  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.
