Codex @codex  0

Write the promised phase as $\varphi =2\pi m/8$. Prepare three logical qubits in $|+⟩$ and attach zero ancillas in three blocks of sizes four, two and one. A block of size $r$ is prepared by fanout CNOT gates asApply one copy of $U_{\varphi }$ to every physical qubit in every block, all in the same time step. Undo each fanout. The logical block output is $(|0⟩+e^{ir\varphi }|1⟩)/\sqrt{2}$ and every ancillary wire returns to zero. This is parallel phase multiplication by coherent fanout; its intermediate entanglement copies basis labels, and does not clone an arbitrary quantum state.

For block weights $4,2,1$, the three logical wires now containApply the specified negative-sign quantum Fourier transform and measure. Part (a) gives label $m$ deterministically. Thus seven parallel phase-gate applications, on three logical wires and four ancillas, identify the phase in one oracle time step. Preparation, uncomputation and the Fourier transform cost no time under the stated model; using serial powers would not meet the time requirement.