Quantum control theory and applications: A survey <s> Optimal control <s> The optimal control of the path to a specified final state of a quantum-mechanical system is investigated. The problem is formulated as a minimization problem over appropriate function spaces, and the well-posedness of this problem is is established by proving the existence of an optimal solution. A Lagrange-multiplier technique is used to reduce the problem to an equivalent optimization problem and to derive necessary conditions for a minimum. These necessary conditions form the basis for a gradient iterative procedure to search for a minimum. A numerical scheme based on finite differences is used to reduce the infinite-dimensional minimization problem to an approximate finite-dimensional problem. Numerical examples are provided for final-state control of a diatomic molecule represented by a Morse potential. Within the context of this optimal control formulation, numerical results are given for the optimal pulsing strategy to demonstrate the feasibility of wave-packet control and finally to achieve a specified dissociative wave packet at a given time. The optimal external optical fields generally have a high degree of structure, including an early time period of wave-packet phase adjustment followed by a period of extensive energy deposition to achieve the imposed objective. Constraints on the form of the molecular dipole (e.g., a linear dipole) are shown to limit the accessibility (i.e., controllability) of certain types of molecular wave-packet objectives. The nontrivial structure of the optimal pulse strategies emphasizes the ultimate usefulness of an optimal-control approach to the steering of quantum systems to desired objectives. <s> BIB001 </s> Quantum control theory and applications: A survey <s> Optimal control <s> A new iteration method is presented for achieving quantum optimal control over the expectation value of a positive definite operator. Theoretical analysis shows that this new algorithm exhibits quadratic and monotonic convergence. Numerical calculations verify that for this new algorithm, within a few steps, the optimized objective functional comes close to its converged limit. <s> BIB002 </s> Quantum control theory and applications: A survey <s> Optimal control <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> BIB003 </s> Quantum control theory and applications: A survey <s> Optimal control <s> This review puts into perspective the present state and prospects for controlling quantum phenomena in atoms and molecules. The topics considered include the nature of physical and chemical control objectives, the development of possible quantum control rules of thumb, the theoretical design of controls and their laboratory realization, quantum learning and feedback control in the laboratory, bulk media influences, and the ability to utilize coherent quantum manipulation as a means for extracting microscopic information. The preview of the field presented here suggests that important advances in the control of molecules and the capability of learning about molecular interactions may be reached through the application of emerging theoretical concepts and laboratory technologies. <s> BIB004 </s> Quantum control theory and applications: A survey <s> Optimal control <s> In this paper, we study the design of pulse sequences for nuclear magnetic resonance spectroscopy as a problem of time optimal control of the unitary propagator. Radio-frequency pulses are used in coherent spectroscopy to implement a unitary transfer between states. Pulse sequences that accomplish a desired transfer should be as short as possible in order to minimize the effects of relaxation and to optimize the sensitivity of the experiments. Here, we give an analytical characterization of such time optimal pulse sequences applicable to coherence transfer experiments in multiple-spin systems. We have adopted a general mathematical formulation, and present many of our results in this setting, mindful of the fact that new structures in optimal pulse design are constantly arising. From a general control theory perspective, the problems we want to study have the following character. Suppose we are given a controllable right invariant system on a compact Lie group. What is the minimum time required to steer the system from some initial point to a specified final point? In nuclear magnetic resonance (NMR) spectroscopy and quantum computing, this translates to, what is the minimum time required to produce a unitary propagator? We also give an analytical characterization of maximum achievable transfer in a given time for the two-spin system. <s> BIB005 </s> Quantum control theory and applications: A survey <s> Optimal control <s> A closed loop learning control concept is introduced for teaching lasers to manipulate quantum systems for the purpose of optimally identifying Hamiltonian information. The closed loop optimal identification