Comparing the randomized benchmarking figure with the average infidelity of a quantum gate-set

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. [Phys. Rev. Lett. 119 (2017) 130502] demonstrated a practically relevant example where the RB measurements give a number [Formula: see text], very different from the actual average gate-set infidelity [Formula: see text], despite past theoretical assurances that the two should be equal. Here, we derive formulas for [Formula: see text], and for [Formula: see text] from the RB protocol, in a manner permitting easy comparison of the two. We show in general that, indeed, [Formula: see text], i.e. RB does not measure average infidelity, and, in fact, neither one bounds the other. We give several examples, all plausible in real experiments, to illustrate the differences in [Formula: see text] and [Formula: see text]. Many recent papers on experimental implementations of QIP have claimed the ability to perform high-fidelity gates because they demonstrated small [Formula: see text] values using RB. Our analysis shows that such a statement from RB alone has to be interpreted with caution.