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Due to the experimental time constraints of state of the art quantumsimulations with trapped ions, the direct preparation of the ground state byadiabatically ramping the field of a transverse field Ising model becomes moreand more difficult as the number of particles increase. We propose aspectroscopy protocol that intentionally creates excitations through diabaticramping of the transverse field and measures a low-noise observable as afunction of time for a constant field to reveal the structure of the coherentdynamics of the resulting many-body states. To simulate the experimental data,noise from counting statistics and decoherence error are added. Compressivesensing is then applied to Fourier transform the simulated data into thefrequency domain and extract the the low-lying energy excitation spectrum. Byusing compressive sensing, the amount of data in time needed to extract thisenergy spectrum is sharply reduced making such experiments feasible withcurrent technology.

Diabatic ramping spectroscopy of many-body excited states for trapped-ion quantum simulators