  We demonstrate experimental implementation of robust phase estimation (RPE)
to learn the phases of X and Y rotations on a trapped $\textrm{Yb}^+$ ion
qubit. We estimate these phases with uncertainties less than $4\cdot10^{-4}$
radians using as few as 176 total experimental samples per phase, and our
estimates exhibit Heisenberg scaling. Unlike standard phase estimation
protocols, RPE neither assumes perfect state preparation and measurement, nor
requires access to ancillae. We cross-validate the results of RPE with the more
resource-intensive protocol of gate set tomography.
