Known-amplitude amplification with integer rounding

Input a coherent sampler with an exactly represented known success weight $a\in [0,1]$. For the finite rational-input procedure, $a$ is given as a binary rational; exact algebraic inputs are also allowed when the supplied representation supports exact comparisons. If $a=0$, return fail without normalizing a good component. If $a=1$, prepare $A\mathrm{\mid }0⟩$, apply the declared good/bad projective instrument, and return its good conditional state (or, in the diagonal classical-output specialization, measure and verify the output).

For $0<a<1$, set $\theta :=\arcsin \sqrt{a}\in (0,\pi \mathrm{/}2)$, using the principal inverse-sine and nonnegative-square-root conventions of Principal inverse sine and inverse cosine and Square roots exist: a unique $\sqrt{a}\ge 0$ with $(\sqrt{a}{)}^2=a$; the positives are $\{x^2:x\mathrm{\ne }0\}$. Then $\theta$ is the unique angle with ${\sin }^2\theta =a$. Set $K:=\lceil 1\mathrm{/}\sqrt{a}\rceil$, equivalently the least positive integer $K$ with $K^{2}a\ge 1$. For each integer $k=0,\dots ,K$, compute $p_k={\sin }^2((2k+1)\theta )$ exactly by the rational recurrence $x_0=1,x_1=1-2a,x_{n+1}=2(1-2a)x_n-x_{n-1}(n\ge 1),p_k=(1-x_{2k+1})\mathrm{/}2.$ Indeed, writing $x_n=\cos (2n\theta )$, double-angle gives $x_0=1$, $x_1=\cos (2\theta )=1-2a$, and cosine addition gives the displayed recurrence. Also ${\sin }^2((2k+1)\theta )=(1-\cos ((4k+2)\theta ))\mathrm{/}2=(1-x_{2k+1})\mathrm{/}2$. Choose the least $k$ attaining the maximum of this finite list. Run the fixed-$k$ procedure Amplitude amplification with a fixed integer iteration count, verify when a classical candidate is requested, and return the good conditional output or fail. For rational $a$, every $x_n$ and $p_k$ is rational, so the comparisons are exact. No time bound for this classical preprocessing is claimed.

For a general real $a$, the same mathematical maximizer is defined, but a finite implementation requires an exact representation and a procedure that decides all finitely many comparisons, including ties. The item makes no computation claim for an arbitrary unrepresented real.