The (classical) lower bound on Fast Fourier transform is still open question. The complexity of $\mathcal{O}(N\log(N))$ (due to Cooley-Tukey) is not known to be optimal. (Here, $N$ is the vector size.)

In the quantum counterpart, there is an $\mathcal{O}(n^2)$ algorithm due to Coppersmith, 1994, and also Deutsch (unpublished). Here $n$ is number of qubits.

Is this result known to be optimal?

There has been some approximate QFT algorithm that do better - see, $\mathcal{O}(n\log(n))$ due to Hales, 2002. But, I am restricting to exact algorithms only for this question.