Randomized Benchmarking as Convolution: Fourier Analysis of Gate Dependent Errors

TL;DR

This paper demonstrates that randomized benchmarking can be analyzed as a convolution in Fourier space, providing new insights into gate-dependent errors and their impact on quantum gate fidelity.

Contribution

It introduces a Fourier analysis framework for RB, offering an alternative proof of error bounds and explicit characterization of decay eigenvalues for gate-dependent noise.

Findings

RB sequences exhibit exponential decay with two dominant eigenvalues.

The framework constructs gauges for depolarizing error parameters.

Close to the Clifford group, errors are well-characterized by this method.

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.