Landscapes of optimization of quantum gates and observables

In the tasks of control of quantum systems, the study of the landscapes of the underlying objective functionals is of great interest [1]. In the works [2,3], various methods for studying landscapes for generating two-qubit quantum gates in open quantum systems with coherent and incoherent controls were developed. Such examples of quantum gates as C-NOT, SWAP and C-PHASE were considered. Three types of objective functionals were consructed that exploit either the full basis or three special states, and the study and comparison of the corresponding control landscapes was carried out. For this, varous methods for studying the control landscapes in such systems were applied such as based on the inGRAPE (Incoherent GRAPE) local search algorithm, previously developed in [4], and on the global stochastic optimization in the double annealing algorithm. Explicit expressions were obtained for gradients and Hessians of the constructed objective functionals. In [5], landscapes of the control problems for generating single-qubit quantum gates H and T ($\pi/8$) were investigated. For Hadamard gate H, a smooth distribution with one peak for the best obtained using the inGRAPE values of the fidelity was found, while for the T gate, the constructed distribution was found to have two separate pikes and moreover, the corresponding controls were found to be substantially separated in the control space. In contrast to the distribution for the single-qubit gate T and similarly to the distribution for the gate H, for the two-qubit gates C-NOT and C-PHASE smooth distributions with one pike were obtained (with possible minor exception for the C-PHASE gate $\pi/2$), indicating the landscape structure without traps [2]. Using analytical and numerical methods, the absence of traps was established for generation of single-qubit phase shift gates in a closed quantum system for small times [6].