Quantum control theory and applications: A survey <s> Conclusions and perspectives <s> The dynamics of a decohering two-level system driven by a suitable control Hamiltonian is studied. The control procedure is implemented as a sequence of radio-frequency pulses that repetitively flip the state of the system, a technique that can be termed quantum ``bang-bang'' control after its classical analog. Decoherence introduced by the system's interaction with a quantum environment is shown to be washed out completely in the limit of continuous flipping and greatly suppressed provided the interval between the pulses is made comparable to the correlation time of the environment. The model suggests a strategy to fight against decoherence that complements existing quantum error-correction techniques. <s> BIB001 </s> Quantum control theory and applications: A survey <s> Conclusions and perspectives <s> The problem of feedback control of quantum systems by means of weak measurements is investigated in detail. When weak measurements are made on a set of identical quantum systems, the single-system density matrix can be determined to a high degree of accuracy while affecting each system only slightly. If this information is fed back into the systems by coherent operations, the single-system density matrix can be made to undergo an arbitrary nonlinear dynamics, including for example a dynamics governed by a nonlinear Schr\"odinger equation. We investigate the implications of such nonlinear quantum dynamics for various problems in quantum control and quantum information theory, including quantum computation. The nonlinear dynamics induced by weak quantum feedback could be used to create a novel form of quantum chaos in which the time evolution of the single-system wave function depends sensitively on initial conditions. <s> BIB002 </s> Quantum control theory and applications: A survey <s> Conclusions and perspectives <s> We solve exactly the non-Markovian dynamics of a cavity mode in the presence of a feedback loop based on homodyne measurements, in the case of a nonzero feedback delay time. With an appropriate choice of the feedback parameters, this scheme is able to significantly increase the decoherence time of the cavity mode, even for delay times not much smaller than the decoherence time itself. <s> BIB003 </s> Quantum control theory and applications: A survey <s> Conclusions and perspectives <s> Quantum feedback can stabilize a two-level atom against decoherence (spontaneous emission), putting it into an arbitrary (specified) pure state. This requires perfect homodyne detection of the atomic emission, and instantaneous feedback. Inefficient detection was considered previously by two of us. Here we allow for a non-zero delay time tau in the feedback circuit. Because a two-level atom is a non-linear optical system, an analytical solution is not possible. However, quantum trajectories allow a simple numerical simulation of the resulting non-Markovian process. We find the effect of the time delay to be qualitatively similar to chat of inefficient detection. The solution of the non-Markovian quantum trajectory will not remain fixed, so that the time-averaged state will be mixed, not pure. In the case where one tries to stabilize the atom in the excited state, an approximate analytical solution to the quantum trajectory is possible. The result, that the purity (P = 2Tr[rho (2)] - 1) of the average state is given by P = 1 - 4y tau (where gamma is the spontaneous emission rate) is found to agree very well with the numerical results. (C) 2001 Elsevier Science B.V. All rights reserved. <s> BIB004 </s> Quantum control theory and applications: A survey <s> Conclusions and perspectives <s> We describe a protocol for continuously protecting unknown quantum states from decoherence that incorporates design principles from both quantum error correction and quantum feedback control. Our protocol uses continuous measurements and Hamiltonian operations, which are weaker control tools than are typically assumed for quantum error correction. We develop a cost function appropriate for unknown quantum states and use it to optimize our state-estimate feedback. Using Monte Carlo simulations, we study our protocol for the three-qubit bit-flip code in detail and demonstrate that it can improve the fidelity of quantum states beyond what is achievable using quantum error correction when the time between quantum error-correction cycles is limited. <s> BIB005 </s> Quantum control theory and applications: A survey <s> Conclusions and perspectives <s> The theory and practice of control over quantum mechanical phenomena is receiving increasing attention, underscored by striking experimental successes. Nevertheless, many questions of fundamental and practical relevance to the field remain unresolved. With the aim of stimulating further development, this paper formulates a number of theoretical questions, divided into three categories. First, questions related to control law design are discussed, with an emphasis on controllability and optimal control theory. This leads to the second category of open problems relevant to closed loop laboratory implementation of quantum control, including learning and feedback methods. The sensitive dependence of control on basic quantum mechanical interactions motivates the third section, which treats coherent dynamical techniques for identifying the system Hamiltonian. An open issue overarching all of these directions is the need to discover general rules for the control of quantum systems. Although the list of issues raised in this paper is extensive, it should be viewed not as a complete menu for exploration, but rather