Existing algorithms for the optimal control of quantum observables are based on locally optimal steps in the space of control fields, or as in the case of genetic algorithms, operate on the basis of heuristics that do not explicitly take into account details pertaining to the geometry of the search space. We present globally efficient algorithms for quantum observable control that follow direct or close-to-direct paths in the domain of unitary dynamical propagators, based on partial reconstruction of these propagators at successive points along the search trajectory through orthogonal observable measurements. These algorithms can be implemented experimentally and offer an alternative to the adaptive learning control approach to optimal control experiments (OCE). Their performance is compared to that of local gradient-based control optimization.
Existing algorithms for the optimal control of quantum observables are based on locally optimal steps in the space of control fields, or as in the case of genetic algorithms, operate on the basis of heuristics that do not explicitly take into account details pertaining to the geometry of the search space. We present globally efficient algorithms for quantum observable control that follow direct or close-to-direct paths in the domain of unitary dynamical propagators, based on partial reconstruction of these propagators at successive points along the search trajectory through orthogonal observable measurements. These algorithms can be implemented experimentally and offer an alternative to the adaptive learning control approach to optimal control experiments (OCE). Their performance is compared to that of local gradient-based control optimization.
The laboratory closed-loop optimal control of quantum phenomena, expressed as minimizing a suitable cost functional, is currently implemented through an optimization algorithm coupled to the experimental apparatus. In practice, the most commonly used search algorithms are variants of genetic algorithms. As an alternative choice, a direct search deterministic algorithm is proposed in this paper. For the simple simulations studied here, it outperforms the existing approaches. An additional algorithm is introduced in order to reveal some properties of the cost functional landscape.
The optimal control of unitary transformations is a fundamental problem in quantum control theory and quantum information processing. The feasibility of performing such optimizations is determined by the computational and control resources required, particularly for systems with large Hilbert spaces. Prior work on unitary transformation control indicates that (i) for controllable systems, local extrema in the search landscape for optimal control of quantum gates have null measure, facilitating the convergence of local search algorithms, but (ii) the required time for convergence to optimal controls can scale exponentially with the Hilbert space dimension. Depending on the control-system Hamiltonian, the landscape structure and scaling may vary. This work introduces methods for quantifying Hamiltonian-dependent and kinematic effects on control optimization dynamics in order to classify quantum systems according to the search effort and control resources required to implement arbitrary unitary transformations.
This paper explores an adaption of the coordinate ascent approach to quantum control problems. It was motivated by the observation that several of the existing monotone-converging schemes for quantum control may be viewed as approximations of the well-known coordinate ascent method. Our implementation employs line searches in coordinate directions in control space using only evaluations of the performance functional, without invoking its derivatives. It is based on recasting the performance functional as a local tracking function which gives the „future” quality of a control at each moment in time. Back propagation of a basis of the target space enables these performance functional (i.e., tracking function) evaluations to be done efficiently. The performance functional may include, or not include, regularization terms as needed. Convergence of the resulting algorithm and its relation to previous schemes are discussed, and numerical examples are provided. In our tests, the coordinate ascent algorithm exhibited rapid convergence to high yield controls. Moreover, it is readily adapted to different restrictions on the controls imposed by the physics of the system under study, e.g., controls with spectrum restrictions in laser control problems, and discrete controls on a specified time grid in electronically controlled nanodevices, or in NMR.
We describe algorithms and experimental strategies for the Pareto optimal control problem of simultaneously driving an arbitrary number of quantum observable expectation values to their respective extrema. Conventional quantum optimal control strategies are less effective at sampling points on the Pareto frontier of multiobservable control landscapes than they are at locating optimal solutions to single observable control problems. The present algorithms facilitate multiobservable optimization by following direct paths to the Pareto front, and are capable of continuously tracing the front once it is found to explore families of viable solutions. The numerical and experimental methodologies introduced are also applicable to other problems that require the simultaneous control of large numbers of observables, such as quantum optimal mixed state preparation.
The dynamics of closed quantum systems may be manipulated by using an applied field to achieve a control objective value for a physical goal. The functional relationship between the applied field and the objective value forms a quantum control landscape, and the optimization process consists of a guided climb up the landscape from the bottom to the top. Two classes of landscape features are important for understanding the ease of finding an optimal control field. The first class of topological landscape features has been proven to be especially simple in that no suboptimal local maxima exist (upon satisfaction of certain assumptions), which partially accounts for the ease of finding optimal fields. Complementary to the topology, the second class of features entails the landscape structure, characterizing the sinuous nature of the paths leading to an optimal control field. Previous work found that the landscape structure is also particularly simple, as excursions up the landscape guided by a gradient algorithm correspond to nearly straight paths through the space of control fields. In this paper we take an alternative approach to examining landscape structure by constructing, and then following, exactly straight trajectories in control space. Each trajectory starts at a corresponding point on the bottom of the landscape and ends at an associated point on the top, with the observable values taken either as the state-to-state transition probability, the expectation value of a general observable, or the distance from a desired unitary transformation. In some cases the starting point is at a suboptimal critical-point saddle, with the goal, again, of following a straight field path to the optimal objective yield or another suboptimal critical point. We find that the objective value almost always rises monotonically upon following a straight control path from one critical point to another, which shows that landscape structure is very simple, being devoid of rough bumps and gnarled ``twists and turns''. An analysis reveals that the generally featureless nature of quantum control landscapes can be understood in terms of the occurrence of many interfering quantum pathways contributing while traversing the landscape, essentially smoothing out the terrain. These results also provide a basis for further studies to seek a new efficient algorithm to discover optimal fields by means of taking into account the inherently smooth landscape structure.
The broad success of theoretical and experimental quantum optimal control is intimately connected to the topology of the underlying control landscape. For several common quantum control goals, including the maximization of an observable expectation value, the landscape has been shown to lack local optima if three assumptions are satisfied: (i) the quantum system is controllable, (ii) the Jacobian of the map from the control field to the evolution operator is full rank, and (iii) the control field is not constrained. In the case of the observable objective, this favorable analysis shows that the associated landscape also contains saddles, i.e., critical points that are not local suboptimal extrema. In this paper, we investigate whether the presence of these saddles affects the trajectories of gradient-based searches for an optimal control. We show through simulations that both the detailed topology of the control landscape and the parameters of the system Hamiltonian influence whether the searches are attracted to a saddle. For some circumstances with a special initial state and target observable, optimizations may approach a saddle very closely, reducing the efficiency of the gradient algorithm. Encounters with such attractive saddles are found to be quite rare. Neither the presence of a large number of saddles on the control landscape nor a large number of system states increases the likelihood that a search will closely approach a saddle. Even for applications that encounter a saddle, well-designed gradient searches with carefully chosen algorithmic parameters will readily locate optimal controls.