On Comparison of Quantum Control Landscapes with Different Trapping Features

In this work, we consider a general method for numerically comparing the efficiency of optimization over control landscapes with different trapping features, both for unconstrained (using GRAPE) and constrained (using gradient projection method) optimization. Traps are controls which locally appear optimal while not being globally optimal. Such controls can significantly affect the efficiency of finding controls close to optimal with local search, e.g. gradient-based, algorithms. As an explicit example, we apply the general method to study unconstrained control landscapes for two three-level quantum Ξ systems which are widely used in quantum optics and known to have different trapping features: a non-degenerate Ξ -system for which the null control is a third-order trap, and a degenerate Ξ -system for which the null control is a seventh-order trap. Using the general approach, we perform a comparative study of the impact of the null control on the efficiency of optimization and estimate the dependence of this impact on the maximal allowed number of iterations to check how the optimization is sensitive to this number. The results show that while quantitative behavior is different, the qualitative difference between the two control landscapes remains similar.