Parallel phase multiplication by coherent fanout ($U_{\varphi }^{\otimes r}:|0^r⟩+|1^r⟩\mapsto |0^r⟩+e^{ir\varphi }|1^r⟩$)

Use CNOT gates to encode one logical qubit and $r-1$ zero ancillas into the repetition subspace spanned by $|0^r⟩,|1^r⟩$. 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 $1,2,4,\dots ,2^{n-1}$ prepare a dyadic Fourier phase state using $2^n-1$ parallel oracle applications.