State variable and parameter estimations are important for signal sensing and feedback control in both traditional engineering systems and quantum systems. The Kalman filter, which is one of the most popular signal recovery techniques in classical systems for decades, has now been connected to the stochastic master equations of linear quantum mechanical systems. Various studies have invested effort on mapping the state evolution of a quantum system into a set of classical filtering equations. However, establishing proper evolution models with symmetry to classical filter equation for quantum systems is not easy. Here, we review works that have successfully built a Kalman filter model for quantum systems and provide an improved method for optimal estimations. We also discuss a practical scenario involving magnetic field estimations in quantum systems, where non-linear Kalman filters could be considered an estimation solution.
State variable and parameter estimations are important for signal sensing and feedback control in both traditional engineering systems and quantum systems. The Kalman filter, which is one of the most popular signal recovery techniques in classical systems for decades, has now been connected to the stochastic master equations of linear quantum mechanical systems. Various studies have invested effort on mapping the state evolution of a quantum system into a set of classical filtering equations. However, establishing proper evolution models with symmetry to classical filter equation for quantum systems is not easy. Here, we review works that have successfully built a Kalman filter model for quantum systems and provide an improved method for optimal estimations. We also discuss a practical scenario involving magnetic field estimations in quantum systems, where non-linear Kalman filters could be considered an estimation solution.
The Quantum Kalman Filter (QKF) emerges as an innovative approach to address the challenges of state estimation in quantum systems, adapting classical Kalman filtering to the complexities of quantum mechanics. This paper examines the theoretical framework, practical implementation hurdles, and potential applications of QKF in quantum information processing and sensing. By incorporating quantum measurement principles and stochastic dynamics, QKF offers a solution for optimal state estimation in noisy quantum environments. The study presents simulation results and comparative analyses with classical Kalman filtering for tracking single targets, underscoring the advantages and limitations of QKF in advancing quantum technology applications.
The purpose of this tutorial is to give a brief introduction to linear quantum control systems. The mathematical model of linear quantum control systems is presented first, then some fundamental control-theoretic notions such as stability, controllability and observability are given, which are closely related to several important concepts in quantum information science such as decoherence-free subsystems, quantum non-demolition variables, and back-action evasion measurements. After that, quantum Gaussian states are introduced, in particular, an information-theoretic uncertainty relation is presented which often gives a better bound for mixed Gaussian states than the well-known Heisenberg uncertainty relation. The quantum Kalman filter is presented for quantum linear systems, which is the quantum analogy of the Kalman filter for classical (namely, non-quantum-mechanical) linear systems. The quantum Kalman canonical decomposition for quantum linear systems is recorded, and its application is illustrated by means of a recent experiment. As single- and multi-photon states are useful resources in quantum information technology, the response of quantum linear systems to these types of input is presented. Finally, coherent feedback control of quantum linear systems is briefly introduced, and a recent experiment is used to demonstrate the effectiveness of quantum linear systems and networks theory.
Quantum algorithms offer significant speed-ups over their classical counterparts in various applications. In this paper, we develop quantum algorithms for the Kalman filter widely used in classical control engineering using the block encoding method. The entire calculation process is achieved by performing matrix operations on Hamiltonians based on the block encoding framework, including addition, multiplication, and inversion, which can be completed in a unified framework compared to previous quantum algorithms for solving control problems. We demonstrate that the quantum algorithm exponentially accelerates the computation of the Kalman filter compared to traditional methods. The time complexity can be reduced from O(n3) to O(κpolylog(n/ε)log(1/ε′)), where n represents the matrix dimension, κ denotes the condition number for the matrix to be inverted, ε indicates desired precision in block encoding, ε′ signifies desired precision in matrix inversion. This paper provides a comprehensive quantum solution for implementing the Kalman filter and serves as an attempt to broaden the scope of quantum computation applications. Finally, we present an illustrative example implemented in Qiskit (a Python-based open-source toolkit) as a proof-of-concept.
Quantum state estimation, based on the numerical integration of stochastic master equations (SMEs), provides estimates for the evolution of quantum systems subject to continuous weak measurements. The approach is similar to classical state estimation methods in that the ‘quantum trajectories’ produced by solving the SME are conditioned on continuous classical measurement signals. In this paper, we explore the use of classical state estimation for a candidate quantum system, one based on an experimentally realisable system: a material object undergoing continuous feedback cooling in an optical trap. In particular, we demonstrate that classical tracking methods based on particle filters can be used to track quantum states, and are particularly useful for higher temperature regimes where quantum state estimation becomes computationally demanding.
This book examines quantum neural networks through renormalization techniques, supersymmetric field theory, and noisy harmonic oscillator systems. The book's analysis covers adaptive beamforming applications, brain modeling, gravitational control mechanisms, and mixed-state dynamics in superstring theory, and also includes: Comprehensive analysis of quantum neural networks through renormalization techniques and supersymmetric field theory applications in computational modeling Investigation of quantum field dynamics with noise integration, filtering mechanisms, and scattering processes in curved spacetime environments Study of adaptive beamforming methodologies combined with quantum neural networks for brain modeling and evolving field system applications Examination of mixed-state dynamics in superstring theory frameworks with emphasis on quantum noisy fields and supersymmetric effects Analysis of extended Kalman filter integration with quantum neural networks for transmission line control and field estimation optimization The work explores extended Kalman filter methodologies for transmission line control, field estimation, and symmetry-broken dynamics in signal processing systems for advanced computational modeling applications. This title has been co-published with Manakin Press. T&F does not sell or distribute the print editions in Bangladesh, Bhutan, India, Nepal, Pakistan, and Sri lanka.
In most of the machine learning problems based on classical neural networks, we try to identify a nonlinear system which may or may not have memory by adapting the weights of the neural network so that for the same input process applied to the system as well as to the neural network, the network output matches that of the true system output in the mean square sense or in the sense of some other metric on the space of output processes. If the network is feed forward, then this model can be used to simulate a memory-less nonlinear system. If however the actual system has memory and is described by a stochastic difference equation for its state followed by an output equation that is a function of the state, then we can use a recurrent neural network (RNN) whose state dynamics are described by a nonlinear difference equation involving weights as parameters and the state and weights of the RNN can be estimated from noisy measurements of the actual system output by means of the extended Kalman filter. If however, we are interested in estimating the joint probability density of a vector valued process like the transmission line current and voltage at different points along the line at all times, then a classical neural network will not work. This is because a probability density has several constraints like non-negativity and integrability to unity, and so on, and a classical neural network does not take care of these constraints.