Abstract This chapter discusses a novel aspect of quantum control, namely quantum feedback. It explains the crucial distinction between measurement-based feedback and autonomous feedback. In the former, a detector performs a measurement of the variable to be controlled, and information passes from the quantum system to a classical system, which processes it to produce a correction signal sent to an actuator acting back on the quantum system. In the latter, the detector-processor-actuator part of the controller is another quantum system coupled to the controlled system, and no information need pass through classical channels. The chapter treats the useful formalism of the quantum stochastic master equation, a basic element of knowledge essential to the detailed understanding of measurement-based feedback, and the design of the filters for the corresponding controller. The autonomous aspect of quantum feedback is illustrated by a discussion of analog, continuous error correction of a qubit by quantum optics elements.
Abstract This chapter discusses a novel aspect of quantum control, namely quantum feedback. It explains the crucial distinction between measurement-based feedback and autonomous feedback. In the former, a detector performs a measurement of the variable to be controlled, and information passes from the quantum system to a classical system, which processes it to produce a correction signal sent to an actuator acting back on the quantum system. In the latter, the detector-processor-actuator part of the controller is another quantum system coupled to the controlled system, and no information need pass through classical channels. The chapter treats the useful formalism of the quantum stochastic master equation, a basic element of knowledge essential to the detailed understanding of measurement-based feedback, and the design of the filters for the corresponding controller. The autonomous aspect of quantum feedback is illustrated by a discussion of analog, continuous error correction of a qubit by quantum optics elements.
Introduction In the preceding chapter we introduced quantum feedback control, devoting most space to the continuous feedback control of a localized quantum system. That is, we considered feeding back the current resulting from the monitoring of that system to control a parameter in the system Hamiltonian. We described feedback both in terms of Heisenberg-picture operator equations and in terms of the stochastic evolution of the conditional state. The former formulation was analytically solvable for linear systems. However, the latter could also be solved analytically for simple linear systems, and had the advantage of giving an explanation for how well the feedback could perform. In this chapter we develop further the theory of quantum feedback control using the conditional state. The state can be used not only as a basis for understanding feedback, but also as the basis for the feedback itself. This is a simple but elegant idea. The conditional state is, by definition, the observer's knowledge about the system. In order to control the system optimally, the observer should use this knowledge. Of course a very similar idea was discussed in Section 2.5 in the context of adaptive measurements. There, one's joint knowledge of a quantum system and a classical parameter was used to choose future measurements so as to increase one's knowledge of the classical parameter. The distinction is that in this chapter we consider state-based feedback to control the quantum system itself. This chapter is structured as follows. Section 6.2 introduces the idea of state-based feedback by discussing the first experimental implementation of a state-based feedback protocol to control a quantum state.
Introduction In the preceding chapter we introduced quantum trajectories: the evolution of the state of a quantum system conditioned on monitoring its outputs. As discussed in the preface, one of the chief motivations for modelling such evolution is for quantum feedback control. Quantum feedback control can be broadly defined as follows. Consider a detector continuously producing an output, which we will call a current. Feedback is any process whereby a physical mechanism makes the statistics of the present current at a later time depend upon the current at earlier times. Feedback control is feedback that has been engineered for a particular purpose, typically to improve the operation of some device. Quantum feedback control is feedback control that requires some knowledge of quantum mechanics to model. That is, there is some part of the feedback loop that must be treated (at some level of sophistication) as a quantum system. There is no implication that the whole apparatus must be treated quantum mechanically. The structure of this chapter is as follows. The first quantum feedback experiments (or at least the first experiments specifically identified as such) were done in the mid 1980s by two groups [WJ85a, MY86]. They showed that the photon statistics of a beam of light could be altered by feedback. In Section 5.2 we review such phenomena and give a theoretical description using linearized operator equations. Section 5.3 considers the changes that arise when one allows the measurement to involve nonlinear optical processes.
