  Coherent errors in quantum operations are ubiquitous. Whether arising from
spurious environmental couplings or errors in control fields, such errors can
accumulate rapidly and degrade the performance of a quantum circuit
significantly more than an average gate fidelity may indicate. As Hastings [1]
and Campbell [2] have recently shown, by replacing the deterministic
implementation of a quantum gate with a randomized ensemble of implementations,
on can dramatically suppress coherent errors. Our work begins by reformulating
the results of Hastings and Campbell as a quantum optimal control problem. We
then discuss a family of convex programs designed to improve the performance,
implementability, and robustness of the resulting mixed quantum gates. Finally,
we implement these mixed quantum gates on a superconducting qubit and discuss
randomized benchmarking results consistent with a marked reduction in the
coherent error.
  [1] M. B. Hastings, Quantum Information & Computation 17, 488 (2017).
  [2] E. Campbell, Physical Review A 95, 042306 (2017).
