I have some trouble understanding some aspects of the Quantum Amplification Algorithm (QAA). I am using this reference for the following discussion (section 4).

The QAA applies a certain operator $Q := -\mathcal{A}U_0\mathcal{A}U_f$ onto a superimposed state $$\mathcal{A}|0\rangle^{\otimes n}:=|s\rangle = \sqrt{1-a^2}|s'\rangle + a|w\rangle$$ such that it amplifies the amplitude of the target state $|w\rangle$.

Confusion: It was my understanding that after applying the operator $Q$ a total of $t \sim O(1/N^{-2})$ times one can estimate $a$ simply by measuring the state.

However, we read in the reference provided previously, that Boussard et. al. need to implement Quantum Phase Estimation (QPE) in order to estimate $a$.

Question: Why is this the case? Why can one not just stay with QAA to perform the measurement and needs to implement QPE?

Note that QPE requires ancillary states to be applied. This relates to my previous question as to whether QAA actually requires ancillas (for the oracle) or not in order to estimate $a$.



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