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Let's say, I want to perform amplitude amplification on the quantum state $$|s\rangle=\sum_{i=0}^{2^n-1}m_i |i\rangle\,,$$ And the aiming state is $|x\rangle\,.$

Usually, when I perform amplitude amplification with the Grover operator $$G=(2|x\rangle\langle x|-I)H^{\otimes n}(2|0\rangle\langle 0|-I)H^{\otimes n}\,,$$ for $k$ times, the amplitude of $|x\rangle$ become $$\sin(2k\alpha+\alpha_0)\,.$$ We would like $k$ to be something that makes $$2k\alpha+\alpha_0=\frac{\pi}{2}\,.$$ But how can we determine $k$, the number of Grover iterations, when we don't know the initial amplitude of aiming state, i.e. we don't know the value of $\alpha_0$?

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Crudely, what you want to do is think of your grover iterator $G$ as a unitary, and your unknown rotation angle $\alpha$ corresponds to two eigenvalues of $G$, $e^{\pm i\alpha}$, and you can prepare an initial state that is a superposition of the two eigenvectors. So, these are all the ingredients that you need in order to perform phase estimation to approximate the angle $\alpha$, and hence the number of steps that you require.

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