Suppose we are given a (univariate) polynomial $P$ of degree $d$, and we wish to determine if $P$ is identically $0$. A standard way to do this is to use a classical PRG to randomly sample $n$ bits, drawing a number $r$ uniformly from $[0,S]$; we can plug $r$ into $P$ to see if $P(r)=0$. If we only perform the above test one time, we would be "fooled" into thinking the polynomial is $0$ when it's not only $d/S$ times. However, we can rinse and repeat $k$ times to amplify our success probability, and our probability of being "fooled" is at most $(d/S)^k$.
As an alternative to drawing uniformly from $[0,S]$, we could also perform random circuit sampling on an $n$-qubit random quantum circuit $U$. That is, we measure $U\vert 00\cdots0\rangle$ for some fixed random quantum circuit $U$.
For each sample, the output of $U$ will most definitely not be uniformly distributed; it's more likely that most states are never sampled, a good number of states having a reasonable probability of being sampled, and a small number of states having a large probability of being sampled. This is according to the Porter-Thomas distribution as I understand it.
If we were to sample $n$ qubits $k$ times from a random quantum circuit $U$, and use these $r_1,r_2,\cdots, r_k$ as a test that a polynomial is $0$, what is our soundness probability?
Could it be more efficient to repeatedly sample $n$ qubits from the same random quantum quantum circuit to determine guesses $r_1,r_2,\cdots,r_k$ drawn from the Porter-Thomas distribution determined by $U|00\cdots 0\rangle$ than to repeatedly sample $n$ bits at random to determine guesses $r_1,r_2,\cdots,r_k$ drawn from the uniform distribution?
