Experimental demonstration of cheap and accurate phase estimation

We demonstrate experimental implementation of robust phase estimation (RPE) to learn the phases of X and Y rotations on a trapped ${\text{Yb}}^{+}$ ion qubit. We estimate these phases with uncertainties less than $4\cdot {10}^{-4}$ 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.