= Parallel phase multiplication by coherent fanout
{title2=$U_\phi^{\otimes r}:\ |0^r\rangle+|1^r\rangle\mapsto |0^r\rangle+e^{ir\phi}|1^r\rangle$}
Use <CNOT gates> to encode one logical <qubit> and $r-1$ zero ancillas into the repetition subspace spanned by $|0^r\rangle,|1^r\rangle$. Apply the <phase gate> in parallel to all $r$ physical <qubits>, then undo the encoding. The logical phase is multiplied by $r$ in one oracle-time layer. This copies computational-basis labels coherently rather than cloning an arbitrary <quantum state>. Blocks of sizes $1,2,4,\ldots,2^{n-1}$ prepare a dyadic Fourier phase state using $2^n-1$ parallel oracle applications.
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