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arXiv · 1804.05951

Randomized Benchmarking as Convolution: Fourier Analysis of Gate Dependent Errors

Seth T. Merkel·Emily J. Pritchett·Bryan H. Fong
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Abstract

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.

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Seth T. Merkel, Emily J. Pritchett, Bryan H. Fong. 2018-04-16. Randomized Benchmarking as Convolution: Fourier Analysis of Gate Dependent Errors. https://doi.org/10.22331/q-2021-11-16-581

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