algorithm operates by revealing the distribution of Hamiltonians consistent with the data. The control laser is guided to perform additional experiments, based on minimizing the dispersion of the distribution. Operation of such an apparatus is simulated for two model finite dimensional quantum systems. <s> BIB006 </s> Quantum control theory and applications: A survey <s> Optimal control <s> Preface.Preliminaries of the Interaction of Light with Matter.Weak-Field Photodissociation.Weak-Field Coherent Control.Optimal Control Theory.Decoherence and Loss of Control.Case Studies in Coherent Control.Coherent Control of Bimolecular Processes.Coherent Control of the Synthesis and Purification of Chiral Molecules.Coherent Control Beyond the Weak-Field Regime: Bound States and Resonances.Photodissociation Beyond the Weak-Field Regime.Coherent Control Beyond the Weak-Field Regime: The Continuum.Strong-Field Coherent Control.Case Studies in Optimal Control.References.Author Index.Subject Index. <s> BIB007 </s> Quantum control theory and applications: A survey <s> Optimal control <s> Experiments in coherent spectroscopy correspond to control of quantum mechanical ensembles guiding them from initial to final target states. The control inputs (pulse sequences) that accomplish these transformations should be designed to minimize the effects of relaxation and to optimize the sensitivity of the experiments. For example in nuclear magnetic resonance (NMR) spectroscopy, a question of fundamental importance is what is the maximum efficiency of coherence or polarization transfer between two spins in the presence of relaxation. Furthermore, what is the optimal pulse sequence which achieves this efficiency? In this paper, we give analytical answers to the above questions. Unexpected gains in sensitivity are reported for one of the most commonly used experimental building blocks in NMR spectroscopy. Surprisingly, in the case when longitudinal relaxation is small, the relaxation optimized pulse elements (ROPE) that transfer maximum polarization between coupled spins are longer than conventional sequences. <s> BIB008 </s> Quantum control theory and applications: A survey <s> Optimal control <s> A large number of experimental studies and simulations show that it is surprisingly easy to find excellent quality control over broad classes of quantum systems. We now prove that for controllable quantum systems with no constraints placed on the controls, the only allowed extrema of the transition probability landscape correspond to perfect control or no control. Under these conditions, no suboptimal local extrema exist as traps that would impede the search for an optimal control. The identified landscape structure is universal for all controllable quantum systems of the same dimension when seeking to maximize the same transition probability, regardless of the detailed nature of the system Hamiltonian. The presence of weak control field noise or environmental decoherence is shown to preserve the general structure of the control landscape, but at lower resolution. <s> BIB009 </s> Quantum control theory and applications: A survey <s> Optimal control <s> Many applications of magnetic resonance are limited by rapid loss of spin coherence caused by large transverse relaxation rates. In NMR of large proteins, increased relaxation losses lead to poor sensitivity of experiments and increased measurement time. In this article, we develop broadband relaxation-optimized pulse sequences that approach fundamental limits of coherence transfer efficiency in the presence of very general relaxation mechanisms that include cross-correlated relaxation. These broadband transfer schemes use techniques of chemical shift refocusing (specific trajectory adapted refocusing echoes) that are tailored to specific trajectories of coupled spin evolution. We present simulations and experimental data indicating significant enhancement in the sensitivity of multidimensional NMR experiments of large molecules through these methods. <s> BIB010 </s> Quantum control theory and applications: A survey <s> Optimal control <s> Fifty years of developments in nuclear magnetic resonance (NMR) have resulted in an unrivaled degree of control of the dynamics of coupled two-level quantum systems. This coherent control of nuclear spin dynamics has recently been taken to a new level, motivated by the interest in quantum information processing. NMR has been the workhorse for the experimental implementation of quantum protocols, allowing exquisite control of systems up to seven qubits in size. This article surveys and summarizes a broad variety of pulse control and tomographic techniques which have been developed for, and used in, NMR quantum computation. Many of these will be useful in other quantum systems now being considered for the implementation of quantum information processing tasks. <s> BIB011 </s> Quantum control theory and applications: A survey <s> Optimal