It is well-known that bounded error quantum query complexity of the function $OR(x_1,x_2,\ldots, x_n)$ is $\Theta(\sqrt{n})$. Now the question is what if we want our quantum algorithm to succeed for every input with probability $1-\epsilon$ rather than the usual $2/3$. Now in terms of $\epsilon$ what would be the appropriate upper and lower bounds?
It is immediate that $O(\sqrt{n} \log(1/\epsilon))$ queries suffices for this task by repeating the Grover algorithm. But from what I recall this is not at all optimal as even plain Grover algorithm if run carefully, i.e. for appropriate number of iterations, can achieve something like $\epsilon=O(1/n)$ with just $O(\sqrt{n})$ iterations. And hence using that one can get an improvement for all $\epsilon$'s. On the other hand, I don't expect that $\Omega(\sqrt{n})$ be the right answer for very small $\epsilon$'s.
But I am interested to see what one can show in terms of $\epsilon$-dependent upper and lower bounds for different ranges of $\epsilon$ especially when $\epsilon$ is very small say $\epsilon= \exp(-\Omega(n))$ or $\epsilon=1/n^k$ for large $k$'s.
(To give some context, the general phenomenon I am getting at is the amplification in context of quantum query complexity. )