Randomized benchmarking (RB) is a popular procedure used to gauge the performance of a set of gates useful for quantum information processing (QIP). Recently, Proctor et al. [PRL 119,130502 (2017)] demonstrated a practically relevant example where the RB measurements give a number r very different from the actual average gate-set infidelity e, despite past theoretical assurances that the two should be equal. Here, we derive formulas for e, and for r from the RB protocol, in a manner permitting easy comparison of the two. We show that r is not equal to e, i.e., RB does not measure average infidelity, and, in fact, neither one bounds the other. We give several examples, all plausible in experiments, to illustrate the differences in e and r. Recently, many papers on experimental implementations of QIP gates have argued that they are close to the fault tolerance noise threshold because they demonstrated small r values using RB. Our anlaysis shows that such a conclusion cannot be drawn from r alone.
This is based on work in an upcoming manuscript by Jiaan Qi (Dept Physics, NUS) and HKN.