Complexity of Hidden Subgroup problems
If $G=\mathbb{Z}_{2}^{n}$ and the hidden subgroup $H$ has order $2$, then finding the hidden subgroup is equivalent to finding two elements in the same coset of $H$. The latter is in turn a birthday paradox-type problem, so should require $\Theta(\sqrt{|G|})$ queries for a randomized algorithm and $\Theta(|G|)$ queries for a deterministic algorithm. Note that $|G|=2^{n}$ is exponential in the input size.