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

TL;DR

This paper analyzes the relationship between randomized benchmarking results and actual average gate-set infidelity, revealing that RB measurements do not directly reflect infidelity and can be misleading in assessing quantum gate performance.

Contribution

The authors derive formulas comparing RB output and true infidelity, demonstrating their differences and providing examples to clarify the limitations of RB as an infidelity measure.

Findings

RB measurement r does not equal average infidelity ε

RB does not bound or is bounded by ε in general

High-fidelity claims based on RB r values can be misleading

Abstract

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, 130502 (2017)] demonstrated a practically relevant example where the RB measurements give a number $r$ very different from the actual average gate-set infidelity $\epsilon$, despite past theoretical assurances that the two should be equal. Here, we derive formulas for $\epsilon$, and for $r$ from the RB protocol, in a manner permitting easy comparison of the two. We show in general that, indeed, $r\neq \epsilon$, 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 $\epsilon$ and $r$. Many recent papers on experimental implementations of QIP have claimed the ability to perform high-fidelity gates because they demonstrated small $r$ values using RB. Our analysis shows that such a statement from RB alone has to be interpreted with caution.