# Relation between approximate counting and sampling

Consider the following statement of Stockmeyer counting theorem.

Given as input a function $$f:\{0, 1\}^{n} \rightarrow \{0, 1\}^{m}$$ and $$y \in \{0, 1\}^{m}$$, there is a procedure that runs in randomized polynomial time, with access to an $$\text{NP}^{f}$$ oracle (that is, in $$\text{FBPP}^{\text{NP}^{f}})$$, and output an estimate $$\alpha$$ such that $$$$(1 - \epsilon)p \leq \alpha \leq (1 + \epsilon)p,$$$$ for $$$$p= \frac{1}{2^{n}}\sum_{x \in \{0, 1\}^{n}}f(x).$$$$

This version can be found in Theorem 21 this paper.

Here are my two questions:

1. Consider a quantum circuit $$C$$ run on the all $$|0^{n}\rangle$$. For $$x \in \{0, 1\}^{n}$$, consider an output probability $$$$p_x = |\langle x|C|0^{n}\rangle|^{2}.$$$$ Because of Stockmeyer's theorem, for any quantum circuit $$C$$ can we multiplicatively estimate $$p_x$$ to error $$\epsilon$$, for any $$x \in \{0, 1\}^{n}$$, in $$\text{FBQP}^{\text{NP}^{S}}$$, where $$S$$ is an exact sampler from the output distribution of $$C$$?

Note that it is trivial to sample from the output distribution of a quantum circuit in quantum randomized polynomial time, so, if the answer to this question is yes, does it mean we can compute this estimate in $$\text{FBQP}$$ itself and not need the oracle?

2. Can we multiplicatively estimate $$p_x$$, to error $$\epsilon$$, for a uniformly randomly chosen $$x$$, in $$\text{FBQP}^{\text{NP}^{S}}$$? The paper I linked seems to indicate that we can, in Lemma 23, but I do not see how.

Basically, I do not understand what $$f$$ of Stockmeyer's counting theorem is in either of these cases. How does the existence of a sampler $$S$$, for the output distribution of a quantum circuit, relate to the $$f$$ in Stockmeyer's theorem --- can we construct an explicit $$f$$ using the circuit and the sampler $$S$$? Does sampling imply approximate counting by Stockmeyer's argument?