Estimation of Reachable Sets for a Two-Level Quantum System Driven by Coherent and Incoherent Controls. A Comparison of Optimization Methods

The problem of describing reachable sets of an open two-level quantum system, which depends on coherent and incoherent controls, is considered. The Bloch parametrization allows to describe reachable sets of the given system in the terms of reachable sets of the system which states are Bloch vectors. For a particular class of controls, exact descriptions of some reachable tubes were derived. For the general class of controls, parallelepipedal and pointwise estimations of reachable sets in the Bloch ball were obtained via solving series of optimal control problems. With respect to these problems, the following optimization methods were considered and compared with each other: (a) reduction to finite-dimensional optimization with using the Dual Annealing method being a zero-order stochastic method from the global optimization theory; (b) GRAPE (GRadient Ascent Pulse Engineering), i.e. reduction to finite-dimensional optimization with using, in our case, the gradient projection method and the projection version of the heavy-ball method; (c) first order Krotov method with some regularization. Based on the numerical experiments, it was shown how the obtained estimations of a number of reachable sets depend on initial states, constraints for controls, and on final times. The talk uses the results of the article being in preparing: O.V. Morzhin, A.N. Pechen, "Estimation of Reachable Sets for a Two-Level Quantum System by Optimizing Coherent and Incoherent Controls".