Quantum control theory and applications: A survey <s> Robust control <s> Recently, systematic optimal control theory has been successfully applied to design optical fields which act on a molecule to guide or transform it into a specifically prescribed state. Under realistic laboratory conditions and for real molecules, it is inevitable that there will exist disturbances and uncertainties in the control optical field, in the molecular Hamiltonian, and in the field-coupling coefficient (e.g., the dipole moment), etc. Therefore, it is important to design an optimal field which is robust to disturbances. In this paper, we provide a method through which the worst possible disturbance to the control process, under certain possible constraints, can be calculated and a corresponding optimal control field designed to be robust to the disturbances. Under disturbances of small amplitude, a perturbation solution to this problem can be applied. Illustrations are given for control of the wave-packet evolution of a diatomic molecule in the presence of disturbances. <s> BIB001 </s> Quantum control theory and applications: A survey <s> Robust control <s> In this paper we analyze the controllability of quantum systems arising in molecular dynamics. We model these systems as systems with finite numbers of levels, and examine their controllability. To do this we pass to their unitary generators and use results on the controllability of invariant systems on Lie groups. Examples of molecular systems, modeled as finite-dimensional control systems, are provided. A simple algorithm to detect the controllability of a molecular system is provided. Finally, we apply this algorithm to a five-level system. <s> BIB002 </s> Quantum control theory and applications: A survey <s> Robust control <s> It was shown recently [D.A. Lidar et al., Phys. Rev. Lett. 81, 2594 (1998)] that within the framework of the semigroup Markovian master equation, decoherence-free (DF) subspaces exist which are stable to first order in time to a perturbation. Here this result is extended to the non-Markovian regime and generalized. In particular, it is shown that within both the semigroup and the non-Markovian operator sum representation, DF subspaces are stable to all orders in time to a symmetry-breaking perturbation. DF subspaces are thus ideal for quantum memory applications. For quantum computation, however, the stability result does not extend beyond the first order. Thus, to perform robust quantum computation in DF subspaces, they must be supplemented with quantum error correcting codes. <s> BIB003 </s> Quantum control theory and applications: A survey <s> Robust 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> BIB004 </s> Quantum control theory and applications: A survey <s> Robust control <s> The impact of control field fluctuations on the optimal manipulation of quantum dynamics phenomena is investigated. The quantum system is driven by an optimal control field, with the physical focus on the evolving expectation value of an observable operator. A relationship is shown to exist between the system dynamics and the control field fluctuations, wherein the process of seeking optimal performance assures an inherent degree of system robustness to such fluctuations. The presence of significant field fluctuations breaks down the evolution of the observable expectation value into a sequence of partially coherent robust steps. Robustness occurs because the optimization process reduces sensitivity to noise-driven quantum system fluctuations by taking advantage of the observable expectation value being bilinear in the evolution operator and its adjoint. The consequences of this inherent robustness are discussed in the light of recent experiments and numerical simulations on the optimal control of quantum phenomena. The analysis in this paper bodes well for the future success of closed-loop quantum optimal control experiments, even in the presence of reasonable levels of field fluctuations. <s> BIB005 </s> Quantum control theory and applications: A survey <s> Robust control <s> This paper gives a unified approach to feedback control theory of quantum mechanical systems of bosonic modes described by noncommutative operators. A quantum optical closed-loop, including a plant and controller, is developed and its fundamental structural properties are analyzed extensively from a purely quantum mechanical point of view, in order to facilitate the use of control theory in microscopic world described by quantum theory. In particular, an input-output description of quantum mechanical systems which is essential in describing the behavior of the feedback systems is fully formulated and developed. This would then provide a powerful tool for quantum control and pave an avenue that connects control theory to quantum dynamics. This paper is divided into two parts. The first part is devoted to the basic formulation of quantum feedback control via quantum communication and local operations