A theoretical analysis of quantum control landscapes

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 H ${}_{2}$ O-D ${}_{2}$ O 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.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 H${}_{2}$O-D${}_{2}$O 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.

arXiv:1502.04045Physical Review AMar 12, 201422 citations

Exploring Quantum Control Landscapes

Vincent Beltrani

Jan 1, 20123 citations

Control landscapes for observable preparation with open quantum systems

Re-Bing Wu, Alexander Nikolaevich Pechen, HERSCHEL A. RABITZ

A quantum control landscape is defined as a function(al) of the system control variables that are to be optimized, and it was introduced to understand the successes in quantum control experiments. This paper explores the control landscape for observable preparation under the circumstance where the environment has an influence on the quantum system to be controlled. Provided that the open system is fully controllable, the analysis proves that there are no false traps to be encountered on the way to finding the absolute landscape maximum. This provides theoretical support for the observed ease of obtaining high quality controls in the laboratory, and opens up perspectives to develop more efficient quantum control algorithms.A quantum control landscape is defined as a function(al) of the system control variables that are to be optimized, and it was introduced to understand the successes in quantum control experiments. This paper explores the control landscape for observable preparation under the circumstance where the environment has an influence on the quantum system to be controlled. Provided that the open system is fully controllable, the analysis proves that there are no false traps to be encountered on the way to finding the absolute landscape maximum. This provides theoretical support for the observed ease of obtaining high quality controls in the laboratory, and opens up perspectives to develop more efficient quantum control algorithms.

IEEE Conference on Decision and Control (CDC)Jan 1, 20072 citations

Exploring the top and bottom of the quantum control landscape

Aug 18, 201627 citations

Experimental observation of saddle points over the quantum control landscape of a two-spin system

Qiuyang Sun, István Pelczer, Gregory Riviello +2 more

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 $^{13}\mathrm{C}$ -labeled chloroform $({}^{13}{\mathrm{CHCl}}_{3})$ 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.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 $^{13}\mathrm{C}$-labeled chloroform $({}^{13}{\mathrm{CHCl}}_{3})$ 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.

Physical Review AApr 17, 201518 citations

Directions in the Theory of Quantum Control

Domenico D’Alessandro

Springer eBooksAug 10, 20077 citations

{}_{2}

O-D

{}_{2}

O 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.

^{13}\mathrm{C}

-labeled chloroform

({}^{13}{\mathrm{CHCl}}_{3})

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.