Unknown-amplitude randomized search

This is a sequential randomized procedure, not a bounded-$T$ circuit family. Fix $c=3\mathrm{/}2$. For rounds $\mathrm{\ell }=1,2,\dots$, compute the finite integer $M_{\mathrm{\ell }}=\lceil c^{\mathrm{\ell }}\rceil$ by exact rational arithmetic. In classical-output mode, require the diagonal-predicate specialization of Prepared unitary and success subspace for amplitude amplification and a costed verifier that accepts exactly the basis labels designated good by ${\mathrm{\Pi }}_{\mathrm{g}\mathrm{o}\mathrm{o}\mathrm{d}}$; measure the designated output basis only after the good/bad instrument and then run the verifier.

1. On a fresh register prepare $A\mathrm{\mid }0⟩$ and perform the declared good/bad projective instrument. On a bad outcome, continue to step 2. On a good outcome, return the conditional quantum state in quantum-output mode; in classical-output mode, measure the designated output basis, verify the candidate, and return it if accepted.

2. Prepare $A\mathrm{\mid }0⟩$ on a second fresh register. Draw $j$ uniformly from the finite set $\{1,\dots ,M_{\mathrm{\ell }}\}$, apply exactly $j$ standard $Q$-iterates, and perform the good/bad instrument. On a good outcome, return the conditional quantum state in quantum-output mode; in classical-output mode, measure the designated output basis, verify the candidate, and return it if accepted. If the outcome is bad or verification rejects, discard the registers and continue to round $\mathrm{\ell }+1$.

The random choice is fresh and independent in each round; the schedule depends only on $\mathrm{\ell }$, never on the unknown $a$. Every round is finite. When $a=0$, the loop runs forever and does not decide whether the good set is empty.

The expected query cost of this unbounded sequential procedure is interpreted using finite execution prefixes as in Expected query cost from finite execution prefixes. A zero-probability good branch is never normalized.