Control through operators for quantum chemistry

We consider the problem of operator identification in quantum control. The free Hamiltonian and the dipole moment are searched such that a given target state is reached at a given time. A local existence result is obtained. As a by-product, our works reveals necessary conditions on the laser field to make the identification feasible. In the last part of this work, some algorithms are proposed to compute effectively these operators

Analysis of a greedy reconstruction algorithm

A novel and detailed convergence analysis is presented for a greedy algorithm that was previously introduced for operator reconstruction problems in the field of quantum mechanics. This algorithm is based on an offline/online decomposition of the reconstruction process and on an ansatz for the unknown operator obtained by an a priori chosen set of linearly independent matrices. The presented convergence analysis focuses on linear-quadratic (optimization) problems governed by linear differential systems and reveals the strong dependence of the performance of the greedy algorithm on the observability properties of the system and on the ansatz of the basis elements. Moreover, the analysis allows us to use a precise (and in some sense optimal) choice of basis elements for the linear case and led to the introduction of a new and more robust optimized greedy reconstruction algorithm. This optimized approach also applies to nonlinear Hamiltonian reconstruction problems, and its efficiency is demonstrated by numerical experiments

On the Convergence of a Greedy Algorithm for Operator Reconstruction

In their publication "A greedy algorithm for the identification of quantum systems" from 2009, Yvon Maday and Julien Salomon introduce an algorithm for the reconstruction of operators in the context of quantum systems. This algorithm shows good results in the numerical application. However, no convergence theory has been developed so far. In this work we investigate the algorithm in the setting of a linear ordinary differential equation, present problematic cases and establish assumptions under which convergence is guaranteed. We also develop working improvements to the algorithm, demonstrate their performance in numerical examples and discuss numerical instabilities. Hereafter, we give a detailed description of the algorithm in its original form and setting. In this context we introduce monotonic schemes as an efficient method to solve optimal control problems for quantum systems. Finally, we apply one of the improvements from the linear setting and show, with the aid of numerical experiments, that it also has a positive effect on the algorithm’s performance for the solution of reconstruction problems governed by quantum systems.publishe