Amplitude amplification with independently chosen phase rotations

Input a prepared sampler $A$, a success projection ${\mathrm{\Pi }}_{\mathrm{g}\mathrm{o}\mathrm{o}\mathrm{d}}$, an integer $m\ge 0$, and phases $\varphi ,\phi \in [0,2\pi )$. In addition to $A,A^{†}$, the access contract must supply the exact phase operations $S_0(\varphi )=I+(e^{i\varphi }-1)\mathrm{\mid }0⟩⟨0\mathrm{\mid },S_{\mathrm{g}\mathrm{o}\mathrm{o}\mathrm{d}}(\phi )=I+(e^{i\phi }-1){\mathrm{\Pi }}_{\mathrm{g}\mathrm{o}\mathrm{o}\mathrm{d}},$ and charge each call. Such access does not follow from a $\pi$-phase reflection alone. Prepare $A\mathrm{\mid }0⟩$, apply $Q(\varphi ,\phi ):=-A{S}_0(\varphi )A^{†}{S}_{\mathrm{g}\mathrm{o}\mathrm{o}\mathrm{d}}(\phi )$ exactly $m$ times, and apply the declared good/bad projective instrument. Return the conditional quantum output on a positive-probability good branch or fail on the bad branch. A classical output additionally needs the diagonal-predicate and costed-verifier conditions in Prepared unitary and success subspace for amplitude amplification.

The outer minus sign is retained in the operator. An uncontrolled $m$-fold sequence differs from the sequence without that scalar only by the global factor $(-1{)}^m$. It cannot be dropped inside a controlled use: the controlled block treats the two control branches differently, as in A Controlled Global Phase Becomes a Relative Control Phase. Controlled access to any factor is separately supplied and charged.