No, it's a different type of decision problem, it doesn't fall into what is usually called "bounded-error complexity" at all. You can't bound the running time by any function, not just a polynomial. 

Suppose you have an algorithm that guaranties to run in time $<f(n)$, i.e. makes no more than $f(n)$ queries to the oracle $U$. And it's required it should output an  answer with the error probability no more than $1/3$. 

It doesn't matter how big $f(n)$ is, there is always a state $|\psi\rangle$ which is very close to the boundary of the set of entangled states. For such a state your algorithm won't be able to decide with the error less than  $1/3$.   

In other words, the time bound $f(n)$ can't be independent of $|\psi\rangle$.

If a state $|\psi\rangle$ is given by a mathematical description, then the problem is easy.