Randomized Benchmarking as Convolution: Fourier Analysis of Gate Dependent Errors

Seth T. Merkel; Emily J. Pritchett; Bryan H. Fong

doi:10.22331/q-2021-11-16-581

Randomized Benchmarking as Convolution: Fourier Analysis of Gate Dependent Errors

Seth T. Merkel¹, Emily J. Pritchett¹, and Bryan H. Fong^1,1

¹HRL Laboratories, LLC 3011 Malibu Canyon Road, Malibu, CA 90265 USA

Published:	2021-11-16, volume 5, page 581
Eprint:	arXiv:1804.05951v3
Doi:	https://doi.org/10.22331/q-2021-11-16-581
Citation:	Quantum 5, 581 (2021).

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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 [19], we provide an alternative proof of Wallman's [32] and Proctor's [28] 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.

► BibTeX data

@article{Merkel2021randomized,
  doi = {10.22331/q-2021-11-16-581},
  url = {https://doi.org/10.22331/q-2021-11-16-581},
  title = {Randomized {B}enchmarking as {C}onvolution: {F}ourier {A}nalysis of {G}ate {D}ependent {E}rrors},
  author = {Merkel, Seth T. and Pritchett, Emily J. and Fong, Bryan H.},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {5},
  pages = {581},
  month = nov,
  year = {2021}
}

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