Some of the limitations of popular optimal control algorithms such as Lyapunov control, the Krotov method and other iterative algorithms based on functional gradients are considered. Lyapunov control, while elegant and generally effective for ideal systems, struggles for realistic systems. Iterative techniques such as the Krotov method tend to be more effective, but the inclusion of an energy penalty term implies that the resulting solutions are not critical points of the objective functional alone, and do not maximize it. These problems can be avoided by using functional gradients without penalty terms, but even these techniques fail for some problems.
Some of the limitations of popular optimal control algorithms such as Lyapunov control, the Krotov method and other iterative algorithms based on functional gradients are considered. Lyapunov control, while elegant and generally effective for ideal systems, struggles for realistic systems. Iterative techniques such as the Krotov method tend to be more effective, but the inclusion of an energy penalty term implies that the resulting solutions are not critical points of the objective functional alone, and do not maximize it. These problems can be avoided by using functional gradients without penalty terms, but even these techniques fail for some problems.
Mathematical problems of optimal control of quantum systems attract high interest in connection with both fundamental questions and existing and prospective applications. An important problem is the development of methods for constructing controls for quantum systems. One of the widely used methods is the Krotov method. The article is the first part of the review devoted to mathematical aspects of this method for optimal control of quantum systems. Beyond quantum control this method was suggested in the articles [V.F. Krotov, I.N. Feldman, Eng. Cybern. 1983. Vol. 21, 123], [A.I. Konnov, V.F. Krotov, Automat. Remote Control, 1999], and then adopted to optimal quantum control in papers by V.A. Kazakov, V.F. Krotov, R. Kosloff, C.P. Koch, D.J. Tannor, and others. The review considers this method, including cases with constraints on control spectrum and on states of the quantum system, various ways for choosing the improvement function, in particular, linear or linear-quadratic form, etc. We also discuss GRAPE (GRadient Ascent Pulse Engineering) and CRAB (Chopped Random-Basis) methods, as well as combined methods such as the method suggested by I. Maday and G. Turinici for generalizing the 1st order Krotov method and the Zhu-Rabitz method. Nowadays, various versions of the Krotov method are used as a key tool in optimal quantum control, and the review describes applications of this method to mathematical control problems for molecular dynamics, Bose-Einstein condensate, nuclear-magnetic resonance, quantum gates generation, etc. The first part of the review is devoted to closed quantum systems.
Efficient algorithms are of fundamental importance for the discovery of optimal control designs for coherent control of quantum processes. One important class of algorithms is sequential update algorithms, generally attributed to Krotov. Although widely and often successfully used, the associated theory is often involved and leaves many crucial questions unanswered, from the monotonicity and convergence of the algorithm to discretization effects, leading to the introduction of ad hoc penalty terms and suboptimal update schemes detrimental to the performance of the algorithm. We present a general framework for sequential update algorithms including specific prescriptions for efficient update rules with inexpensive dynamic search length control, taking into account discretization effects and eliminating the need for ad hoc penalty terms. The latter, despite being necessary for regularizing the problem in the limit of infinite time resolution, i.e. the continuum limit, are shown to be undesirable and unnecessary in the practically relevant case of finite time resolution. Numerical examples show that the ideas underlying many of these results extend even beyond what can be rigorously proved.
One of the most popular methods for solving numerical optimal control problems is the Krotov method, adapted for quantum control by Tannor and coworkers. The Krotov method has the following three appealing properties: (1) monotonic increase of the objective with iteration number, (2) no requirement for a line search, leading to a significant savings over gradient (first-order) methods, and (3) macrosteps at each iteration, resulting in significantly faster growth of the objective at early iterations than in gradient methods where small steps are required. The principal drawback of the Krotov method is slow convergence at later iterations, which is particularly problematic when high fidelity is desired. We show here that, near convergence, the Krotov method degenerates to a first-order gradient method. We then present a variation on the Krotov method that has all the advantages of the original Krotov method but with significantly enhanced convergence (second-order or quasi-Newton) as the optimal solution is approached. We illustrate the method by controlling the three-dimensional dynamics of the valence electron in the Na atom.
Abstract The mathematics of optimal control of quantum systems is of great interest in connection with fundamental problems of physics as well as with existing and prospective applications to quantum technologies. One important problem is the development of methods for constructing controls for quantum systems. One of the commonly used methods is the Krotov method, which was initially proposed outside of quantum control theory in articles by Krotov and Feldman (1978, 1983). This method was used to develop a novel approach to finding optimal controls for quantum systems in [64] (Tannor, Kazakov, and Orlov, 1992), [65] (Somlói, Kazakov, and Tannor, 1993), and in many other works by various scientists. Our survey discusses mathematical aspects of this method for optimal control of closed quantum systems. It outlines various modifications with different forms of the improvement function (for example, linear or linear-quadratic), different constraints on the control spectrum and on the admissible states of the quantum system, different regularisers, and so on. The survey describes applications of the Krotov method to controlling molecular dynamics and Bose–Einstein condensates, and to quantum gate generation. This method is compared with the GRAPE (GRadient Ascent Pulse Engineering) method, the CRAB (Chopped Random-Basis) method, and the Zhu–Rabitz and Maday–Turinici methods. Bibliography: 158 titles.
We investigate the problem of control of quantum systems. Various notions of controllability of quantum systems are discussed and Lie algebraic techniques are employed to derive necessary and sufficient conditions for the resulting degrees of controllability. The results are employed to study the degree of controllability for various model systems. The problem of optimal control is formulated as a problem of maximisation of the expectation value of an observable of the system subject to dynamical constraints and costs. A class of iterative algorithms is discussed, and shown to converge to a solution of the Euler-Lagrange equations, which is a necessary condition for optimality of a solution. A numerical realisation of the algorithm is presented and potential implementation of the resulting control fields in the laboratory is discussed. The algorithm is applied to find optimal control fields for a variety of applications including selective and simultaneous excitation of individual quantum dots in a globally addressed ensemble, creation of superposition states, and selective excitation of certain vibrational modes of a simple molecule such as Hydrogen Fluoride. In each case we systematically explore the parameter space of the algorithm to select solutions with desirable physical features from the multitude of possible choices. As there are many possible optimal control fields which give similar yields the question of whether some are more robust than others is investigated. A number of optimal fields are analysed quantitatively in terms of robustness with regard to temporal and spectral noise affecting the fields, uncertainty in the model parameters and the effect of experimental limitations such as restrictions on the bandwidth of the fields. It is shown that whilst the solutions tend to be sensitive to uncertainty in the energy levels chosen for the model, they appear to be surprisingly tolerant to even severe bandwidth limitations.