Quantum projection filter for a highly nonlinear model in cavity QED

Both in classical and quantum stochastic control theory a major role is played by the filtering equation, which recursively updates the information state of the system under observation. Unfortunately, the theory is plagued by infinite-dimensionality of the information state which severely limits its practical applicability, except in a few select cases (e.g. the linear Gaussian case.) One solution proposed in classical filtering theory is that of the projection filter. In this scheme, the filter is constrained to evolve in a finite-dimensional family of densities through orthogonal projection on the tangent space with respect to the Fisher metric. Here we apply this approach to the simple but highly nonlinear quantum model of optical phase bistability of a stongly coupled two-level atom in an optical cavity. We observe near-optimal performance of the quantum projection filter, demonstrating the utility of such an approach.Comment: 19 pages, 6 figures. A version with high quality images can be found at http://minty.caltech.edu/papers.ph

Always-On Quantum Error Tracking with Continuous Parity Measurements

We investigate quantum error correction using continuous parity measurements to correct bit-flip errors with the three-qubit code. Continuous monitoring of errors brings the benefit of a continuous stream of information, which facilitates passive error tracking in real time. It reduces overhead from the standard gate-based approach that periodically entangles and measures additional ancilla qubits. However, the noisy analog signals from continuous parity measurements mandate more complicated signal processing to interpret syndromes accurately. We analyze the performance of several practical filtering methods for continuous error correction and demonstrate that they are viable alternatives to the standard ancilla-based approach. As an optimal filter, we discuss an unnormalized (linear) Bayesian filter, with improved computational efficiency compared to the related Wonham filter introduced by Mabuchi [New J. Phys. 11, 105044 (2009)]. We compare this optimal continuous filter to two practical variations of the simplest periodic boxcar-averaging-and-thresholding filter, targeting real-time hardware implementations with low-latency circuitry. As variations, we introduce a non-Markovian ``half-boxcar'' filter and a Markovian filter with a second adjustable threshold; these filters eliminate the dominant source of error in the boxcar filter, and compare favorably to the optimal filter. For each filter, we derive analytic results for the decay in average fidelity and verify them with numerical simulations.Comment: 34 pages, 7 figures, published versio

A nonlinear filtering approach to changepoint detection problems:direct and differentialg-geometric methods

A benchmark change detection problem is considered which involves the detection of a change of unknown size at an unknown time. Both unknown quantities are modelled by stochastic variables, which allows the problem to be formulated within a Bayesian framework. It turns out that the resulting nonlinear filtering problem is much harder than the well-known detection problem for known sizes of the change, and in particular that it can no longer be solved in a recursive manner. An approximating recursive filter is therefore proposed, which is designed using differential-geometric methods in a suitably chosen space of unnormalized probability densities. The new nonlinear filter can be interpreted as an adaptive version of the celebrated Shiryayev-Wonham equation for the detection of a priori known changes, combined with a modified Kalman filter structure to generate estimates of the unknown size of the change. This intuitively appealing interpretation of the nonlinear filter and its excellent performance in simulation studies indicate that it may be of practical use in realistic change detection problems

Always-On Quantum Error Tracking with Continuous Parity Measurements

We investigate quantum error correction using continuous parity measurements to correct bit-flip errors with the three-qubit code. Continuous monitoring of errors brings the benefit of a continuous stream of information, which facilitates passive error tracking in real time. It reduces overhead from the standard gate-based approach that periodically entangles and measures additional ancilla qubits. However, the noisy analog signals from continuous parity measurements mandate more complicated signal processing to interpret syndromes accurately. We analyze the performance of several practical filtering methods for continuous error correction and demonstrate that they are viable alternatives to the standard ancilla-based approach. As an optimal filter, we discuss an unnormalized (linear) Bayesian filter, with improved computational efficiency compared to the related Wonham filter introduced by Mabuchi [New J. Phys. 11, 105044 (2009)]. We compare this optimal continuous filter to two practical variations of the simplest periodic boxcar-averaging-and-thresholding filter, targeting real-time hardware implementations with low-latency circuitry. As variations, we introduce a non-Markovian ``half-boxcar\u27\u27 filter and a Markovian filter with a second adjustable threshold; these filters eliminate the dominant source of error in the boxcar filter, and compare favorably to the optimal filter. For each filter, we derive analytic results for the decay in average fidelity and verify them with numerical simulations

