Feedback control of quantum systems via continuous measurement involves complex nonlinear dynamics. Except in very special cases, even for a single qubit optimal feedback protocols are unknown. Not even do intuitive candidates exist for choosing the measurement basis, which is the primary non-trivial ingredient in the feedback control of a qubit. Here we present a series of arguments that suggest a particular form for the optimal protocol for a broad class of noise sources in the regime of good control. This regime is defined as that in which the control is strong enough to keep the system close to the desired state. With the assumption of this form the remaining parameters can be determined via a numerical search. The result is a non-trivial feedback protocol valid for all feedback strengths in the regime of good control. We conjecture that this protocol is optimal to leading order in the small parameters that define this regime. The protocol can be described relatively simply, and as a notable feature contains a discontinuity as a function of the feedback strength.
Feedback control of quantum systems via continuous measurement involves complex nonlinear dynamics. Except in very special cases, even for a single qubit optimal feedback protocols are unknown. Not even do intuitive candidates exist for choosing the measurement basis, which is the primary non-trivial ingredient in the feedback control of a qubit. Here we present a series of arguments that suggest a particular form for the optimal protocol for a broad class of noise sources in the regime of good control. This regime is defined as that in which the control is strong enough to keep the system close to the desired state. With the assumption of this form the remaining parameters can be determined via a numerical search. The result is a non-trivial feedback protocol valid for all feedback strengths in the regime of good control. We conjecture that this protocol is optimal to leading order in the small parameters that define this regime. The protocol can be described relatively simply, and as a notable feature contains a discontinuity as a function of the feedback strength.
We derive the equations of motion describing the feedback control of quantum systems in the regime of ``good control", in which the control is sufficient to keep the system close to the desired state. One can view this regime as the quantum equivalent of the ``linearized" regime for feedback control of classical nonlinear systems. Strikingly, while the dynamics of a single qubit in this regime is indeed linear, that of all larger systems remains nonlinear, in contrast to the classical case. As a first application of these equations, we determine the steady-state performance of feedback protocols for a single qubit that use unbiased measurements.
We derive the equations of motion describing the feedback control of quantum systems in the regime of "good control", in which the control is sufficient to keep the system close to the desired state. One can view this regime as the quantum equivalent of the "linearized" regime for feedback control of classical nonlinear systems. Strikingly, while the dynamics of a single qubit in this regime is indeed linear, that of all larger systems remains nonlinear, in contrast to the classical case. As a first application of these equations, we determine the steady-state performance of feedback protocols for a single qubit that use unbiased measurements.
The advance of experimental techniques in circuit quantum electrodynamics over last decadehas made it possible to monitor the evolution of qubits under continuous measurements in\nreal time . This kind of measurements, as compared to the discrete ones, provides a stream\nof information for feedback to alter the system's quantum dynamics continuously in a desired\nmanner, thus allowing broader possibilities. We obtain the information at a cost, the backaction on the states. However, the back-action turns out not to be just something disturbing \nand harmful to the control but instead can be harnessed, together with the control, to steer the system to our targets. \nThis thesis discusses one such application, namely to use measurements to create entanglement from remote qubits under locally optimal control. It is illustrated that by carefully designing symmetric measurement and feedback operators, we can construct efficient Markovian or non-Markovian protocols which generate N-qubit W,\ngeneral Dicke and Greenberger-Horne-Zeilinger states with high fidelity above 94% up to\nN = 100. In some cases, the fidelity even approaches unity. Following that, we describe how\nto take advantages of the continuous stream of information to protect the unitary dynamics of\na quantum system subject to noises. We develop a practical protocol for continuous operation\nof a quantum error correcting code for protection of coherent evolution due to an encoded\nHamiltonian against environmental errors, using the three qubit bit-flip code and bit-flip\nerrors as a canonical example. We demonstrate, for both quantum memory and adiabatic\nHamiltonians, that the protocol, once optimized, can significantly reduce logical errors with\na performance which is close to the theoretical limit of the bit-flip code's capability. In the\nadiabatic Hamiltonian case, we also propose some novel correction operators instead of the\nconventional ones to further reduce the logical errors.
Measurements in quantum mechanics cannot perfectly distinguish all states and necessarily disturb the measured system. We present and analyze a proposal to demonstrate fundamental limits on quantum control of a single qubit arising from these properties of quantum measurements. We consider a qubit prepared in one of two nonorthogonal states and subsequently subjected to dephasing noise. The task is to use measurement and feedback control to attempt to correct the state of the qubit. We demonstrate that projective measurements are not optimal for this task, and that there exists a nonprojective measurement with an optimum measurement strength which achieves the best trade-off between gaining information about the system and disturbing it through measurement backaction. We study the performance of a quantum control scheme that makes use of this weak measurement followed by feedback control, and demonstrate that it realizes the optimal recovery from noise for this system. We contrast this approach with various classically inspired control schemes.
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.