Randomized Benchmarking as Convolution: Fourier Analysis of Gate Dependent Errors
Abstract
We show that the Randomized Benchmarking (RB) protocol is a convolution amenable to Fourier space analysis. By adopting the mathematical framework of Fourier transforms of matrix-valued functions on groups established in recent work from Gowers and Hatami [Sbornik: Mathematics 208, 1784 (2017)], we provide an alternative proof of Wallman's [Quantum 2, 47 (2018)] and Proctor's [Phys. Rev. Lett. 119, 130502 (2017)] bounds on the effect of gate-dependent noise on randomized benchmarking. We show explicitly that as long as our faulty gate-set is close to the targeted representation of the Clifford group, an RB sequence is described by the exponential decay of a process that has exactly two eigenvalues close to one and the rest close to zero. This framework also allows us to construct a gauge in which the average gate-set error is a depolarizing channel parameterized by the RB decay rates, as well as a gauge which maximizes the fidelity with respect to the ideal gate-set.
Cite
@article{arxiv.1804.05951,
title = {Randomized Benchmarking as Convolution: Fourier Analysis of Gate Dependent Errors},
author = {Seth T. Merkel and Emily J. Pritchett and Bryan H. Fong},
journal= {arXiv preprint arXiv:1804.05951},
year = {2021}
}