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 \cite{GH15}, we provide an alternative proof of Wallman's \cite{Wallman2018} and Proctor's \cite{Proctor17} 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.
Highlights
Randomized benchmarking (RB) [11, 12, 20, 22, 23] is a workhorse of the quantum characterization community
We believe that in the latter case this is the first such construction. The outline of this manuscript is as follows: in Section 2, we review the basics of randomized benchmarking and show that an RB sequence can be thought of as a convolution; in Section 3, we review matrix-valued Fourier transforms; in Section 4 we apply this Fourier transformation to the super-operator representation of the Clifford group; in Section 5, we compactly reproduce Wallman’s proof of the effects of gate-dependent noise; in Section 6, we show how the eigenvectors of the Fourier transform can be used to construct gauges; in Section 7, we apply this Fourier technique to reproduce examples from Proctor [28] and Wallman [32], as well as an example of our own exploring leakage characterization and the relevance of global phases to the Clifford group
As we look at longer RB sequences, or raise the Fourier transforms to higher powers, our spectrum will be dominated by these two eigenvalues to
Summary
Randomized benchmarking (RB) [11, 12, 20, 22, 23] is a workhorse of the quantum characterization community. We believe that in the latter case this is the first such construction The outline of this manuscript is as follows: in Section 2, we review the basics of randomized benchmarking and show that an RB sequence can be thought of as a convolution; in Section 3, we review matrix-valued Fourier transforms; in Section 4 we apply this Fourier transformation to the super-operator representation of the Clifford group; in Section 5, we compactly reproduce Wallman’s proof of the effects of gate-dependent noise; in Section 6, we show how the eigenvectors of the Fourier transform can be used to construct gauges; in Section 7, we apply this Fourier technique to reproduce examples from Proctor [28] and Wallman [32], as well as an example of our own exploring leakage characterization and the relevance of global phases to the Clifford group
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