as a springboard to new challenges as the field evolves. <s> BIB006 </s> Quantum control theory and applications: A survey <s> Conclusions and perspectives <s> PREFACE ACKNOWLEDGEMENTS PART 1: PROBABILITY IN CLASSICAL AND QUANTUM MECHANICS 1. Classical probability theory and stochastic processes 2. Quantum Probability PART 2: DENSITY MATRIX THEORY 3. Quantum Master Equations 4. Decoherence PART 3: STOCHASTIC PROCESSES IN HILBERT SPACE 5. Probability distributions on Hilbert space 6. Stochastic dynamics in Hilbert space 7. The stochastic simulation method 8. Applications to quantum optical systems PART 4: NON-MARKOVIAN QUANTUM PROCESSES 9. Projection operator techniques 10. Non-Markovian dynamics in physical systems PART 5: RELATIVISTIC QUANTUM PROCESSES 11. Measurements in relativistic quantum mechanics 12. Open quantum electrodynamics <s> BIB007 </s> Quantum control theory and applications: A survey <s> Conclusions and perspectives <s> We show that quantum feedback control can be used as a quantum error correction process for errors induced by weak continuous measurement. In particular, when the error model is restricted to one, perfectly measured, error channel per physical qubit, quantum feedback can act to perfectly protect a stabilizer codespace. Using the stabilizer formalism we derive an explicit scheme, involving feedback and an additional constant Hamiltonian, to protect art (n-1)-qubit logical state encoded in n physical qubits. This works for both Poisson (jump) and white-noise (diffusion) measurement processes. As an example, we show that detected-spontaneous emission error correction with a driving Hamiltonian can greatly reduce the amount of redundancy required to protect a state from that which has been previously postulated. <s> BIB008 </s> Quantum control theory and applications: A survey <s> Conclusions and perspectives <s> We describe a scheme for quantum-error correction that employs feedback and weak measurement rather than the standard tools of projective measurement and fast controlled unitary gates. The advantage of this scheme over previous protocols [for example, Ahn et al. Phys. Rev. A 65, 042301 (2001)], is that it requires little side processing while remaining robust to measurement inefficiency, and is therefore considerably more practical. We evaluate the performance of our scheme by simulating the correction of bit flips. We also consider implementation in a solid-state quantum-computation architecture and estimate the maximal error rate that could be corrected with current technology. <s> BIB009 </s> Quantum control theory and applications: A survey <s> Conclusions and perspectives <s> In this paper, we design an optimal control law to suppress decoherence effects in Markovian open quantum systems. The optimal control law is subject to the tracking precision of the trajectory governed by the free system, which is ideally free from decoherence. We observe from numerical simulation that the undesired decohering dynamics can be partially squeezed out in most systems. Moreover, we observe the existence of sinusoidally oscillating resonant modes that play dominant roles in the controlled trajectory, which can be easily realized by continuous wave pulses. These key features are strictly demonstrated in subsequent analysis under proper assumptions. For systems in which the coherent control does not work, we suggest a feedback control strategy to extend the applicability of control to wider class of systems. <s> BIB010 </s> Quantum control theory and applications: A survey <s> Conclusions and perspectives <s> A pair of spins is the most simple quantum system that can possess entanglement, a non-classical property playing an essential role in quantum information technologies. In this paper, feedback control problems of the two-spin system conditioned on a continuous measurement are considered. In order to make some useful formulas in stochastic control theory directly applicable, we first derive a two-dimensional description of the system. We then prove that a feedback controller stabilizes an entangled state of the two spins almost globally with probability one. Furthermore, it is shown that another entangled state, which corresponds to a non-equilibrium point of the dynamics, is stabilized via feedback in the sense that the expectation of the distance from the target can be made arbitrarily small. <s> BIB011 </s> Quantum control theory and applications: A survey <s> Conclusions and perspectives <s> A nonequilibrium, generally time-dependent, environment whose form is deduced by optimal learning control is shown to provide a means for incoherent manipulation of quantum systems. Incoherent control by the environment (ICE) can serve to steer a system from an initial state to a target state, either mixed or in some cases pure, by exploiting dissipative dynamics. Implementing ICE with either incoherent radiation or a gas as the control is explicitly considered, and the environmental control is characterized by its distribution function. Simulated learning control experiments are performed with simple illustrations to find the shape of the optimal nonequilibrium distribution function that best affects the posed dynamical objectives. <s> BIB012 </s> Quantum control theory and applications: A survey <s> Conclusions and perspectives <s> We present quantum feedback control for deterministic entanglement generation at the single-photon level. The protocol of controlling both total photon number and phase difference is based on the cascade structure of cavities placed in an optical closed loop, quantum nondemolition measurement with cross-Kerr interactions, and Lyapunov stability for feedback