Exact success amplification by attenuating the good branch

Input a prepared sampler $A$, its inverse $A^{†}$, an orthogonal good projection ${\mathrm{\Pi }}_{\mathrm{g}\mathrm{o}\mathrm{o}\mathrm{d}}$, and an exactly represented known $a\in [0,1]$ equal to the success weight. The model must supply the coherent flag unitary $C_{\mathrm{g}\mathrm{o}\mathrm{o}\mathrm{d}}={\mathrm{\Pi }}_{\mathrm{g}\mathrm{o}\mathrm{o}\mathrm{d}}\otimes X+(I-{\mathrm{\Pi }}_{\mathrm{g}\mathrm{o}\mathrm{o}\mathrm{d}})\otimes I$ and its inverse, a clean flag qubit $F$, a clean temporary qubit $T$, external clean implementation workspace, and exact real arithmetic and controlled flag-rotation access, including inverse rotations. These are explicit capabilities; a bare phase reflection does not supply $C_{\mathrm{g}\mathrm{o}\mathrm{o}\mathrm{d}}$, and no finite-bit gate synthesis is asserted. The input data space is the possibly abstract space $H$ and specified vector $\mathrm{\mid }0⟩$ of Prepared unitary and success subspace for amplitude amplification; no full classical description or computational-basis test for that vector is inferred. No controlled access to $A$ is required.

If $a=0$, return fail without normalizing a good component. If $a=1$, prepare $A\mathrm{\mid }0⟩$ and apply $\{{\mathrm{\Pi }}_{\mathrm{g}\mathrm{o}\mathrm{o}\mathrm{d}},I-{\mathrm{\Pi }}_{\mathrm{g}\mathrm{o}\mathrm{o}\mathrm{d}}\}$. Supply and charge that instrument directly, or construct it by applying $C_{\mathrm{g}\mathrm{o}\mathrm{o}\mathrm{d}}$ to data and a clean scratch bit, measuring only that bit, and applying $C_{\mathrm{g}\mathrm{o}\mathrm{o}\mathrm{d}}^{†}$; this costs two flag calls and one bit measurement and resets the scratch. Return the good conditional state; in the diagonal classical-output case, measure the designated output and verify it.

For $0<a<1$, set $\theta :=\arcsin \sqrt{a}\in (0,\pi \mathrm{/}2)$, using Principal inverse sine and inverse cosine and Square roots exist: a unique $\sqrt{a}\ge 0$ with $(\sqrt{a}{)}^2=a$; the positives are $\{x^2:x\mathrm{\ne }0\}$, and define $m=\lceil \frac{\pi }{4\theta }-\frac{1}{2}\rceil ,{\theta }^{\mathrm{\prime }}=\frac{\pi }{4m+2},r=\frac{{\sin }^2{\theta }^{\mathrm{\prime }}}{a}\mathrm{.}$ Since $m\ge \pi \mathrm{/}(4\theta )-1\mathrm{/}2$, we have $4m+2\ge \pi \mathrm{/}\theta$, hence $0<{\theta }^{\mathrm{\prime }}\le \theta <\pi \mathrm{/}2$. Sine is strictly increasing on $[0,\pi \mathrm{/}2]$, so $0<{\sin }^2{\theta }^{\mathrm{\prime }}\le {\sin }^2\theta =a$, proving $0<r\le 1$; the square roots in the following rotation are the unique nonnegative roots. On the enlarged data, flag, and temporary registers, construct $A^{\mathrm{\prime }}$ by the following finite gate sequence:

1. Start every added register in $\mathrm{\mid }0⟩$ and apply $A$ to the data.
2. Apply $C_{\mathrm{g}\mathrm{o}\mathrm{o}\mathrm{d}}$ to the data and temporary bit.
3. Controlled on the temporary bit being $1$, apply the flag rotation $R_r=\left(\begin{array}{cc}\sqrt{1-r}& -\sqrt{r}\\ \sqrt{r}& \sqrt{1-r}\end{array}\right),R_r\mathrm{\mid }0⟩=\sqrt{1-r}\mathrm{\mid }0⟩+\sqrt{r}\mathrm{\mid }1⟩\mathrm{.}$
4. Apply $C_{\mathrm{g}\mathrm{o}\mathrm{o}\mathrm{d}}^{†}$ to uncompute the temporary bit. This full unitary is $A^{\mathrm{\prime }}$; its inverse reverses these operations and uses $A^{†}$.

