# Can one obliviously reflect about the *initial* state in fixed-point amplitude amplification?

It is normal to extend fixed-point amplitude amplification to an oblivious version, i.e., $$1 - (1-e^{i \beta})|t\rangle \langle t | \rightarrow 1 - (1-e^{i \beta}) 1 \otimes |0\rangle \langle 0|$$, and one uses an additional ancilla to apply a phase to the target state $$|t\rangle$$.

However typically the other reflection, $$1 - (1-e^{-i\alpha}) |0\rangle \langle 0|$$ is about the all-zero initial state. But suppose you are given an unknown initial state, $$|\psi_0 \rangle$$. You know you can construct a unitary $$U$$ that does $$U |\psi_0\rangle |0\rangle = \sqrt{a}|A_0\rangle |0\rangle + \sqrt{1-a}|A_1\rangle |1\rangle,$$ and so could amplify the probability of $$|A_{0}\rangle |0\rangle$$ to 1, but only if you could implement $$1 - (1-e^{-i\alpha})|\psi_0\rangle\langle \psi_0 | \otimes |0\rangle \langle 0|$$, and you don't know how $$|\psi_{0}\rangle$$ came about. Is it possible instead to construct a unitary $$U'$$ that will give you $$U' |\psi_0\rangle |0\rangle = \sqrt{a}|A_0\rangle |0\rangle + \sqrt{1-a}|A_1\rangle |1\rangle,$$ while simultaneously only using $$1 - (1-e^{-i\alpha})1\otimes |0\rangle \langle 0|$$ for the initial state reflection during amplitude amplification? It is not clear to me how one would construct a $$U'$$ that moves between the two bases $$|A_0\rangle |0\rangle$$, $$|A_1\rangle |1\rangle$$, and $$|\psi_0\rangle |0\rangle$$, $$|\psi_{1}\rangle |1\rangle$$, for some $$|\psi_1 \rangle |1\rangle$$.

Yes, it is possible to combine fixed-point amplitude amplification with oblivious amplitude amplification, provided that $$a$$ in the definition of $$U$$ is independent of the input state $$|\psi_0\rangle$$, as discussed in this paper by Dalzell, Yoder, and Chuang (cf. Section VI.B in particular). Importantly, that means oblivious amplitude amplification does not work if $$U$$ is non-unitary. This paper by Guerreschi also discusses fixed-point oblivious amplitude amplification.