Data-driven extrapolation methods aim to extend the dynamics of quantum observables from measurements, but they often lack guarantees on prediction accuracy. We introduce a framework based on atomic norm minimization that can certify whether the spectral model learned by a forecasting algorithm -- i.e., Bohr frequencies and amplitudes -- is consistent with unitary quantum time evolution. Certification holds when the dynamics are governed by a small number of well-separated Bohr frequencies. We validate the approach on multiple forecasting algorithms applied to spin-chain Hamiltonians with 8--20 sites. Comparing with exact diagonalization, certified models yield an average forecasting error below 0.1 (observable range [−1,1]) in 97\% of cases and below 0.05 in 91--99\% of cases. Even in the presence of noise, certified models remain robust at the 0.1 error threshold.
@article{arxiv.2510.14897,
title = {Forecasting Quantum Observables: A Compressed Sensing Approach with Performance Guarantees},
author = {Víctor Valls and Albert Akhriev and Olatz Sanz Larrarte and Javier Oliva del Moral and Štěpán Šmíd and Josu Etxezarreta Martinez and Sergiy Zhuk and Dmytro Mishagli},
journal= {arXiv preprint arXiv:2510.14897},
year = {2026}
}