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By adopting the mathematical framework of Fourier transforms of matrix-valued functions on groups established in recent work from Gowers and Hatami \\cite{GH15}, we provide an alternative proof of Wallman's \\cite{Wallman2018} and Proctor's \\cite{Proctor17} 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.<\/jats:p>","DOI":"10.22331\/q-2021-11-16-581","type":"journal-article","created":{"date-parts":[[2021,11,16]],"date-time":"2021-11-16T13:31:25Z","timestamp":1637069485000},"page":"581","update-policy":"https:\/\/doi.org\/10.22331\/q-crossmark-policy-page","source":"Crossref","is-referenced-by-count":24,"title":["Randomized Benchmarking as Convolution: Fourier Analysis of Gate Dependent Errors"],"prefix":"10.22331","volume":"5","author":[{"given":"Seth T.","family":"Merkel","sequence":"first","affiliation":[{"name":"HRL Laboratories, LLC 3011 Malibu Canyon Road, Malibu, CA 90265 USA"}]},{"given":"Emily J.","family":"Pritchett","sequence":"additional","affiliation":[{"name":"HRL Laboratories, LLC 3011 Malibu Canyon Road, Malibu, CA 90265 USA"}]},{"given":"Bryan H.","family":"Fong","sequence":"additional","affiliation":[{"name":"HRL Laboratories, LLC 3011 Malibu Canyon Road, Malibu, CA 90265 USA"},{"name":"HRL Laboratories, LLC 3011 Malibu Canyon Road, Malibu, CA 90265 USA"}]}],"member":"9598","published-online":{"date-parts":[[2021,11,16]]},"reference":[{"key":"0","unstructured":"Supplementary Mathematica notebook included as ancillary file of this arxiv submission."},{"key":"1","unstructured":"The GAP Group, GAP \u2014 Groups, Algorithms, and Programming, version 4.10.0, 2018. 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