  Due to the experimental time constraints of state of the art quantum
simulations with trapped ions, the direct preparation of the ground state by
adiabatically ramping the field of a transverse field Ising model becomes more
and more difficult as the number of particles increase. We propose a
spectroscopy protocol that intentionally creates excitations through diabatic
ramping of the transverse field and measures a low-noise observable as a
function of time for a constant field to reveal the structure of the coherent
dynamics of the resulting many-body states. To simulate the experimental data,
noise from counting statistics and decoherence error are added. Compressive
sensing is then applied to Fourier transform the simulated data into the
frequency domain and extract the the low-lying energy excitation spectrum. By
using compressive sensing, the amount of data in time needed to extract this
energy spectrum is sharply reduced making such experiments feasible with
current technology.
