficient quantum control methods.
In order to reduce the computational cost for online calculation of the quantum filter, in this paper, a linear quantum projection filtering scheme is developed by projecting the time evolution trajectory of quantum system state onto a lower-dimensional linear submanifold spanned by a set of basis matrices. The lower-dimensional submanifold is obtained through an extended singular value decomposition algorithm. Compared with the existing works on quantum projection filtering, the proposed approach requires no prior knowledge on the structure of the submanifold, and thus can be used to deal with general open quantum systems. Simulation results demonstrate that the proposed linear projection filter is able to approximate the high-dimensional quantum filter within acceptable accuracy.
This article develops a quantum projection filtering approach to approximating the quantum filter equation based on Ito^\hat{\rm o}o^ stochastic Taylor expansions and quantum information geometric techniques. The proposed approximation scheme is designed so that the truncated Taylor expansion of the difference between the true quantum trajectory, and its approximation on a lower dimensional differential submanifold is minimized, through an orthogonal projection operation in the quantum Fisher metric. In addition, a convenient design for a special class of open quantum systems is formulated. Simulation results from a numerical example demonstrate the approximation capability of the proposed quantum projection filter.
Bayesian quantum estimation provides a robust formalism for quantum technologies, particularly in scenarios with limited data and minimal prior information. Yet, its application to continuous-variable systems has remained limited and largely numerical due to the difficulty of the underlying parameter integrals. Here, we introduce a variational framework that restricts the optimisation over all measurements and estimators to a finite-dimensional subspace, reducing the problem to a linear system of equations with closed-form solutions. These solutions have a geometric interpretation as orthogonal projections of the global optimum onto the chosen subspace. For Gaussian states, a natural subspace is spanned by operators polynomial in the canonical quadratures, whose moments are computable from the first and second moments of the state. We further derive a necessary and sufficient condition for global optimality, and show that post-processing with the posterior mean, rather than using the estimator obtained from the subspace, improves performance towards the global optimum. Through single-shot examples, we show that the framework yields experimentally feasible strategies based on Gaussian operations and quadrature measurements that are either optimal or near-optimal.
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In this letter, we consider an open quantum system undergoing imperfect and indirect measurement. For quantum non-demolition (QND) measurement, we show that the system evolves on an appropriately chosen manifold and we express the exact solution of the quantum filter equation in terms of the solution of a lower dimensional stochastic differential equation. In order to further reduce the dimension of the system under study, we consider the projection on the lower dimensional manifold originally introduced in Gao et al. (2019) for the case of perfect measurements. An error analysis is performed to evaluate the precision of this approximate quantum filter, focusing on the case of QND measurement. Simulations suggest the efficiency of the proposed quantum projection filter, even in presence of a stabilizing feedback control which depends on the projection filter.