Monotonically convergent optimization in quantum control using Krotov's method

Abstract

The non-linear optimization method developed by A. Konnov and V. Krotov [Autom. Remote Cont. (Engl. Transl.) 60, 1427 (1999)] has been used previously to extend the capabilities of optimal control theory from the linear to the non-linear Schrödinger equation [S. E. Sklarz and D. J. Tannor, Phys. Rev. A 66, 053619 (2002)]. Here we show that based on the Konnov-Krotov method, monotonically convergent algorithms are obtained for a large class of quantum control problems. It includes, in addition to nonlinear equations of motion, control problems that are characterized by non-unitary time evolution, nonlinear dependencies of the Hamiltonian on the control, time-dependent targets, and optimization functionals that depend to higher than second order on the time-evolving states. We furthermore show that the nonlinear (second order) contribution can be estimated either analytically or numerically, yielding readily applicable optimization algorithms. We demonstrate monotonic convergence for an optimization functional that is an eighth-degree polynomial in the states. For the "standard" quantum control problem of a convex final-time functional, linear equations of motion and linear dependency of the Hamiltonian on the field, the second-order contribution is not required for monotonic convergence but can be used to speed up convergence. We demonstrate this by comparing the performance of first- and second-order algorithms for two examples.

Fully efficient time-parallelized quantum optimal control algorithm

“…We study the efficiency of the parallelization method in the cases where the optimization solver consists in one step of either a monotonic algorithm or a Newton method. We start with a simulation using monotonic algorithm (see [8,47] for details about this method). Given a target value ε of the figure of merit, we measure the computational time necessary to obtain it.…”

“…For the space discretization, we consider a uniform grid composed of 50 points. The time discretization is achieved with a time grid of 2 9 points, and we use Strang's splitting (8) to compute the trajectories. The optimization solver in Step 2c consists in one iteration of the constant step gradient descent method, see Eq.…”

“…In recent years, the advances in quantum control have emerged through the introduction of appropriate and powerful tools coming from mathematical control theory and by the use of sophisticated experimental techniques to shape the corresponding control fields [3][4][5][6]. In this framework, different numerical optimal control algorithms [7][8][9] have been developed and applied to a large variety of quantum systems. Optimal control was used in physical chemistry in order to steer chemical reactions [3], but also for spin systems [10,11] with applications in Nuclear Magnetic Resonance [7,[12][13][14][15][16] and Magnetic Resonance Imaging [17][18][19].…”

Discrete-valued-pulse optimal control algorithms: Application to spin systems

“…Nowadays, Optimal Control Theory (OCT) reveals to be a highly efficient and versatile tool to bring answers to the different issues raised by the experimental setups [2,[10][11][12][13][14][15][16][17][18][19][20][21][22]. For the past few years, there has been an intense theoretical activity in developing new optimal control procedures able to build high quality control fields in presence of some experimental imperfections and constraints [2,13,[23][24][25].…”

“…Nowadays, Optimal Control Theory (OCT) reveals to be a highly efficient and versatile tool to bring answers to the different issues raised by the experimental setups [2,[10][11][12][13][14][15][16][17][18][19][20][21][22]. For the past few years, there has been an intense theoretical activity in developing new optimal control procedures able to build high quality control fields in presence of some experimental imperfections and constraints [2,13,[23][24][25]. These include spectral constraints [26][27][28][29], amplitude and phase transients [30], non-linear interactions between the system and the control field [31][32][33], robustness against experimental uncertainties and errors [20,34].…”