  A disordered system of interacting particles exhibits localized behavior when
the disorder is large compared to the interaction strength. Studying this
phenomenon on a quantum computer without error correction is challenging
because even weak coupling to a thermal environment destroys most signatures of
localization. Fortunately, spectral functions of local operators are known to
contain features that can survive the presence of noise. In these spectra,
discrete peaks and a soft gap at low frequencies compared to the thermal phase
indicate localization. Here, we present the computation of spectral functions
on a trapped-ion quantum computer for a one-dimensional Heisenberg model with
disorder. Further, we design an error-mitigation technique which is effective
at removing the noise from the measurement allowing clear signatures of
localization to emerge as the disorder increases. Thus, we show that spectral
functions can serve as a robust and scalable diagnostic of many-body
localization on the current generation of quantum computers.
