Background
In this paper (section 10), the linear cross-entropy benchmark (fidelity) for a random circuit $C$ and samples $s_i$ taken from that circuit is defined as
$$\chi := \frac{2^n}{k}\sum^k_{i=1}|\langle 0^n|C|s_i\rangle|^2$$
In that section, Scott Aaronson writes: "One can show that ideal sampling, with a noiseless quantum computer, would yield an expected value of $χ \approx 2$."
My question
Why does a random circuit produce $\chi \approx 2$?
I understand why a uniform distribution (classical random guessing) will produce $\chi \approx 1$. But I don't understand the random quantum circuit case.