control <s> Given two linearly independent matrices in so(3), Z"1 and Z"2, every rotation matrix, X"f@?SO(3), can be written as the product of alternate elements from the one-dimensional subgroups corresponding to Z"1 and Z"2, namely X"f=e^Z^"^1^t^"^1e^Z^"^2^t^"^2e^Z^"^1^t^"^3...e^Z^"^1^t^"^s. The parameters t"i, i=1,...,s are called Generalized Euler Angles. In this paper, the minimum number of factors required for the factorization of X"f@?SO(3), as a function of X"f, is evaluated. An algorithm is given to determine the generalized Euler angles, in the optimal factorization. The results can be applied to the bang-bang control, with minimum number of switches, of some classical and quantum systems. <s> BIB012 </s> Quantum control theory and applications: A survey <s> Optimal control <s> In this paper, we give a characterization of all the unitary transformations that can be synthesized in a given time for a two-qubit system in the presence of general time varying coupling tensor. This characterization helps to compute the minimum time and the shortest pulse sequence for generating a general two-qubit transformation under nonstationary interactions. The methods presented here can be applied in design of time optimal pulse sequences for transferring coherence and polarization between coupled spins with time varying couplings as in solid-state NMR under magic angle spinning. <s> BIB013 </s> Quantum control theory and applications: A survey <s> Optimal control <s> In this paper we consider the minimum time population transfer problem for the z component of the spin of a (spin 1/2) particle, driven by a magnetic field, that is constant along the z axis and controlled along the x axis, with bounded amplitude. On the Bloch sphere (i.e., after a suitable Hopf projection), this problem can be attacked with techniques of optimal syntheses on two-dimensional (2-D) manifolds. Let (−E,E) be the two energy levels, and ∣Ω(t)∣≤M the bound on the field amplitude. For each couple of values E and M, we determine the time optimal synthesis starting from the level −E, and we provide the explicit expression of the time optimal trajectories, steering the state one to the state two, in terms of a parameter that can be computed solving numerically a suitable equation. For M∕E≪1, every time optimal trajectory is bang-bang and, in particular, the corresponding control is periodic with frequency of the order of the resonance frequency ωR=2E. On the other side, for M∕E>1, the time optimal ... <s> BIB014 </s> Quantum control theory and applications: A survey <s> Optimal control <s> The control of quantum dynamics via specially tailored laser pulses is a long-standing goal in physics and chemistry. Partly, this dream has come true, as sophisticated pulse shaping experiments allow to coherently control product ratios of chemical reactions. The theoretical design of the laser pulse to transfer an initial state to a given final state can be achieved with the help of quantum optimal control theory (QOCT). This tutorial provides an introduction to QOCT. It shows how the control equations defining such an optimal pulse follow from the variation of a properly defined functional. We explain the most successful schemes to solve these control equations and show how to incorporate additional constraints in the pulse design. The algorithms are then applied to simple quantum systems and the obtained pulses are analyzed. Besides the traditional final-time control methods, the tutorial also presents an algorithm and an example to handle time-dependent control targets. <s> BIB015 </s> Quantum control theory and applications: A survey <s> Optimal control <s> We propose an analysis of the time-optimal control of a dissipative two-level quantum system whose dynamics is governed by the Lindblad equation. This simple system allows one to use tools of geometric control theory and to construct its optimal synthesis, i.e. to determine the set of all the optimal trajectories starting from a given initial point. We study different processes such as conversion of a pure state into a mixed state and purification of a mixed state. In particular cases, we show that dissipation is not undesirable and can help accelerating the control. <s> BIB016 </s> Quantum control theory and applications: A survey <s> Optimal control <s> Numerous lines of experimental, numerical and analytical evidence indicate that it is surprisingly easy to locate optimal controls steering quantum dynamical systems to desired objectives. This has enabled the control of complex quantum systems despite the expense of solving the Schrodinger equation in simulations and the complicating effects of environmental decoherence in the laboratory. Recent work indicates that this simplicity originates in universal properties of the solution sets to quantum control problems that are fundamentally different from their classical counterparts. Here, we review studies that aim to systematically