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

TL;DR

This paper develops gradient projection and stochastic search methods for optimal control of quantum spin chains, incorporating constraints and control classes, with applications demonstrated through genetic algorithms.

Contribution

It introduces adapted gradient projection and stochastic search techniques for constrained quantum control problems with spin chains, extending previous methods.

Findings

Gradient projection methods effectively handle control constraints.

Genetic algorithms successfully optimize control signals.

Methods are applicable to spin chains of arbitrary length.

Abstract

This article (I) considers the known optimal control model of a quantum information transfer along a spin chain with controlled external parabolic magnetic field, with an arbitrary length. The article adds certain lower and upper pointwise constraints on controls, adds the problem of keeping the signal at the last spin, considers various classes of controls. For these problems under piecewise continuous controls, the projection-type linearized Pontryagin maximum principle, gradient projection method's constructions in its one- and two- and three-step forms were adapted by analogy with [Morzhin O.V., Pechen A.N. J. Phys. A: Math. Theor., 2025]. Moreover, an example with a genetic algorithm's successful use under a special class of controls is given.