An important step in building a quantum computer is calibrating
experimentally implemented quantum gates to produce operations that are close
to ideal unitaries. The calibration step involves estimating the error in gates
and then using controls to correct the implementation. Quantum process
tomography is a standard technique for estimating these errors, but is both
time consuming, (when one only wants to learn a few key parameters), and
requires resources, like perfect state preparation and measurement, that might
not be available. With the goal of efficiently estimating specific errors using
minimal resources, we develop a parameter estimation technique, which can gauge
two key parameters (amplitude and off-resonance errors) in a single-qubit gate
with provable robustness and efficiency. In particular, our estimates achieve
the optimal efficiency, Heisenberg scaling. Our main theorem making this
possible is a robust version of the phase estimation procedure of Higgins et
al. [B. L. Higgins, New J. Phys. 11, 073023 (2009)].
