TY - JOUR

T1 - Topological and statistical properties of quantum control transition landscapes

AU - Hsieh, Michael

AU - Wu, Rebing

AU - Rosenthal, Carey

AU - Rabitz, Herschel Albert

PY - 2008/4/14

Y1 - 2008/4/14

N2 - A puzzle arising in the control of quantum dynamics is to explain the relative ease with which high-quality control solutions can be found in the laboratory and in simulations. The emerging explanation appears to lie in the nature of the quantum control landscape, which is an observable as a function of the control variables. This work considers the common case of the observable being the transition probability between an initial and a target state. For any controllable quantum system, this landscape contains only global maxima and minima, and no local extrema traps. The probability distribution function for the landscape value is used to calculate the relative volume of the region of the landscape corresponding to good control solutions. The topology of the global optima of the landscape is analysed and the optima are shown to have inherent robustness to variations in the controls. Although the relative landscape volume of good control solutions is found to shrink rapidly as the system Hilbert space dimension increases, the highly favourable landscape topology at and away from the global optima provides a rationale for understanding the relative ease of finding high-quality, stable quantum optimal control solutions.

AB - A puzzle arising in the control of quantum dynamics is to explain the relative ease with which high-quality control solutions can be found in the laboratory and in simulations. The emerging explanation appears to lie in the nature of the quantum control landscape, which is an observable as a function of the control variables. This work considers the common case of the observable being the transition probability between an initial and a target state. For any controllable quantum system, this landscape contains only global maxima and minima, and no local extrema traps. The probability distribution function for the landscape value is used to calculate the relative volume of the region of the landscape corresponding to good control solutions. The topology of the global optima of the landscape is analysed and the optima are shown to have inherent robustness to variations in the controls. Although the relative landscape volume of good control solutions is found to shrink rapidly as the system Hilbert space dimension increases, the highly favourable landscape topology at and away from the global optima provides a rationale for understanding the relative ease of finding high-quality, stable quantum optimal control solutions.

UR - http://www.scopus.com/inward/record.url?scp=42549088322&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=42549088322&partnerID=8YFLogxK

U2 - 10.1088/0953-4075/41/7/074020

DO - 10.1088/0953-4075/41/7/074020

M3 - Article

AN - SCOPUS:42549088322

SN - 0953-4075

VL - 41

JO - Journal of Physics B: Atomic, Molecular and Optical Physics

JF - Journal of Physics B: Atomic, Molecular and Optical Physics

IS - 7

M1 - 074020

ER -