on an optical device, cavity, that can be regarded as a unit of quantum dynamics of bosonic modes. The formulation introduced in this paper presents the feature intrinsic in quantum feedback systems based on quantum stochastic differential equations. The input-output description provides a basis for developing quantum feedback control through the transfer function representation of quantum feedback systems. In the follow-up paper, the quantum mechanical representation of feedback is further elaborated to yield the control theoretical representation of fundamental notions of quantum theory, uncertainty principle, e.g., and some applications are presented. <s> BIB006 </s> Quantum control theory and applications: A survey <s> Robust control <s> Based on the stochastic differential equation of quantum mechanical feedback obtained in the first part of this paper, detailed control concepts and applications are discussed for quantum systems interacting with a noncommutative noise source. A feedback system in our framework is purely nonclassical in the sense that feedback control is performed via local operation and quantum communication through a quantum channel. The role of the controller is to alter the quantum dynamic characteristics of the plant through entanglement, shared between the plant and controller by sending quantum states, that is modulated by the Hamiltonian on the controller. The input-output relation of quantum systems provides a natural extension of control theory to the quantum domain. This enables one to present a control theoretical interpretation of some fundamental quantum mechanical notions such as the uncertainty principle, but also to find applications of ideas and tools of control theory. One of the most important applications is the production of squeezed states, which has been an important issue of quantum theory in relation to quantum computation and quantum communication. The method proposed here reduces the application to a conventional noise reduction problem with feedback. The H/sub /spl infin// control then leads to complete squeezing. <s> BIB007 </s> Quantum control theory and applications: A survey <s> Robust control <s> We are currently in the midst of a second quantum revolution. The first quantum revolution gave us new rules that govern physical reality. The second quantum revolution will take these rules and use them to develop new technologies. In this review we discuss the principles upon which quantum technology is based and the tools required to develop it. We discuss a number of examples of research programs that could deliver quantum technologies in coming decades including; quantum information technology, quantum electromechanical systems, coherent quantum electronics, quantum optics and coherent matter technology. <s> BIB008 </s> Quantum control theory and applications: A survey <s> Robust 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> BIB009 </s> Quantum control theory and applications: A survey <s> Robust control <s> A Lyapunov-based approach for trajectory tracking of the Schrodinger equation is proposed. In the finite dimensional case, convergence is precisely analyzed. Connection between the controllability of the linearized system around the reference trajectory and asymptotic tracking is studied. When the linearized system is controllable, such a feedback ensures almost global asymptotic convergence. When the linearized system is not controllable, the stability of the closed-loop system is not asymptotic. To overcome such lack of convergence, we propose, when the reference trajectory is an eigenstate, a modification based on adiabatic invariance. Simulations illustrate the simplicity and also the interest for trajectory generation. <s> BIB010 </s> Quantum control theory and applications: A survey <s> Robust control <s> In this paper we are concerned with the problem of stability for quantum feedback networks. We demonstrate in the context of quantum optics how stability of quantum feedback networks can be guaranteed using only simple gain inequalities for network components and algebraic relationships determined by the network. Quantum feedback networks are shown to be stable if the loop gain is less than one--this is an extension of the famous small gain theorem of classical control theory. We illustrate the simplicity and power of the small gain approach with applications to important problems of robust stability and robust stabilization. <s> BIB011 </s> Quantum control theory and applications: A survey <s> Robust control <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> BIB012 </s> Quantum control theory and applications: A survey <s> Robust control <s> This paper proposes a new robust control method for quantum systems with uncertainties involving sliding mode control (SMC). Sliding mode control is a widely used approach in classical control theory and industrial applications. We show that SMC is also a useful method for robust control of quantum systems. In this paper, we define two specific classes of sliding modes (i.e., eigenstates and state subspaces) and propose two novel methods combining unitary control and periodic projective measurements for the design of quantum sliding mode control systems. Two examples including a two-level system and a three-level system are presented to demonstrate the proposed SMC method. One of main features of the proposed method is that the designed control laws can guarantee desired control performance in the presence of uncertainties in the system Hamiltonian. This sliding mode control approach provides a useful control theoretic tool for robust quantum information processing with uncertainties. <s> BIB013
