Exact success for known positive amplitude with explicit dilution

The algorithm sets $m=\lceil \pi \mathrm{/}(4\theta )-1\mathrm{/}2\rceil$, so $m\ge \pi \mathrm{/}(4\theta )-1\mathrm{/}2$. Multiplying by $4\theta >0$ gives $4m\theta +2\theta \ge \pi$, hence $4m+2\ge \pi \mathrm{/}\theta$ and $0<{\theta }^{\mathrm{\prime }}=\pi \mathrm{/}(4m+2)\le \theta <\pi \mathrm{/}2$. By sine monotonicity in [F6], $0<{\sin }^2{\theta }^{\mathrm{\prime }}\le {\sin }^2\theta =a$, so $0<r={\sin }^2{\theta }^{\mathrm{\prime }}\mathrm{/}a\le 1$ and the two nonnegative square roots used in the supplied flag rotation are defined.

In the constructed zero reflection, on fixed logical flag/temporary bits $f,t$, the four $X$ gates and two Toffolis compute and then erase $q=(1-f)(1-t)$. The data sees $S_0$ exactly when $f=t=0$, and identity otherwise. This is $I-2\mathrm{\mid }0⟩⟨0{\mathrm{\mid }}_H\otimes \mathrm{\mid }00⟩⟨00{\mathrm{\mid }}_{F,T}=S_0^{\mathrm{\prime }}$. In the constructed good reflection, split any data vector as $v_g+v_b$ with $v_g={\mathrm{\Pi }}_{\mathrm{g}\mathrm{o}\mathrm{o}\mathrm{d}}v$. The first flag call records $1$ on $v_g$ and $0$ on $v_b$ in $q$; $CZ(q,F)$ multiplies only $v_g$ with $F=1$ by $-1$, and the inverse flag call clears $q$. Thus it is $I-2{\mathrm{\Pi }}_{\mathrm{g}\mathrm{o}\mathrm{o}\mathrm{d}}\otimes \mathrm{\mid }1⟩⟨1{\mathrm{\mid }}_F\otimes I_T=S_{\mathrm{g}\mathrm{o}\mathrm{o}\mathrm{d}}^{\mathrm{\prime }}$ for arbitrary $T$. Linearity extends both calculations to every logical vector with clean external scratch. Direct modes supply these same operators by hypothesis.

For the optional constructed terminal instrument, the coherent computation into $d$ records exactly the predicate defining ${\mathrm{\Pi }}^{\mathrm{\prime }}$. Measuring only $d$ therefore leaves the unnormalized logical branches ${\mathrm{\Pi }}^{\mathrm{\prime }}\rho {\mathrm{\Pi }}^{\mathrm{\prime }}$ and $(I-{\mathrm{\Pi }}^{\mathrm{\prime }})\rho (I-{\mathrm{\Pi }}^{\mathrm{\prime }})$; reversing the conjunction and flag computation clears $d,q$ within either branch without measuring the good component separately. This is the stated projective instrument. The direct mode supplies it by hypothesis.

In the prepared state, the good and bad components are $\sqrt{a}\mathrm{\mid }G⟩$ and $\sqrt{1-a}\mathrm{\mid }B⟩$. The coherent flag operation sets the temporary bit exactly on the good component; the controlled rotation places flag-$1$ amplitude $\sqrt{r}$ on that component and leaves the bad component with flag $0$; uncomputing the temporary bit returns it to $0$. Thus $A^{\mathrm{\prime }}\mathrm{\mid }{0}^{\mathrm{\prime }}⟩$ has good component $\sqrt{ar}\mathrm{\mid }G⟩\mathrm{\mid }1⟩\mathrm{\mid }0⟩$. Since $ar={\sin }^2{\theta }^{\mathrm{\prime }}$, the modified pair $(A^{\mathrm{\prime }},{\mathrm{\Pi }}^{\mathrm{\prime }})$ has success weight ${\sin }^2{\theta }^{\mathrm{\prime }}$.

By step 1.2 the selected access modes implement the exact modified-pair iterates with external scratch reset after each call. The two-dimensional amplification plane is spanned by the good and bad components of $A^{\mathrm{\prime }}\mathrm{\mid }{0}^{\mathrm{\prime }}⟩$, both with $T=0$ by step 2.1; it therefore retains $T=0$. The rotation theorem [F2] applied after $m$ iterates gives the following good probability. [given, F1, F2, F3, F4, F5, step 2.1, step 1.2, step 1.3, algebra] ${\sin }^2((2m+1){\theta }^{\mathrm{\prime }})={\sin }^2\text{\hspace{0.17em}⁣}((2m+1)\frac{\pi }{4m+2})={\sin }^2(\pi \mathrm{/}2)=1.$ By the projective outcome law [F3], its good branch occurs with probability one. Its conditional output has support in $\mathrm{ran}{\mathrm{\Pi }}^{\mathrm{\prime }}$ by [F4]; the success flag is $1$ and the temporary bit was uncomputed to $0$ in step 2.1.

If $a=1$, the algorithm's direct branch prepares a state in $\mathrm{ran}{\mathrm{\Pi }}_{\mathrm{g}\mathrm{o}\mathrm{o}\mathrm{d}}$, so [F3] gives good-outcome probability one and [F4] gives its conditional support. If $a=0$, it returns fail without normalizing a zero good component and makes no claim that a good state exists.