In the black-box model, the problem of determining the output of a BPP machine $M(x,r)$ on input $x$ is the approximate counting problem of determining $E_r M(x,r)$ with additive error 1/3 (say).
Is there a similar problem for BQP? This comment by Ken Regan suggests such a problem
You can reduce a BPP question to approximating a single #P function, but with BQP what you get is the difference of two #P functions, call them $f$ and $g$. Approximating $f$ and $g$ separately does not help you approximate $f - g$ when $f - g$ is near zero!
BQP does give you a little help: When the answer to the BQP question on an input $x$ is yes, you get that $f(x) - g(x)$ is close to the square root of $2^m$, where the counting predicates defining $f$ and $g$ have m binary variables after you substitute for $x$. (There are no absolute-value bars; “magically” you always get $f(x) > g(x)$. Under common representations of quantum circuits for BQP, $m$ becomes the number of Hadamard gates.) When the answer is no, the difference is close to 0.
Can you precisely formulate such a problem as close as possible to BQP? I am hoping for something like: given black-box access to functions $f,g$ mapping $X$ to $Y$, with the promise that ..., estimate $f-g$ to within $\varepsilon$.