In practical applications, it is unavoidable that quantum systems are subject to all kinds of disturbances and uncertainties BIB001 . Many instances of incomplete knowledge and unknown errors can also be treated as uncertainties. Hence, robustness has been identified as an important aspect for the practical application of quantum technology BIB011 , BIB003 , BIB005 . Recently, several robust control methods in classical control theory have been extended into quantum domain. For example, the small gain theorem has been applied to the stability analysis of quantum feedback networks BIB011 . A transfer function approach has been applied to the feedback and robust control for single input single output quantum systems BIB006 , BIB007 . An H ∞ controller synthesis problem has been formulated for a class of linear quantum stochastic systems BIB012 . A sliding mode control approach has been applied to robust control design for quantum systems where bounded uncertainties exist in the system Hamiltonian BIB013 . Some open-loop and feedback approaches to quantum control can lead to a certain degree of robustness. Since feedback plays a key role in robust control, in this survey we present robust control within the framework of quantum feedback control. In the following, we will only present several specific results on H ∞ control for quantum systems, which is based on the reference BIB012 . Consider the quantum system to be controlled to be described by the following LQSDE defined in an analogous way to BIB008 : where the input w(t) represents a disturbance signal of the form BIB002 . Also, dv(t) represents any additional quantum noise in the system. The control input u(t) can be written as where β u (t) is the adapted, self-adjoint part of u(t) andũ(t) is the noise part of u(t). The vectors v(t),w(t), andũ(t) are quantum noises with positive semidefinite Hermitian Itô matrices F v , Fw and Fũ. The controller is assumed to be of the following form where ξ(t) = [ξ 1 (t) . . . ξ n (t)] T is a vector of self-adjoint controller variables. The noise T is a vector of noncommutative Wiener processes with canonical Hermitian Itô matrix F v K . By identifying β u = C K ξ(t) and interconnecting BIB004 and (37), we can obtain the resulting closed-loop system (see BIB012 ). The goal of the H ∞ controller synthesis problem is to find a controller (37) for a given disturbance attenuation parameter g > 0 such that the resulting closed-loop system satisfies (for details, see BIB012  for some real constants ǫ, µ 1 , µ 2 > 0, where Furthermore, we suppose that the system (35) satisfies the following assumption ] is full rank for all ω ≥ 0; 4) The matrix [ ] is full rank for all ω ≥ 0. The results on quantum H ∞ control will be stated in terms of the following pair of algebraic Riccati equations: where X and Y are positive definite symmetric matrices. The solutions to these Riccati equations will be required to satisfy the following assumption (Assumption B): is a stability matrix; 3) The matrix XY has a spectral radius strictly less than one. If the Riccati equations (39) and (40) have solutions satisfying Assumption B, a controller of the form (37) will solve the H ∞ control problem under consideration and the system matrices of the controller can be constructed from the Riccati solutions as follows: The necessary and sufficient conditions on H ∞ controller synthesis can be described as follows: Theorem 12 Necessity. Consider the system BIB004 and suppose that Assumption A is satisfied. If there exists a controller of the form BIB009 such that the resulting closedloop system is strictly bounded real with disturbance attenuation g, then the Riccati equations (39) and (40) will have stabilizing solutions X ≥ 0 and Y ≥ 0 satisfying Assumption B. Sufficiency. Suppose the Riccati equations (39) and (40) have stabilizing solutions X ≥ 0 and Y ≥ 0 satisfying Assumption B. If the controller (37) is such that the matrices A K , B K , C K are as defined in BIB010 , then the resulting closed-loop system will be strictly bounded real with disturbance attenuation g. It is worth noting that when the controller which is constructed via this approach is a quantum controller, we need to take into account the physical realizability conditions using Theorem 11.