Gradient projection method and stochastic search for some optimal control models with spin chains

To the known model of optimal control for transferring quantum information along a spin chain the point-to-point constraints of a specific type on the controls are added. A model for maintaining the signal on the last spin is formulated, with different objective functions and classes of controls considered. For piecewise-continuous controls, by analogy with the article [Morzhin O.V., Pechen A.N. J. Phys. A: Math. Theor., 2025] devoted to other quantum problems, the gradient projection method (GPM) is adapted in one-, two-, and three-step forms using infinite-dimensional gradients. Then, based on these gradients, finite-dimensional gradients are derived, and corresponding versions of the GPM are considered. The projection form of the linearized Pontryagin maximum principle, as well as a finite-dimensional projection necessary condition for local optimality, are examined. For simple specialized parametrized classes of controls (these classes are narrower than the class of piecewise-constant controls), introduced for signal transfer and retention, a genetic algorithm was successfully applied, achieving good performance metrics for transfer and retention. The overall numerical results demonstrate that the developed approaches, including the specialized control classes, allow achieving good values for the target metrics. It is shown, in particular, that the two- and three-step GPM can be many times faster compared to the single-step GPM. It is important to note that the acceleration is based on ideas underlying B.T. Polyak's heavy-ball method and the finite-dimensional forms of GPM in the works of A.S. Antipin and A. Nedić. At the end of the report, we plan to briefly list the main conclusions and some directions for future work.