Learning Fermionic Correlations by Evolving with Random Translationally Invariant Hamiltonians

Schemes of classical shadows have been developed to facilitate the readout of digital quantum devices, but similar tools for analog quantum simulators are scarce and experimentally impractical. In this Letter, we provide a measurement scheme for fermionic quantum devices that estimates second and fourth order correlation functions by means of free fermionic, translationally invariant evolutions-or quenches-and measurements in the mode occupation number basis. We precisely characterize what correlation functions can be recovered and equip the estimates with rigorous bounds on sample complexities, a particularly important feature in light of the difficulty of getting good statistics in reasonable experimental platforms, with measurements being slow. Finally, we demonstrate how our procedure can be approximately implemented with just nearest-neighbor, translationally invariant hopping quenches, a very plausible procedure under current experimental requirements and requiring only random time evolution with respect to a single native Hamiltonian. On a conceptual level, this Letter brings the idea of classical shadows to the realm of large scale analog quantum simulators.