20 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 in the presence of singular critical points
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.Comment: 14 pages, 2 figure
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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
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
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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