The Hitchhiker's Guide to Nonlinear Filtering

Nonlinear filtering is the problem of online estimation of a dynamic hidden variable from incoming data and has vast applications in different fields, ranging from engineering, machine learning, economic science and natural sciences. We start our review of the theory on nonlinear filtering from the simplest `filtering' task we can think of, namely static Bayesian inference. From there we continue our journey through discrete-time models, which is usually encountered in machine learning, and generalize to and further emphasize continuous-time filtering theory. The idea of changing the probability measure connects and elucidates several aspects of the theory, such as the parallels between the discrete- and continuous-time problems and between different observation models. Furthermore, it gives insight into the construction of particle filtering algorithms. This tutorial is targeted at scientists and engineers and should serve as an introduction to the main ideas of nonlinear filtering, and as a segway to more advanced and specialized literature.Comment: 64 page

Nonlinear Stochastic Systems And Controls: Lotka-Volterra Type Models, Permanence And Extinction, Optimal Harvesting Strategies, And Numerical Methods For Systems Under Partial Observations

This dissertation focuses on a class of stochastic models formulated using stochastic differential equations with regime switching represented by a continuous-time Markov chain, which also known as hybrid switching diffusion processes. Our motivations for studying such processes in this dissertation stem from emerging and existing applications in biological systems, ecosystems, financial engineering, modeling, analysis, and control and optimization of stochastic systems under the influence of random environments, with complete observations or partial observations. The first part is concerned with Lotka-Volterra models with white noise and regime switching represented by a continuous-time Markov chain. Different from the existing literature, the Markov chain is hidden and canonly be observed in a Gaussian white noise in our work. We use a Wonham filter to estimate the Markov chain from the observable evolution of the given process, and convert the original system to a completely observable one. We then establish the regularity, positivity, stochastic boundedness, and sample path continuity of the solution. Moreover, stochastic permanence and extinction using feedback controls are investigated. The second part develops optimal harvest strategies for Lotka-Volterra systems so as to establish economically, ecologically, and environmentally reasonable strategies for populations subject to the risk of extinction. The underlying systems are controlled regime-switching diffusions that belong to the class of singular control problems. We construct upper bounds for the value functions, prove the finiteness of the harvesting value, and derive properties of the value functions. Then we construct explicit chattering harvesting strategies and the corresponding lower bounds for the value functions by using the idea of harvesting only one species at a time. We further show that this is a reasonable candidate for the best lower bound that one can expect. In the last part, we study optimal harvesting problems for a general systems in the case that the Markov chain is hidden and can only be observed in a Gaussian white noise. The Wonham filter is employed to convert the original problem to a completely observable one. Then we treat the resulting optimal control problem. Because the problem is virtually impossible to solve in closed form, our main effort is devoted to developing numerical approximation algorithms. To approximate the value function and optimal strategies, Markov chain approximation methods are used to construct a discrete-time controlled Markov chain. Convergence of the algorithm is proved by weak convergence method and suitable scaling

Reduced Complexity Filtering with Stochastic Dominance Bounds: A Convex Optimization Approach

This paper uses stochastic dominance principles to construct upper and lower sample path bounds for Hidden Markov Model (HMM) filters. Given a HMM, by using convex optimization methods for nuclear norm minimization with copositive constraints, we construct low rank stochastic marices so that the optimal filters using these matrices provably lower and upper bound (with respect to a partially ordered set) the true filtered distribution at each time instant. Since these matrices are low rank (say R), the computational cost of evaluating the filtering bounds is O(XR) instead of O(X2). A Monte-Carlo importance sampling filter is presented that exploits these upper and lower bounds to estimate the optimal posterior. Finally, using the Dobrushin coefficient, explicit bounds are given on the variational norm between the true posterior and the upper and lower bounds