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Issue 1, 2015
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Exploring the complexity of quantum control optimization trajectories

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Abstract

The control of quantum system dynamics is generally performed by seeking a suitable applied field. The physical objective as a functional of the field forms the quantum control landscape, whose topology, under certain conditions, has been shown to contain no critical point suboptimal traps, thereby enabling effective searches for fields that give the global maximum of the objective. This paper addresses the structure of the landscape as a complement to topological critical point features. Recent work showed that landscape structure is highly favorable for optimization of state-to-state transition probabilities, in that gradient-based control trajectories to the global maximum value are nearly straight paths. The landscape structure is codified in the metric R ≥ 1.0, defined as the ratio of the length of the control trajectory to the Euclidean distance between the initial and optimal controls. A value of R = 1 would indicate an exactly straight trajectory to the optimal observable value. This paper extends the state-to-state transition probability results to the quantum ensemble and unitary transformation control landscapes. Again, nearly straight trajectories predominate, and we demonstrate that R can take values approaching 1.0 with high precision. However, the interplay of optimization trajectories with critical saddle submanifolds is found to influence landscape structure. A fundamental relationship necessary for perfectly straight gradient-based control trajectories is derived, wherein the gradient on the quantum control landscape must be an eigenfunction of the Hessian. This relation is an indicator of landscape structure and may provide a means to identify physical conditions when control trajectories can achieve perfect linearity. The collective favorable landscape topology and structure provide a foundation to understand why optimal quantum control can be readily achieved.

Graphical abstract: Exploring the complexity of quantum control optimization trajectories

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Publication details

The article was received on 27 Aug 2014, accepted on 30 Oct 2014 and first published on 07 Nov 2014


Article type: Paper
DOI: 10.1039/C4CP03853C
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Phys. Chem. Chem. Phys., 2015,17, 334-347

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    Exploring the complexity of quantum control optimization trajectories

    A. Nanduri, O. M. Shir, A. Donovan, T. Ho and H. Rabitz, Phys. Chem. Chem. Phys., 2015, 17, 334
    DOI: 10.1039/C4CP03853C

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