In the beginning of this paper about amplitude amplification algorithm, the author stated the conventions of amplitude amplification which I quoted the important part for your convenience as follows:
Let $|\Psi\rangle=\mathcal{A}|0\rangle$ denote the superposition obtained by applying algorithm $\mathcal{A}$ on the initial state $|0\rangle$. Let $\left|\Psi_{1}\right\rangle=$ $\mathbf{P}_{\text {good }}|\Psi\rangle$ where $\mathbf{P}_{\text {good }}=\sum_{x: \chi(x)=1}|x\rangle\langle x|$ denotes the projection onto the subspace spanned by the good basis states, and similarly let $\left|\Psi_{0}\right\rangle=\mathbf{P}_{\mathrm{bad}}|\Psi\rangle$ where $\mathbf{P}_{\mathrm{bad}}=\sum_{x: \chi(x)=0}|x\rangle\langle x| .$ Let $a=\left\langle\Psi_{1} \mid \Psi_{1}\right\rangle$ denote the probability that a measurement of $|\Psi\rangle=\mathcal{A}|0\rangle$ yields a good state, and let $b=\left\langle\Psi_{0} \mid \Psi_{0}\right\rangle$ denote the probability that a measurement of $|\Psi\rangle$ yields a bad state. We then have that $|\Psi\rangle=\left|\Psi_{1}\right\rangle+\left|\Psi_{0}\right\rangle$ and $1=a+b$. Finally, let angle $\theta$ be so that $0 \leq \theta \leq \pi / 2$ and $a=\sin ^{2}(\theta)$
So from the statement above, I didn't see the requirement that $|\Psi\rangle$ need be state like $\frac{1}{\sqrt{N}}\sum_i|i\rangle$, i.e., the equal superposition state. It seems there's no wrong theoretically, but shouldn't we prepare an equal superposition state considering we don't have any idea which one is the answer? It's kind of weird for me.
Did I misunderstand the Amplitude Amplification algorithm or it is exactly what I said above just for theoretical generality?