characterize these properties, enabling the classification of quantum control mechanisms and the design of globally efficient quantum control algorithms. <s> BIB017 </s> Quantum control theory and applications: A survey <s> Optimal control <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> BIB018 </s> Quantum control theory and applications: A survey <s> Optimal control <s> Transferring the state of a quantum system to a given distribution of populations is an important problem with applications to Quantum Chemistry and Atomic Physics. In this work we consider exact population transfers that minimize the L^2 norm of the control which is typically the amplitude of an electromagnetic field. This problem is analytically and numerically challenging. Except for few exactly solvable cases, there is no general understanding of the nature of optimal controls and trajectories. We find that by examining the limit of large transfer times, we can uncover such general properties. In particular, for transfer times large with respect to the time scale of the free dynamics of the quantum system, the optimal control is a sum of components, each being a Bohr frequency sinusoid modulated by a slow amplitude, i.e. a profile that changes considerably only on the scale of the transfer time. Moreover, we show that the optimal trajectory follows a "mean'' evolution modulated by the fast free dynamics of the system. The calculation of the "mean'' optimal trajectory and the slow control profiles is done via an "averaged'' two-point boundary value problem which we derive and which is much easier to solve than the one expressing the necessary conditions for optimality of the original optimal transfer problem. <s> BIB019 </s> Quantum control theory and applications: A survey <s> Optimal control <s> In this study, in order to generate a sequence of desired quantum states (or quantum bits) for quantum commu- nication and computation, it is more appealing to formulate a quantum control system as a bilinear state reference tracking system. An optimal tracking control is proposed to achieve the state-tracking by solving a Hamilton-Jacobi equation (HJE). In order to avoid the difficulty in solving the HJE with a closed-form solution, the technique of formal tensor power series is employed to treat with the HJE to obtain the optimal tracking control in quantum systems from the ap- proximate design perspective. If the quantum system suffers from stochastic parameter variations, it could be modeled as state-dependent noise. In the situation, stochastic optimal tracking control design is also developed for quantum systems. Finally, several examples are given to illustrate the design procedure and to confirm the performance of the proposed tracking control method. <s> BIB020 </s> Quantum control theory and applications: A survey <s> Optimal control <s> A rapid incoherent control scheme is proposed for driving a quantum system from an initial mixed state to a final state having a high fidelity with a target pure state. This scheme consists of two main steps: rapid purification of the initial mixed state and time-optimal control of the conditional state. The first step rapidly purifies the initial mixed state into an almost-pure state (conditional state) through continuous measurements and feedback control. The second step finds a set of suitable time-optimal controls through the control design of a reference model to drive the quantum system from the conditional state to the final state. The switching time between the two steps is determined by the expected purity or fidelity and the resultant state is an almost-pure state having a high fidelity with the target pure state. As an illustrative example, the authors apply the present rapid incoherent control scheme to a spin system. The potential applications of model predictive control to quantum systems are also discussed briefly. <s> BIB021 </s> Quantum control theory and applications: A survey <s> Optimal control <s> A robust incoherent quantum control scheme via projective measurements plus unitary transformations is proposed for driving a qubit system from an unknown initial mixed state to an arbitrary target pure state. This scheme consists of two main steps: projective measurement on the initial mixed state and optimal control between two pure states. The first step projects the initial state into an eigenstate of the qubit system by projective measurement and guarantees that the proposed scheme is robust to different initial mixed states. The second step finds a set of suitable optimal controls to drive the qubit system from the conditional eigenstate to the target pure state. The connection between the two steps is accomplished by a switching strategy. To accomplish the second step, two approaches are presented in detail. These approaches are time-optimal transition with unbounded control and bang-bang control with minimal switches. The minimal time and minimal number of switches in these approaches can be calculated by simple analytical expressions. The proposed approaches provide two relatively straightforward optimal design methods. <s> BIB022