design. <s> BIB013 </s> Quantum control theory and applications: A survey <s> Conclusions and perspectives <s> We show that in the regime in which feedback control is most effective—when measurements are relatively efficient, and feedback is relatively strong—then, in the absence of any sharp inhomogeneity in the noise, it is always best to measure in a basis that does not commute with the system density matrix than one that does. That is, it is optimal to make measurements that disturb the state one is attempting to stabilize. <s> BIB014 </s> Quantum control theory and applications: A survey <s> Conclusions and perspectives <s> The noise decoupling problem is investigated for general N-level Markovian open quantum systems. First, the concept of Cartan decomposition of the Lie algebra su(N) is introduced as a tool of designing control Hamiltonians. Next, under certain assumptions, it is shown that a part of variables of the coherence vector of the system density matrix can be asymptotically decoupled from the environmental noises. The resulting noise decoupling scheme is applied to one-qubit, qutrit, and two-qubit quantum systems, by which the coherence evolution of the one-qubit and qutrit systems can always be asymptotically preserved, while, for two-qubit systems, our findings indicate that evolution of some variables can be preserved only for some initial states. <s> BIB015 </s> Quantum control theory and applications: A survey <s> Conclusions and perspectives <s> In this work we propose a strategy using techniques from systems theory to completely eliminate decoherence and also provide conditions under which it can be done. A construction employing an auxiliary system, the bait, which is instrumental to decoupling the system from the environment is presented. Our approach to decoherence control in contrast to other approaches in the literature involves the bilinear input affine model of quantum control system which lends itself to various techniques from classical control theory, but with nontrivial modifications to the quantum regime. The elegance of this approach yields interesting results on open loop decouplability and decoherence free subspaces. Additionally, the feedback control of decoherence may be related to disturbance decoupling for classical input affine systems, which entails careful application of the methods by avoiding all the quantum mechanical pitfalls. In the process of calculating a suitable feedback the system must be restructured due to its tensorial nature of interaction with the environment, which is unique to quantum systems. In the subsequent section we discuss a general information extraction scheme to gain knowledge of the state and the amount of decoherence based on indirect continuous measurement. The analysis of continuous measurement on a decohering quantum system has not been extensively studied before. Finally, a methodology to synthesize feedback parameters itself is given, that technology permitting, could be implemented for practical 2-qubit systems to perform decoherence free quantum computing. The results obtained are qualitatively different and superior to the ones obtained via master equations. <s> BIB016 </s> Quantum control theory and applications: A survey <s> Conclusions and perspectives <s> It has recently been shown that finding the optimal measurement on the environment for stationary linear quadratic Gaussian control problems is a semidefinite program. We apply this technique to the control of the Einstein-Podolsky-Rosen correlations between two bosonic modes interacting via a parametric Hamiltonian at steady state. The optimal measurement turns out to be nonlocal homodyne measurement\char22{}the outputs of the two modes must be combined before measurement. We also find the optimal local measurement and control technique. This gives the same degree of entanglement but a higher degree of purity than the local technique previously considered [S. Mancini, Phys. Rev. A 73, 010304(R) (2006)]. <s> BIB017 </s> Quantum control theory and applications: A survey <s> Conclusions and perspectives <s> In this paper we show how weak measurement and local feedback can be used to control entanglement generation between two qubits. To do this, we make use of a decoherence free subspace. Weak measurement and feedback can be used to drive the system into this subspace rapidly. Once within the subspace, feedback can generate entanglement rapidly, or turn off entanglement generation dynamically. We also consider, in the context of weak measurement, some of the differences between purification and generating entanglement. <s> BIB018 </s> Quantum control theory and applications: A survey <s> Conclusions and perspectives <s> The purpose of this paper is to formulate and solve a H∞ controller synthesis problem for a class of non-commutative linear stochastic systems which includes many examples of interest in quantum technology. The paper includes results on the class of such systems for which the quantum commutation relations are preserved (such a requirement must be satisfied in a physical quantum system). A quantum version of standard (classical) dissipativity results are presented and from this a quantum version of the Strict Bounded Real Lemma is derived. This enables a quantum version of the two Riccati solution to the H∞ control problem to be presented. This result leads to controllers which may be realized using purely quantum, purely classical or a mixture of quantum and classical elements. This issue of physical realizability of the controller is examined in detail, and necessary and sufficient conditions are given. Our results are constructive in the sense that we provide