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.

中文翻译：

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作为卷积的随机基准：门相关误差的傅立叶分析

我们表明随机基准 (RB) 协议是一种适合傅立叶空间分析的卷积。通过采用 Gowers 和 Hatami [19] 最近工作中建立的群上矩阵值函数的傅立叶变换的数学框架，我们提供了 Wallman [32] 和 Proctor [28] 边界对门相关影响的替代证明随机基准测试的噪音。我们明确表明，只要我们的错误门集接近 Clifford 群的目标表示，RB 序列就由一个过程的指数衰减来描述，该过程恰好具有两个接近于 1 的特征值，其余的接近于 0。该框架还允许我们构建一个量规，其中平均门设置误差是一个由 RB 衰减率参数化的去极化通道，