  Typical quantum gate tomography protocols struggle with a self-consistency
problem: the gate operation cannot be reconstructed without knowledge of the
initial state and final measurement, but such knowledge cannot be obtained
without well-characterized gates. A recently proposed technique, known as
randomized benchmarking tomography (RBT), sidesteps this self-consistency
problem by designing experiments to be insensitive to preparation and
measurement imperfections. We implement this proposal in a superconducting
qubit system, using a number of experimental improvements including
implementing each of the elements of the Clifford group in single `atomic'
pulses and custom control hardware to enable large overhead protocols. We show
a robust reconstruction of several single-qubit quantum gates, including a
unitary outside the Clifford group. We demonstrate that RBT yields physical
gate reconstructions that are consistent with fidelities obtained by randomized
benchmarking.