explicit formulas for the Hamiltonian function and coupling operator corresponding to the controller. <s> BIB019 </s> Quantum control theory and applications: A survey <s> Conclusions and perspectives <s> We present a theory of dynamical control by modulation for optimal decoherence reduction. The theory is based on the non-Markovian Euler-Lagrange equation for the energy-constrained field that minimizes the average dephasing rate of a qubit for any given dephasing spectrum. <s> BIB020 </s> Quantum control theory and applications: A survey <s> Conclusions and perspectives <s> In this paper, we consider a linear quantum network composed of two distantly separated cavities that are connected via a one-way optical field. When one of the cavities is damped and the other undamped, the overall cavity state obtains a large amount of entanglement in its quadratures. This entanglement, however, immediately decays and vanishes in a finite time. That is, entanglement sudden death occurs. We show that the direct measurement feedback method proposed by Wiseman can avoid this entanglement sudden death, and, further, enhance the entanglement. It is also shown that the entangled state under feedback control is robust against signal loss in a realistic detector, indicating the reliability of the proposed direct feedback method in practical situations. <s> BIB021 </s> Quantum control theory and applications: A survey <s> Conclusions and perspectives <s> We investigate the optimal control problem for a non-Markovian open, dissipative quantum system. Optimal control using the Pontryagin maximum principle is specifically derived. The influences of ohmic reservoir with Lorentz-Drude regularization are numerically studied in a two-level system under the following three conditions: ${\ensuremath{\omega}}_{0}⪡{\ensuremath{\omega}}_{c}$, ${\ensuremath{\omega}}_{0}\ensuremath{\approx}{\ensuremath{\omega}}_{c}$, or ${\ensuremath{\omega}}_{0}⪢{\ensuremath{\omega}}_{c}$, where ${\ensuremath{\omega}}_{0}$ is the characteristic frequency of the quantum system of interest, and ${\ensuremath{\omega}}_{c}$ the cutoff frequency of the ohmic reservoir. The optimal control process shows its remarkable influences on the decoherence dynamics. The temperature is a key factor in the decoherence dynamics. We analyze the optimal decoherence control in high temperature, intermediate temperature, and low temperature reservoirs, respectively. It implies that designing some engineered reservoirs with the controlled coupling and state of the environment can slow down the decoherence rate and delay the decoherence time. Moreover, we compare the non-Markovian optimal decoherence control with the Markovian one and find that with non-Markovian the engineered artificial reservoirs are better than with the Markovian approximation in controlling the open, dissipative quantum system's decoherence. <s> BIB022 </s> Quantum control theory and applications: A survey <s> Conclusions and perspectives <s> The purpose of this paper is to present simple and general algebraic methods for describing series connections in quantum networks. These methods build on and generalize existing methods for series (or cascade) connections by allowing for more general interfaces, and by introducing an efficient algebraic tool, the series product. We also introduce another product, which we call the concatenation product, that is useful for assembling and representing systems without necessarily having connections. We show how the concatenation and series products can be used to describe feedforward and feedback networks. A selection of examples from the quantum control literature are analyzed to illustrate the utility of our network modeling methodology. <s> BIB023 </s> Quantum control theory and applications: A survey <s> Conclusions and perspectives <s> This paper considers a coherent H∞ control problem for a class of linear quantum systems which can be defined by complex quantum stochastic differential equations in terms of annihilation operators only. For this class of quantum systems, a solution to the H∞ control problem can be obtained in terms of a pair of complex Riccati equations. The paper also considers complex versions of the Bounded Real Lemma, the Strict Bounded Real Lemma and the Lossless Bounded Real Lemma. For the class of quantum systems under consideration, the question of physical realizability is related to the Bounded Real and Lossless Bounded Real properties. <s> BIB024 </s> Quantum control theory and applications: A survey <s> Conclusions and perspectives <s> When a quantum system interacts with its environment, the so-called decoherence effect will normally destroy the coherence in the quantum state and the entanglement between its subsystems. We propose a feedback control strategy based on quantum weak measurements to protect coherence and entanglement of the quantum state against environmental disturbance. For a one-qubit quantum system under amplitude damping and dephasing decoherence channels, our strategy can preserve the coherence based on the measured information about the population difference between its two levels. For a two-qubit quantum system disentangled by independent amplitude damping and dephasing decoherence channels, the designed feedback control can preserve coherence between the ground state and the highest excited states by tuning the coupling strength between the two qubits, and at the same time minimize the loss of entanglement between the two qubits. As a consequence of dynamic symmetry, the generalization of these results derives the concept of control-induced decoherence-free observable subspace, for which several criteria are provided. <s> BIB025
