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IEEE Transactions on Cybernetics

Design of a Discrete-Time Fault-Tolerant Quantum Filter and Fault Detector

Qing Zhong Gao, Daoyi Dong, Ian R. Petersen, Steven X. Ding

Published Mar 1, 2019• 23 citations
Publisher

Abstract

This paper solves the problem of discrete-time fault-tolerant quantum filtering for a class of laser-atom open quantum systems subject to the stochastic faults. We show that by using the discrete-time quantum measurements, optimal estimates of both the atomic observables and the classical fault process can be simultaneously determined in terms of recursive quantum stochastic difference equations. A dispersive interaction quantum system example is used to demonstrate the proposed filtering approach.

DOI
10.1109/tcyb.2019.2899877

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This paper solves the problem of discrete-time fault-tolerant quantum filtering for a class of laser-atom open quantum systems subject to the stochastic faults. We show that by using the discrete-time quantum measurements, optimal estimates of both the atomic observables and the classical fault process can be simultaneously determined in terms of recursive quantum stochastic difference equations. A dispersive interaction quantum system example is used to demonstrate the proposed filtering approach.
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Continuous-time measurements are instrumental for a multitude of tasks in quantum engineering and quantum control, including the estimation of dynamical parameters of open quantum systems monitored through the environment. However, such measurements do not extract the maximum amount of information available in the output state, so finding alternative optimal measurement strategies is a major open problem. In this paper we solve this problem in the setting of discrete-time input-output quantum Markov chains. We present an efficient algorithm for optimal estimation of one-dimensional dynamical parameters which consists of an iterative procedure for updating a `measurement filter' operator and determining successive measurement bases for the output units. A key ingredient of the scheme is the use of a coherent quantum absorber as a way to post-process the output after the interaction with the system. This is designed adaptively such that the joint system and absorber stationary state is pure at a reference parameter value. The scheme offers an exciting prospect for optimal continuous-time adaptive measurements, but more work is needed to find realistic practical implementations.
This work considers the theory underlying a discrete-time quantum filter recently used in a quantum feedback experiment. It proves that this filter taking into account decoherence and measurement errors is optimal and stable. We present the general framework underlying the filter and show that it corresponds to a recursive expression of the least-square optimal estimation of the density operator in the presence of measurement imperfections. By measurement imperfections, we mean in a very general sense unread measurement performed by the environment (decoherence) and active measurement performed by non-ideal detectors. However, we assume to know precisely all the Kraus operators and also the detection error rates. Such recursive expressions combine well known methods from quantum filtering theory and classical probability theory (Bayes' law). We then demonstrate that such a recursive filter is always stable with respect to its initial condition: the fidelity between the optimal filter state (when the initial filter state coincides with the real quantum state) and the filter state (when the initial filter state is arbitrary) is a sub-martingale.