Randomized Benchmarking as Convolution: Fourier Analysis of Gate Dependent Errors

Seth T. Merkel1, Emily J. Pritchett1, and Bryan H. Fong1,1

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

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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.

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