# What are the "higher moments" of the gate fidelity?

Reading the paper Gate fidelity fluctuations and quantum process invariants I came across the concept of higher moments of the gate fidelity, for example in the following excerpt from the introduction:

In this paper, we calculate the variance of the gate fidelity analytically and discuss how it might be measured in experiments (as well as challenges to doing so!). Moreover, we develop a general method for calculating higher moments of the gate fidelity, which can be applied to other purposes.

However, I couldn't find this concept defined anywhere. From some questions on this site, I found out that Cirq uses the concept of moments to define its circuits which are collections of operations that act during the same "abstract time slice". Do moments of the gate fidelity have something to do with this notion of moments? If not, then what are the moments of gate fidelity mentioned in the paper?

Note that in section IV of the same this concept is discussed using what is derived in the earlier parts of the paper. However, I'm looking for a more conceptual definition of higher order moments before going into the mathematical details of them and the paper in general.

• Yes, just to clarify, based off this post in MSE, the moment would be $\overline{\mathcal{F}^2}$ and the variance would be the central moment, right? Sep 8 at 9:38
• And to add on for future people. As @Martin Vesely mentioned in his moment, this concept has to do with skewness, kurtosis, etc. The third central moment measures skewness, the fourth kurtosis, etc. The $m$-th central moment if $\overline{(\mathcal{F}-\overline{F})^m}$ Sep 8 at 9:41