One-step oblivious amplitude amplification

Input: A unitary $W$ on coefficient/work ancillas and system, explicit inverse access to $W$, the reflection $R_0=I-2{\mathrm{\Pi }}_0$ of Oblivious ancilla reflection, and an operator $M$ satisfying $J^{†}WJ=M\mathrm{/}2$ for the clean injection $Jx=\mathrm{\mid }{0}_{\mathrm{a}\mathrm{n}\mathrm{c}}⟩\otimes x$.

Procedure: Chronologically apply $W$, then $R_0$, then $W^{†}$, then $R_0$, then $W$. The resulting unitary is $C:=-W{R}_0{W}^{†}{R}_{0}W\mathrm{.}$ The scalar $-1$ is an explicit phase correction; it is not omitted when this procedure is controlled. The circuit uses two $W$ calls, one $W^{†}$ call, and two reflection calls. A controlled invocation requires the declared controlled forward, inverse, reflection, and phase-correction gates. This procedure makes no measurement and asserts no exact ancilla cleanup for a nonunitary or approximate $M$; the robust clean-input guarantee is proved separately.