We give a concrete description of a controlled quantum stochastic dynamical model corresponding to a quantum system (a cavity mode) undergoing continual quadrature measurements, with a PID controller acting on the filtered estimate for the mode operator. The control has three feedback paths: P (proportional), I (integral) and D (derivative) of the error signal. Central use is made of the input and output pictures when constructing the model: these unitarily equivalent pictures are presented in the paper, and used to transfer concepts relating to the controlled internal dynamics to those relating to measurement output, and vice versa. The approach shows the general principle for investigating mathematically and physically consistent models in which standard control theoretic methods are to be extended to the quantum setting.
We give a concrete description of a controlled quantum stochastic dynamical model corresponding to a quantum system (a cavity mode) undergoing continual quadrature measurements, with a PID controller acting on the filtered estimate for the mode operator. The control has three feedback paths: P (proportional), I (integral) and D (derivative) of the error signal. Central use is made of the input and output pictures when constructing the model: these unitarily equivalent pictures are presented in the paper, and used to transfer concepts relating to the controlled internal dynamics to those relating to measurement output, and vice versa. The approach shows the general principle for investigating mathematically and physically consistent models in which standard control theoretic methods are to be extended to the quantum setting.
Proportional-Integral-Derivative (PID) control is used for automatically regulating a measurable quantity to a desired setpoint. It is widely used in different types of classical control electronics. Here, we show how extending the feedback theory in quantum systems to include the derivative and integral parts influences both the transient and steady-state behavior of the amplitude and squeezing of a mechanical quadrature in an optomechanical system. We show that, in contrast to standard proportional feedback, derivative feedback affects both the conditional and unconditional squeezing. Furthermore, we demonstrate how feedback may be employed to drive a mechanical quadrature to track a desired reference signal. Our findings offer new routes for an improved quantum state control and measurement precision.
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.
In this paper, we develop the theory of the quantum Pontryagin principle for continuous measurements and feedback. The analysis is carried out under the assumption of compatible events in the output channel. The plant is a quantum system, which generally is in a mixed state, coupled to a continuous measurement channel. The Pontryagin maximum principle is derived in both the Schrödinger picture and Heisenberg picture, in particular, in statistical moment coordinates. To avoid solving stochastic equations, we derive a linear-quadratic-Gaussian scheme, which is more suitable for control purposes. Finally, we use the quantum harmonic oscillator as a concrete example to illustrate the performance of the controller.
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.
Abstract Feedback control is an essential component of many modern technologies and provides a key capability for emergent quantum technologies. We extend existing approaches of direct feedback control in which the controller applies a function directly proportional to the output signal (P feedback), to strategies in which feedback determined by an integrated output signal (I feedback), and to strategies in which feedback consists of a combination of P and I terms. The latter quantum PI feedback constitutes the analog of the widely used proportional-integral feedback of classical control. All of these strategies are experimentally feasible and require no complex state estimation. We apply the resulting formalism to two canonical quantum feedback control problems, namely, generation of an entangled state of two remote qubits, and stabilization of a harmonic oscillator under thermal noise under conditions of arbitrary measurement efficiency. These two problems allow analysis of the relative benefits of P, I, and PI feedback control. We find that for the two-qubit remote entanglement generation the best strategy can be a combined PI strategy when the measurement efficiency is less than one. In contrast, for harmonic state stabilization we find that P feedback shows the best performance when actuation of both position and momentum feedback is possible, while when only actuation of position is available, I feedback consistently shows the best performance, although feedback delay is shown to improve the performance of a P strategy here.
This paper introduces a Lyapunov-based control strategy alongside two filtering methods for controlling and estimating the evolution of coherence vector elements from sequential homodyne measurements. The methods include traditional quantum filtering and a novel extended Kalman filter, which explicitly addresses the dynamics of a stochastic master equation with correlated noise, thereby ensuring the quantum properties of the estimated state variable by design. We also explore scenarios where the system’s Hamiltonian is unknown, demonstrating that both filters exhibit reduced performance with increased estimation errors. To address this, we propose a multiple model estimation scheme applicable to either filter. The estimated density operator is then controlled using the proposed switching-based Lyapunov scheme, which guarantees noise-to-state practical stability in probability. We demonstrate the effectiveness of our approach in stabilizing a qubit coupled to a leaky cavity under homodyne detection, even with uncertainty in the resonance frequency.