Optimization search effort over the control landscapes for open quantum systems with Kraus-map evolution
A quantum control landscape is defined as the expectation value of a target observable $Θ$ as a function of the control variables. In this work control landscapes for open quantum systems governed by Kraus map evolution are analyzed. Kraus maps are used as the controls transforming an initial density matrix $ρ_{\rm i}$ into a final density matrix to maximize the expectation value of the observable $Θ$. The absence of suboptimal local maxima for the relevant control landscapes is numerically illustrated. The dependence of the optimization search effort is analyzed in terms of the dimension of the system $N$, the initial state $ρ_{\rm i}$, and the target observable $Θ$. It is found that if the number of nonzero eigenvalues in $ρ_{\rm i}$ remains constant, the search effort does not exhibit any significant dependence on $N$. If $ρ_{\rm i}$ has no zero eigenvalues, then the computational complexity and the required search effort rise with $N$. The dimension of the top manifold (i.e., the set of Kraus operators that maximizes the objective) is found to positively correlate with the optimization search efficiency. Under the assumption of full controllability, incoherent control modelled by Kraus maps is found to be more efficient in reaching the same value of the objective than coherent control modelled by unitary maps. Numerical simulations are also performed for control landscapes with linear constraints on the available Kraus maps, and suboptimal maxima are not revealed for these landscapes.