Most results on the controllability of quantum systems do not provide constructive methods to design a control law for driving a quantum system from an initial state to a predetermined target state. Hence, it is desirable to develop useful methods to design such a control law. A straightforward strategy for constructing a suitable control law is based on a Lie group decomposition, where the desired system evolution operator U f is decomposed as the product of basic unitary operators which are easy to generate using simple controls  ,  ,  . This method is constructive and exact. However, it may be difficult to complete the decomposition for most practical systems. For many practical quantum control problems, optimal control theory is a powerful tool in achieving quantum control objectives BIB001 , BIB015 . In the optimal control approach, the quantum control problem can be formulated as a problem of seeking a set of admissible controls satisfying the system dynamic equations and simultaneously minimizing a cost functional. The cost functional may be different according to the practical requirements of the quantum control problem, such as minimizing the control time BIB005 , BIB016 , the control energy  , BIB019 , the error between the final state and target state, or a combination of these requirements. Many useful tools in traditional optimal control, such as the variational method, the Pontryagin minimum principle and convergent iterative algorithms BIB002 , can be adapted to quantum systems and applied to search for optimal controls. Optimal control techniques have been widely applied to control quantum phenomena in physical chemistry (for details, see, e.g., BIB004 , [11] , BIB007 , BIB009 , BIB017 ) and NMR experiments BIB008 , BIB010 , BIB011 . Here we present a simple example to demonstrate optimal quantum control design. We consider the model (11) and assume u y (t) = 0. We can produce an x rotation of the spin by using radio-frequency (RF) pulses to control the system. The quantum control system becomes From Theorem 5, it is easy to check that the system (24) is operator controllable (completely controllable). It is possible to design a control law which is optimal in some sense to accomplish a specified quantum control task. In some applications, it is desirable to accomplish the control task as soon as possible in order to minimize the relaxation effects BIB005 . For example, it is desirable to perform a quantum computation task with a minimum time using a collection of control resources BIB013 . Hence, the time-optimal control of quantum systems is an interesting practical problem BIB005 , BIB016 . If the control is not bounded, we have the following theorem BIB005 . Theorem 7 Consider the model  . Given any U f ∈ SU (2), there exists a unique , where α, γ ∈ R, and the minimum time required in producing U f is β. If we ignore the global phase, we can denote the initial state |ψ 0 and the target state |ψ f as where θ 0 , θ f ∈ [0, π], ϕ 0 , ϕ f ∈ [0, 2π] and |ψ f = U f |ψ 0 . We can give an analytical expression for the minimal time required to accomplish this control task using geometric manipulations on the Bloch sphere BIB021 . This minimum time β can be calculated according to the following equation: A formal proof of this result can be found in BIB022 . Under the assumption of bounded control (i.e., |u(t)| ≤ V ∈ R + ), an interesting case is bang-bang control where the control u(t) switches between two values ±V . The solutions to many optimal problems take the form of bang-bang control BIB014 . The bang-bang control strategy has been used to suppress decoherence BIB003 (the evolution of pure states to mixed states; e.g., see  ) and to manipulate a class of coherent states  . To reduce the error introduced by the switches, the number of switches can be used as a performance index for optimal control design. Several algorithms have been developed to determine the minimum number of switches required. In one approach, the number of switches is determined by finding the minimum number of factors required in the factorization of U f BIB012 . Another approach involves geometric analysis on the corresponding complex unit sphere (for details, see BIB022 ). From the above example, we can see that, for the quantum system (24), we may wish to optimize different cost functionals depending on the application. We may also use different methods to solve the same optimal quantum control problem. Optimal control is often combined with other strategies, such as closed-loop learning control and quantum feedback control, when manipulating quantum entanglement BIB018 , identifying Hamiltonian parameters BIB006 , controlling chemical reactions BIB004 , BIB007 , and tracking quantum states BIB020 .