primary-metadata-checked Abstract; Physical Review Letters 136, 240603, published 18 June 2026
QET-based function preparation with few ancillas. Approximation, normalization and success probability remain separate costs.
Structure before circuit tricks
Block extraction and spectral filtering
What norm, overlap and polynomial approximation determine success?
\[U_A|0\rangle|\psi\rangle=|0\rangle A|\psi\rangle/\alpha+|\perp\rangle,\quad p=\|A\psi\|_2^2/\alpha^2\]
What this technique preserves
A projected block supplies an unnormalized desired branch. Normalize conditionally and explicitly account for amplification and polynomial-approximation errors.
Hypotheses and hidden contracts
- A nonzero accepted branch
- Known normalization alpha and available inverses/controls as used
- Gap/overlap or filling assumptions for efficiency, not for mere definition
Mathematical proof mechanism
This is an authored reusable derivation guide, not a claim that the full family has been source-assimilated.
- Apply the clean projection to identify the accepted branch.
- Use Born's rule to derive its norm-squared probability.
- Bound approximation and success amplification under the declared access model.
Exact Lean substrates
No local transport theorem is bound to this record. Do not infer formal truth from its position in the atlas.
Do not cross this boundary
Block encoding does not make normalized state preparation deterministic or uniformly cheap.
Related transports
BE + input + overlap gives a state branch — proposal
Source and prior-art ledger
primary-metadata-checked Abstract: initial overlap, spectral-gap promise, energy information and lower bounds
Use the promised overlap/gap model; do not erase these costs in a generic strong-correlation claim.
Copy mathematical mechanism as LaTeX
% Authored mechanism lesson; not a new theorem certificate.
\section*{Block extraction and spectral filtering}
What norm, overlap and polynomial approximation determine success?
\[
U_A|0\rangle|\psi\rangle=|0\rangle A|\psi\rangle/\alpha+|\perp\rangle,\quad p=\|A\psi\|_2^2/\alpha^2
\]
A projected block supplies an unnormalized desired branch. Normalize conditionally and explicitly account for amplification and polynomial-approximation errors.
\paragraph{Hypotheses and contracts.}
\begin{enumerate}
\item A nonzero accepted branch
\item Known normalization alpha and available inverses/controls as used
\item Gap/overlap or filling assumptions for efficiency, not for mere definition
\end{enumerate}
\paragraph{Mathematical proof mechanism.}
This is a reusable derivation guide; exact certified scope is given by the linked Lean signatures.
\begin{enumerate}
\item Apply the clean projection to identify the accepted branch.
\item Use Born's rule to derive its norm-squared probability.
\item Bound approximation and success amplification under the declared access model.
\end{enumerate}
\paragraph{Boundary.} Block encoding does not make normalized state preparation deterministic or uniformly cheap.