On optimization of coherent and incoherent controls in one- and two-qubit open systems

The talk considers some one- and two-qubit open quantum systems driven by coherent and incoherent controls with control goal of maximizing the Hilbert-Schmidt overlap and minimizing the Hilbert-Schmidt distance between the final density matrix and a given target density matrix. The talk discusses several directions related to these control problems. First, for the problem of minimizing the Hilbert-Schmidt distance for one-qubit system, based on the work [Morzhin O.V., Pechen A.N. On optimization of coherent and incoherent controls for two-level quantum systems. Izv. RAN. Ser. Mat. 87:5 (2023) (In press)], we discuss a modification of the two-stage method [Pechen A., Phys. Rev. A., 84, 042106 (2011)] by using the two-step gradient projection method, where at the cost of complicating the first (incoherent) stage we obtain the possibility to decrease duration of this stage at the cost of losing the simplicity of the original method. Second, for the both problems and two-qubit systems, we outline the use of the Pontryagin maximum principle and gradient projection methods [Morzhin O.V., Pechen A.N. Optimal state manipulation for a two-qubit system driven by coherent and incoherent controls (Submitted)], paying the attention on singular controls, etc. Third, for the problem of maximizing the overlap for the two-qubit case, we present a construction, in the terms of density matrices, for the two Krotov's type methods based on the special exact formulas for the increment of the objective functional [Morzhin O.V., Pechen A.N. Nonlocal improvement methods of coherent and incoherent controls for maximizing the overlap of quantum states for an open two-qubit system (Submitted)].