I am trying to figure out equation (2.33) [here][1].

Given that

\begin{gather}
U_{\psi_0}|{0^m}\rangle|{0^n}\rangle=\sqrt{p_0}|{0^m}\rangle|{\psi_0}\rangle+\sqrt{1-p_0}|{\perp}\rangle,\\
(\Pi\otimes I_n)|{\perp}\rangle=0, \quad \Pi=|{0^m}\rangle\langle{0^m}|,
\end{gather}

we call $|\psi_{\rm good}\rangle=|0^m\rangle|\psi_0\rangle$ and would like to prepare $|\psi_0\rangle$ in the second register with $O(1)$ probability. The recipe is to define

\begin{gather}
R_{\mathrm{good}}=(1-2|{0^m}\rangle\langle{0^m}|)\otimes I_n=(1-2\Pi)\otimes I_n,\\
R_{\psi_0}=U_{\psi_0}(2|{0^{m+n}}\rangle\langle{0^{m+n}}|-I)U^{\dagger}_{\psi_0}.
\end{gather}

and to *"apply $G^k=(R_{\mathrm{good}}R_{\psi_0})^k$ to $|\psi_0\rangle$ for $k=O(1\sqrt{p_o})$ times."* **(I think that there's a typo, and we should actually apply $G^k$ to the first equation above, correct?)**

What I do not understand here is why the so-defined $R_{\mathrm{good}}$ is equivalent to $(1-2|\psi_{\rm good}\rangle\langle \psi_{\rm good}|)= (1-\Pi \otimes\Pi_{\rm good})$, which is the original definition requried for the amplitude amplification. The line after (2.33) says: *"This is because $|{\psi_{\mathrm{good}}}\rangle$ can be entirely identified by measuring the ancilla qubits."*

**So at which point in this process do we measure ancillas?** It seems that measuring ancillas each time after we apply $R_{\mathrm{good}}$ would ruin the whole procedure. If we don't measure anything until the very end, then, as I said above, it's unclear why $R_{\mathrm{good}} = (1-2|\psi_{\rm good}\rangle\langle \psi_{\rm good}|)$ and how this construction reduces to amplitude amplification.


  [1]: https://arxiv.org/abs/2201.08309