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 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.

Quantum optimal control has enjoyed wide success for a variety of theoretical and experimental objectives. These favorable results have been attributed to advantageous properties of the corresponding control landscapes, which are free from local optima if three conditions are met: (1) the quantum system is controllable, (2) the Jacobian of the map from the control field to the evolution operator is full rank, and (3) the control field is not constrained. This paper explores how gradient searches for globally optimal control fields are affected by deviations from assumption (2). In some quantum control problems, so-called singular critical points, at which the Jacobian is rank-deficient, may exist on the landscape. Using optimal control simulations, we show that search failure is only observed when a singular critical point is also a second-order trap, which occurs if the control problem meets additional conditions involving the system Hamiltonian and/or the control objective. All known second-order traps occur at constant control fields, and we also show that they only affect searches that originate very close to them. As a result, even when such traps exist on the control landscape, they are unlikely to affect well-designed gradient optimizations under realistic searching conditions.

The success of quantum optimal control for both experimental and theoretical objectives is connected to the topology of the corresponding control landscapes, which are free from local traps if three conditions are met: (1) the quantum system is controllable, (2) the Jacobian of the map from the control field to the evolution operator is of full rank, and (3) there are no constraints on the control field. This paper investigates how the violation of assumption (3) affects gradient searches for globally optimal control fields. The satisfaction of assumptions (1) and (2) ensures that the control landscape lacks fundamental traps, but certain control constraints can still introduce artificial traps. Proper management of these constraints is an issue of great practical importance for numerical simulations as well as optimization in the laboratory. Using optimal control simulations, we show that constraints on quantities such as the number of control variables, the control duration, and the field strength are potentially severe enough to prevent successful optimization of the objective. For each such constraint, we show that exceeding quantifiable limits can prevent gradient searches from reaching a globally optimal solution. These results demonstrate that careful choice of relevant control parameters helps to eliminate artificial traps and facilitate successful optimization.

The growing successes in performing quantum control experiments motivated the development of control landscape analysis as a basis to explain these findings. When a quantum system is controlled by an electromagnetic field, the observable as a functional of the control field forms a landscape. Theoretical

The growing successes in performing quantum control experiments motivated the development of control landscape analysis as a basis to explain these findings. When a quantum system is controlled by an electromagnetic field, the observable as a functional of the control field forms a landscape. Theoretical analyses have predicted the existence of critical points over the landscapes, including saddle points with indefinite Hessians. This paper presents a systematic experimental study of quantum control landscape saddle points. Nuclear magnetic resonance control experiments are performed on acoupled two-spin system in a l3C-labeled chloroform ( I3CHC13) sample. We address the saddles with a combined theoretical and experimental approach, measure the Hessian at each identified saddle point, and study how their presence can influence the search effort utilizing a gradient algorithm to seek an optimal control outcome. The results have significance beyond spin systems, as landscape saddles are expected to be present for the control of broad classes of quantum systems.