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Local topology at limited resource induced suboptimal traps on the quantum control landscape

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Abstract

In a quantum optimal control experiment a system is driven towards a target observable value with a tailored external field. The underlying quantum control landscape, defined by the observable as a function of the control variables, lacks suboptimal extrema upon satisfaction of certain physical assumptions. This favorable topology implies that upon climbing the landscape to seek an optimal control field, a steepest ascent algorithm should not halt prematurely at suboptimal critical points, or traps. One of the important aforementioned assumptions is that no limitations are imposed on the control resources. Constraints on the control restricts access to certain regions of the landscape, potentially preventing optimal performance through convergence to limited resource induced suboptimal traps. This work develops mathematical tools to explore the local landscape structure around suboptimal critical points. The landscape structure may be favorably altered by systematically relaxing the control resources. In this fashion, isolated suboptimal critical points may be transformed into extensive level sets and then to saddle points permitting further landscape ascent. Time-independent kinematic controls are employed as stand-ins for traditional dynamic controls to allow for performing a simpler constrained resource landscape analysis. The kinematic controls can be directly transferred to their dynamic counterparts at any juncture of the kinematic analysis. The numerical simulations employ a family of landscape exploration algorithms while imposing constraints on the kinematic controls. Particular algorithms are introduced to meet the goals of either climbing the landscape or seeking specific changes in the topology of the landscape by relaxing the control resources.

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Acknowledgments

A.D. acknowledges support from the Program in Plasma Science and Technology at Princeton University and thanks David Hocker (Princeton University) for his assistance in generating Fig. 1. The authors thank Professor Carey Rosenthal (Drexel University) for insightful discussions. We also acknowledge support from the DOE (Grant Number DE-FG-02-02ER15344), NSF (Grant Number CHE-1058644), and the ARO (Grant Number W911NF-13-1-0237).

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  1. Department of Chemistry, Princeton University, Princeton, NJ, USA

    Ashley Donovan, Vincent Beltrani & Herschel Rabitz

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  1. Ashley Donovan
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  2. Vincent Beltrani
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  3. Herschel Rabitz
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Correspondence to Herschel Rabitz.

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Donovan, A., Beltrani, V. & Rabitz, H. Local topology at limited resource induced suboptimal traps on the quantum control landscape. J Math Chem 52, 407–429 (2014). https://doi.org/10.1007/s10910-013-0269-x

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