In this paper, we have surveyed developments in quantum control theory and its applications from a control systems perspective. Some results on controllability of quantum systems, open-loop control strategies and closed-loop design methods have been presented. Also, some applications of quantum control theory are briefly mentioned in this survey. Although great progress has been made in this area, quantum control is still in its infancy. Moreover, there exist several essential differences between quantum control theory and classical control theory. For example, controlling quantum entanglement and protecting quantum coherence are important two classes of quantum control tasks. However, no corresponding tasks exist in classical control theory. In classical feedback control, measurement is taken as the main tool of acquiring feedback information to deal with uncertainties and it is assumed that measurement does not affect the measured system. However, measurement in quantum feedback control unavoidably introduces essential uncertainties when we acquire feedback information through measurement to deal with uncertainties of quantum control systems. Hence, it is necessary to develop more new theory and approaches to control quantum phenomena. We now outline several important topics that are worth exploring, emphasizing different aspects of quantum control (a more detailed discussion on mathematical and algorithmic challenges in quantum control can be found in BIB006 ). (1) Quantum incoherent control. Quantum coherent control is a powerful control paradigm in open-loop quantum control and many important results have been achieved. However, it may not be practical to accomplish some quantum control tasks using only coherent control approaches. Recently, incoherent resources have been proven useful for accomplishing some quantum control tasks. An example of this is the fact that quantum measurement can serve as a useful control tool. However, there are only few results on the use of quantum incoherent control. Hence, it is desirable to develop systematic and new approaches to construct quantum incoherent (or coherent/incoherent hybrid) controllers which are easily physically realized BIB012 . (2) Quantum feedback control. Feedback plays a key role in classical control theory. It is desirable to develop systematic feedback control methods for emerging quantum engineering applications. Although many results on quantum feedback control have been presented, they are usually constrained to some special cases (e.g., linear feedback, Markovian approximation, simple quantum systems). Many fundamental problems still need to be considered: How does the weak measurement BIB002 affect quantum systems in quantum feedback control? How does one design feedback controllers for nonlinear quantum systems BIB014 and is it possible to control nonlinear dynamics of quantum systems using linear controllers? What approaches are effective for control of quantum systems with non-Markovian dynamics BIB003 , BIB004 , BIB022 ? What is the capability of feedback to deal with uncertainties in quantum systems? How do we synthesize complex quantum feedback network systems BIB023 ? (3) Robust control of quantum systems. It is inevitable that quantum systems are subject to many kinds of uncertainties (including disturbances and noises). Hence, the requirement of robustness in the presence of uncertainties has been recognized as one of the key tasks for developing practical quantum technologies. Existing results in this area focus mainly on quantum systems whose dynamics can be described by linear equations in the Heisenberg picture BIB019 , BIB024 or whose uncertainties can be approximately described as a perturbation in the system Hamiltonian. It is desirable to develop systematic robust control approaches to deal with more general kinds of uncertainties existing in practical quantum systems. (4) Decoherence control. Decoherence is the main obstacle to the application of quantum information technology. Many quantum control systems are essentially open quantum systems BIB007 ,  and interaction with external environments (including external controls and the measurement apparatus) unavoidably introduces decoherence BIB015 . A typical example of decoherence control is quantum error correction BIB005 , BIB009 , BIB008 . Several decoherence control approaches have been proposed. For example, "bang-bang" control has been proposed for dynamic decoherence suppression in two-state quantum systems BIB001 . A classical feedback strategy has been proposed to eliminate decoherence in open quantum systems BIB016 . Optimal control techniques have been applied to decoherence control of Markovian and non-Markovian open quantum systems BIB022 , BIB010 , BIB020 . In spite of these efforts, there are still many challenges for decoherence control and the development of systematic and practical decoherence control methods is still an important problem. (5) Entanglement control. Quantum entanglement is the most important resource in quantum information technology. Several optimal control and feedback control approaches have been proposed to control quantum entanglement BIB011 , BIB017 , BIB013 , BIB021 , BIB018 , BIB025 ,  . However, existing results are only a first step in entanglement control. Quantum entanglement has many unique characteristics which have no classical counterparts. Hence, it is desirable to develop completely new methods to control quantum entanglement. Moreover, we still lack a suitable theory that can effectively characterize quantum entanglement. This makes the development of a control theory for quantum entanglement more challenging.