The most powerful existing threshold theorems for fault tolerant quantum computing require one- and two-qubit gates that are within 1e-3 to 1e-4 (in diamond norm distance) of ideal. Certifying that an experimental qubit system achieves this threshold thus requires (1) characterizing the full process matrices of its gates, and (2), assigning reliable uncertainty regions. These requirements must be met for both one- and two-qubit gates, with errors that are small in the diamond norm distance. We demonstrate how to achieve all these desiderata using gate set tomography (GST). GST provides a full characterization (including diamond norm) that randomized benchmarking cannot, while avoiding process tomography's reliance on pre-calibrated operations. We show how to put very tight (<1e-4) error bars on any single-qubit gate diamond norm using GST, by incorporating data from long periodic circuits (gate sequences) akin to those that provide Heisenberg scaling in phase estimation. We also extend GST to two-qubit gates, by formalizing several aspects of GST to enable extensive optimizations, and discuss the tricks required to analyze two-qubit data. We benchmark two-qubit GST using simulated and trapped-ion data, achieving similarly tight (<1e-4) error bars on the diamond norm.
