On the Convergence of a Greedy Algorithm for Operator Reconstruction

Abstract

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.