Randomized Benchmarking as Convolution

We show that the standard randomized benchmarking protocol can be described as a convolution, and is thus amenable to Fourier analysis. We utilize a form of Fourier transform that maps matrix-valued functions on group elements to matrix-valued functions of the group’s irreducible representations in order to map the average over sequences of Clifford operations to the power of a single matrix. We can then demonstrate that as long as our faulty gate-set is close to some 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, even though the bounds with respect to any particular representation of the Clifford group may not tightly describe the rate of decay.