25 research outputs found
Experimental exploration over a quantum control landscape through nuclear magnetic resonance
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 revealed many properties of control landscapes, especially
regarding their slopes, curvatures, and topologies. A full experimental
assessment of the landscape predictions is important for future consideration
of controlling quantum phenomena. Nuclear magnetic resonance (NMR) is exploited
here as an ideal laboratory setting for quantitative testing of the landscape
principles. The experiments are performed on a simple two-level proton system
in a HO-DO sample. We report a variety of NMR experiments roving over
the control landscape based on estimation of the gradient and Hessian,
including ascent or descent of the landscape, level set exploration, and an
assessment of the theoretical predictions on the structure of the Hessian. The
experimental results are fully consistent with the theoretical predictions. The
procedures employed in this study provide the basis for future multispin
control landscape exploration where additional features are predicted to exist
Searching for quantum optimal controls under severe constraints
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.Comment: 16 pages, 7 figure
Search complexity and resource scaling for the quantum optimal control of unitary transformations
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
Recommended from our members
Searching for an optimal control in the presence of saddles on the quantum-mechanical observable landscape
Physical Review A.
Volume 95, Issue 6, 22 June 2017, Article number 063418.© 2017 American Physical Society. 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
Recommended from our members
Experimental observation of saddle points over the quantum control landscape of a two-spin system
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 a coupled two-spin system in a C-13-labeled chloroform ((CHCl3)-C-13) 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