Measurement and feedback control are essential features of quantum science, with applications ranging from quantum technology protocols to information-to-work conversion in quantum thermodynamics. Theoretical descriptions of feedback control are typically given in terms of stochastic equations requiring numerical solutions, or are limited to linear feedback protocols. Here we present a formalism for continuous quantum measurement and feedback, both linear and nonlinear. Our main result is a quantum Fokker-Planck master equation describing the joint dynamics of a quantum system and a detector with finite bandwidth. For fast measurements, we derive a Markovian master equation for the system alone, amenable to analytical treatment. We illustrate our formalism by investigating two basic information engines, one quantum and one classical.
This review develops a control-theoretic perspective on measurement-based feedback for continuously monitored open quantum systems, with the main analysis focused on finite-dimensional systems governed by diffusive stochastic master equations. We introduce the relevant state-space, invariant-subspace, and quantum non-demolition structures, interpret quantum filtering as nonlinear observer dynamics, and review open-loop asymptotics, filter stability, state-feedback stabilization, robustness, and reduced-order observer-based control. Particular emphasis is placed on a recurrence--contraction framework, in which Hamiltonian feedback removes non-target invariant obstructions while measurement-induced dynamics provide local exponential contraction. Although the detailed analysis is developed for finite-dimensional diffusive models, the underlying measurement--estimation--feedback architecture is relevant across a broad range of quantum platforms. We further discuss implementation challenges and open problems involving scalable estimation, sampling and delay, adaptation, hybrid and non-Markovian dynamics, practical stability, optimal control, and learning-based design. By organizing these developments around invariance, estimation, recurrence, and contraction, the review provides a tutorial bridge between measurement-based quantum feedback and nonlinear stochastic control.
Abstract Quantum measurement has always been thought of as a formal, sudden projection that changes the system state instantly. However, in real life, the process of measurement involves a physical interaction that takes time to gather the information. The paper discusses the continuous quantum measurement that challenges the traditional instant measurement idea with the detector’s accumulated data continuously uninterruptedly. The paper starts by laying down the basic mathematics, which gradually leads to the modern Stochastic master equations and quantum trajectories through generalized measurements positive operator-valued measures and quantum instruments. The crucial experimental setups like circuit QED, trapped ions, and optomechanics are delineated where these notions have been put into practice. The significant ramifications of ongoing observation are then discussed, starting with the fundamental Zeno effect, measurement backaction, and continuing through the active steering of quantum states and the emergence of measurement-induced phase transitions in many-body systems. One of the main ideas of the paper is the attribute of measurement as a powerful control method which enables real-time feedback, state stabilization, and continuous quantum error correction. Finally, the paper points out the open problems that remain in the areas of many-body systems’ scalability and the coupling of machine learning with real-time quantum control.
Quantum computers hold immense promise for solving problems intractable for classical machines, yet their practical realization is hindered by the fragility of quantum states. Quantum Error Correction (QEC) is indispensable for protecting quantum information from environmental noise and operational imperfections. Traditional QEC schemes often rely on measurement-based feedback, which involves performing measurements, processing the classical outcomes, and then applying corrective operations. This process introduces latency, consumes significant resources, and can itself be a source of errors due to the invasive nature of quantum measurement and the classical-quantum interface. Coherent feedback control offers a compelling alternative by allowing the controller to directly interact with the quantum system using quantum fields, without intermediate measurement and classical processing. This maintains the coherence of the quantum information throughout the feedback loop, potentially leading to more robust and faster error correction. This paper explores the theoretical framework and potential advantages of employing coherent feedback control for autonomous QEC. We review existing QEC paradigms, delve into the principles of coherent feedback, and propose how this approach can be integrated into autonomous QEC architectures. Specifically, we investigate its application to bosonic codes and stabilizer codes, discussing the engineering challenges and the prospects for achieving truly fault-tolerant quantum computation by minimizing the classical overhead and enhancing real-time error suppression.