Topology of the quantum control landscape for observables
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For , 1000 pairs of nondegenerate and were randomly generated. The HQF [Eq. (15)] was computed for each of the 1000 pairs, and the number of positive Hessian eigenvalues were enumerated at the optima. For , there are critical values of the landscape, and graphed is the sample number of positive Hessian eigenvalues averaged over the 100 pairs of and at each of the critical values, ranked from global minimum to global maximum and labeled on the horizontal axis as 1,2,3,…,720 (i.e., each point in the figure corresponds to one of the 720 critical values). With maximization of landscape value as the goal, positive Hessian eigenvalues generally correspond to desirable directions at local saddle optima, leading to advancement toward the optimization objective. This illustration shows a general trend found (in other examples) as well of a decrease in the number of positive Hessian eigenvalues for higher critical landscape values. The bar corresponds to a one standard deviation spread for the 1000 sample points for the 480th critical value, which a typical spread observed for other critical values.
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