Randomized Benchmarking as Convolution: Fourier Analysis of Gate Dependent Errors

We show that the Randomized Benchmarking (RB) protocol is a convolution amenable to Fourier space analysis. By adopting mathematical framework of transforms matrix-valued functions on groups established in recent work from Gowers and Hatami \cite{GH15}, we provide an alternative proof Wallman's \cite{Wallman2018} Proctor's \cite{Proctor17} bounds effect gate-dependent noise randomized benchmarking. explicitly as long our faulty gate-set close targeted representation Clifford group, RB sequence described by exponential decay process has exactly two eigenvalues one rest zero. This also allows us construct gauge which average error depolarizing channel parameterized rates, well maximizes fidelity with respect ideal gate-set.