The logical enlarged space is $H^{\mathrm{\prime }}=H\otimes F\otimes T$, with specified input $\mathrm{\mid }{0}^{\mathrm{\prime }}⟩=\mathrm{\mid }0{⟩}_H\mathrm{\mid }0{⟩}_F\mathrm{\mid }0{⟩}_T$ and success projector ${\mathrm{\Pi }}^{\mathrm{\prime }}={\mathrm{\Pi }}_{\mathrm{g}\mathrm{o}\mathrm{o}\mathrm{d}}\otimes \mathrm{\mid }1⟩⟨1{\mathrm{\mid }}_F\otimes I_T$. Its required reflections are $S_0^{\mathrm{\prime }}=I_{H^{\mathrm{\prime }}}-2\mathrm{\mid }{0}^{\mathrm{\prime }}⟩⟨0^{\mathrm{\prime }}\mathrm{\mid }$ and $S_{\mathrm{g}\mathrm{o}\mathrm{o}\mathrm{d}}^{\mathrm{\prime }}=I_{H^{\mathrm{\prime }}}-2{\mathrm{\Pi }}^{\mathrm{\prime }}$. Select and charge the following access modes before execution.

• Zero reflection. Either explicitly supply one direct $S_0^{\mathrm{\prime }}$ call, or supply controlled access $C(S_0)$ to the original data reflection $S_0=I_H-2\mathrm{\mid }0⟩⟨0\mathrm{\mid }$. In the latter mode, initialize an external scratch qubit $q$ to $0$; apply $X$ to $F,T$, Toffoli$(F,T;q)$, $C(S_0)$ controlled by $q$ on $H$, Toffoli$(F,T;q)$, and $X$ to $F,T$ again. This uses one controlled-$S_0$ call, two Toffolis and four $X$ gates per enlarged zero reflection. Controlled $S_0$ is an additional capability and is not inferred from ordinary $S_0$ access.
• Good reflection. In the constructed mode, apply $C_{\mathrm{g}\mathrm{o}\mathrm{o}\mathrm{d}}$ to $(H,q)$ with external scratch $q=0$, then $CZ(q,F)$, then $C_{\mathrm{g}\mathrm{o}\mathrm{o}\mathrm{d}}^{†}$ to $(H,q)$. This uses two flag calls and one $CZ$ per $S_{\mathrm{g}\mathrm{o}\mathrm{o}\mathrm{d}}^{\mathrm{\prime }}$. An alternative direct mode explicitly supplies and charges $S_{\mathrm{g}\mathrm{o}\mathrm{o}\mathrm{d}}^{\mathrm{\prime }}$ instead. Here $X$, Toffoli and $CZ$ are the exact supplied qubit gates of The Pauli matrices, Toffoli and Fredkin gates and Controlled-Z and SWAP, with their gate costs charged.

The scratch $q$ starts and ends in $0$ for each reflection and may be reused. It is external implementation workspace, not part of the logical input $\mathrm{\mid }{0}^{\mathrm{\prime }}⟩$ or the projector defining $S_0^{\mathrm{\prime }}$. The constructions implement the displayed logical operators on every vector of $H^{\mathrm{\prime }}$ with clean scratch; neither assumes $T=0$ to implement $I_T$ in the good projector.

Prepare $A^{\mathrm{\prime }}\mathrm{\mid }{0}^{\mathrm{\prime }}⟩$, apply exactly $m$ iterates $-A^{\mathrm{\prime }}{S}_0^{\mathrm{\prime }}{A}^{\mathrm{\prime }†}{S}_{\mathrm{g}\mathrm{o}\mathrm{o}\mathrm{d}}^{\mathrm{\prime }}$, then perform the projective instrument $\{{\mathrm{\Pi }}^{\mathrm{\prime }},I-{\mathrm{\Pi }}^{\mathrm{\prime }}\}$. The terminal instrument is an explicitly supplied, separately charged capability. Alternatively construct it with two external clean bits $q,d$: apply $C_{\mathrm{g}\mathrm{o}\mathrm{o}\mathrm{d}}$ on $(H,q)$, Toffoli$(q,F;d)$, measure only $d$ in the computational basis to obtain the good/bad record, then reverse the Toffoli and the flag call. Both scratch bits return to $0$ on either observed branch; this option adds two flag calls, two Toffolis and one bit measurement, separately from the reflection counts. Return the good conditional output, discarding $F,T$; $T$ is clean and $F=1$ on success. In the diagonal classical-output specialization, measure the designated output after the good outcome and use the supplied verifier.

The ideal exact-real and exact-rotation assumptions are part of the input model. A finite-precision implementation requires a separate error budget and theorem.