The control of individual quantum systems is now a reality in a variety of physical settings. Feedback control is an important class of control methods because of its ability to reduce the effects of noise. In this review we give an introductory overview of the various ways in which feedback may be implemented in quantum systems, the theoretical methods that are currently used to treat it, the experiments in which it has been demonstrated to-date, and its applications. In the last few years there has been rapid experimental progress in the ability to realize quantum measurement and control of mesoscopic systems. We expect that the next few years will see further rapid advances in the precision and sophistication of feedback control protocols realized in the laboratory.

Generating a unitary transformation in the shortest possible time is of practical importance to quantum information processing because it helps to reduce decoherence effects and improve robustness to additive control field noise. Many analytical and numerical studies have identified the minimum time necessary to implement a variety of quantum gates on coupled-spin qubit systems. This work focuses on exploring the Pareto front that quantifies the trade-off between the competitive objectives of maximizing the gate fidelity F and minimizing the control time T . In order to identify the critical time T * , below which the target transformation is not reachable, as well as to determine the associated Pareto front, we introduce a numerical method of Pareto front tracking (PFT). We consider closed two-and multi-qubit systems with constant inter-qubit coupling strengths and each individual qubit controlled by a separate time-dependent external field. Our analysis demonstrates that unit fidelity (to a desired numerical accuracy) can be achieved at any T ≥ T * in most cases. However, the optimization search effort rises superexponentially as T decreases and approaches T * . Furthermore, a small decrease in control time incurs a significant penalty in fidelity for T < T * , indicating that it is generally undesirable to operate below the critical time. We investigate the dependence of the critical time T * on the coupling strength between qubits and the target gate transformation. Practical consequences of these findings for laboratory implementation of quantum gates are discussed.

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The success of quantum optimal control for both experimental and theoretical objectives is connected to the topology of the corresponding control landscapes, which are free from local traps if three conditions are met: (1) the quantum system is controllable, (2) the Jacobian of the map from the control field to the evolution operator is of full rank, and (3) there are no constraints on the control field. This paper investigates how the violation of assumption (3) affects gradient searches for globally optimal control fields. The satisfaction of assumptions (1) and (2) ensures that the control landscape lacks fundamental traps, but certain control constraints can still introduce artificial traps. Proper management of these constraints is an issue of great practical importance for numerical simulations as well as optimization in the laboratory. Using optimal control simulations, we show that constraints on quantities such as the number of control variables, the control duration, and the field strength are potentially severe enough to prevent successful optimization of the objective. For each such constraint, we show that exceeding quantifiable limits can prevent gradient searches from reaching a globally optimal solution. These results demonstrate that careful choice of relevant control parameters helps to eliminate artificial traps and facilitate successful optimization.

We propose a strategy to suppress decoherence of a solid-state qubit coupled to non-Markovian noises by attaching the qubit to a chaotic setup with the broad power distribution in particular in the high-frequency domain. Different from the existing decoherence control methods such as the usual dynamics decoupling control, high-frequency components of our control are generated by the chaotic setup driven by a low-frequency field, and the generation of complex optimized control pulses is not necessary. We apply the scheme to superconducting quantum circuits and find that various noises in a wide frequency domain, including low-frequency 1/f , high-frequency Ohmic, sub-Ohmic, and super-Ohmic noises, can be efficiently suppressed by coupling the qubits to a Duffing oscillator as the chaotic setup. Significantly, the decoherence time of the qubit is prolonged approximately 100 times in magnitude. Introduction.-Solid state quantum information processing [1] develops very rapidly in recent years. One of the basic features that makes quantum information unique is the quantum parallelism resulted from quantum coherence and entanglement. However, the inevitable interaction between the qubit and its environment leads to qubit-environment entanglement that deteriorates quantum coherence of the qubit. In solid state systems, the decoherence process is mainly caused by the non-Markovian noises induced, e.g., by the two-level fluctuators in the substrate and the charge and flux noises in the circuits [2][3][4].