Randomized Benchmarking under Different Gatesets

We provide a comprehensive analysis of the differences between two important standards for randomized benchmarking (RB): the Clifford-group RB protocol proposed originally in [1] and [2], and a variant of that RB protocol proposed later by the NIST group in [3]. While these two protocols are frequently conflated or presumed equivalent, we prove that they produce distinct exponential fidelity decays leading to differences of up to a factor of 3 in the estimated error rates under experimentally realistic conditions. These differences arise because the NIST RB protocol does not satisfy the unitary two-design condition for the twirl in the Clifford-group protocol and thus the decay rate depends on non-invariant features of the error model. Our analysis provides an important first step towards developing definitive standards for benchmarking quantum gates and a more rigorous theoretical underpinning for the NIST protocol and other RB protocols lacking a group-structure. We conclude by discussing the potential impact of these differences for estimating fault-tolerant overheads.

[1] J. Emerson et al, Journal of Optics B: Quantum and Semiclassical Optics 7, S347 (2005).
[2] C. Dankert et al, arXiv preprint quant-ph/0606161 (2006).
[3] E. Knill et al, Physical Review A 77, 012307 (2008).