Analysis of a greedy reconstruction algorithm

TL;DR

This paper provides a detailed convergence analysis of a greedy algorithm for operator reconstruction in quantum mechanics, highlighting its dependence on system observability and basis choice, and introduces an optimized, more robust version with demonstrated numerical efficiency.

Contribution

It offers the first comprehensive convergence analysis for this greedy reconstruction algorithm and proposes an optimized version tailored for linear and nonlinear quantum problems.

Findings

Convergence depends strongly on system observability and basis choice.

Optimal basis selection improves the algorithm's performance.

The optimized algorithm shows superior efficiency in numerical experiments.

Abstract

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.