Amplitude amplification with a fixed integer iteration count

Input a coherent sampler $A$, an orthogonal success projection ${\mathrm{\Pi }}_{\mathrm{g}\mathrm{o}\mathrm{o}\mathrm{d}}$, and a fixed integer $k\ge 0$, under the access contract of Prepared unitary and success subspace for amplitude amplification. Additionally supply the good/bad projective instrument $\{{\mathrm{\Pi }}_{\mathrm{g}\mathrm{o}\mathrm{o}\mathrm{d}},I-{\mathrm{\Pi }}_{\mathrm{g}\mathrm{o}\mathrm{o}\mathrm{d}}\}$, charging each measurement and its implementation costs separately. Prepare $\mathrm{\mid }\psi ⟩=A\mathrm{\mid }0⟩$. In the displayed right-to-left operator convention, set $Q:=-A(I-2\mathrm{\mid }0⟩⟨0\mathrm{\mid })A^{†}(I-2{\mathrm{\Pi }}_{\mathrm{g}\mathrm{o}\mathrm{o}\mathrm{d}})\mathrm{.}$ Apply $Q$ exactly $k$ times. Then apply the projective instrument $\{{\mathrm{\Pi }}_{\mathrm{g}\mathrm{o}\mathrm{o}\mathrm{d}},I-{\mathrm{\Pi }}_{\mathrm{g}\mathrm{o}\mathrm{o}\mathrm{d}}\}$. On a positive-probability good outcome return its conditional quantum state; on the bad outcome return fail. The conditional state exists only on a positive-probability branch, as in Conditional state for a projective measurement.

For the classical-output specialization, additionally require ${\mathrm{\Pi }}_{\mathrm{g}\mathrm{o}\mathrm{o}\mathrm{d}}$ to be diagonal in the designated output basis and supply a costed verifier. Measure that basis only after the success-projector instrument has accepted; verify the resulting candidate and return it or fail. No arbitrary-projector-to-classical-predicate conversion is implicit.

This fixed-$k$ procedure makes no controlled-$A$ call. Any controlled or inverse access used by an enclosing procedure must be separately declared and charged. The scalar minus sign in $Q$ is retained exactly.