Forecasting Quantum Observables: A Compressed Sensing Approach with Performance Guarantees

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 realistic shot noise, certified models remain robust at the 0.1 error threshold.