  Randomized benchmarking provides a tool for obtaining precise quantitative
estimates of the average error rate of a physical quantum channel. Here we
define real randomized benchmarking, which enables a separate determination of
the average error rate in the real and complex parts of the channel. This
provides more fine-grained information about average error rates with
approximately the same cost as the standard protocol. The protocol requires
only averaging over the real Clifford group, a subgroup of the full complex
Clifford group, and makes use of the fact that it forms an orthogonal 2-design.
It therefore allows benchmarking of fault-tolerant gates for an encoding which
does not contain the full Clifford group transversally. Furthermore, our
results are especially useful when considering quantum computations on rebits
(or real encodings of complex computations), in which case the real Clifford
group now plays the role of the complex Clifford group when studying stabilizer
circuits.
