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First construct a controlled unitary gate from the uncontrolled oracle. Keep the supplied $|v_0⟩$ in a reference register. Controlled on an extra qubit, swap the data and reference registers, query $U$ on the register that contains $|v_0⟩$ in one branch, and swap back. Because $U|v_0⟩=|v_0⟩$, that branch is unchanged, while the other branch acquires $U$ on the data. Reversing the control convention with Pauli X gates gives controlled-$U$. The reference state is returned unchanged, and each use costs one query to $U$ and $O(n)$ controlled-SWAP gates.

Expand $|b⟩={\sum }_j{b}_j|v_j⟩$ in an eigenbasis of $U$. Exact quantum phase estimation with an $m$-qubit phase register producesApply the available phase gates, controlled by the corresponding bits of $c_j$, to multiply branch $j$ byThen reverse phase estimation. Although no $U^{†}$ oracle was supplied, every eigenvalue obeys ${\lambda }_j^{2^m}=1$, so $U^{-q}=U^{2^m-q}$. The inverse controlled powers can therefore be implemented with forward calls to $U$. Since $2^m=\mathrm{poly}(n)$, phase estimation and its inverse use only $\mathrm{poly}(n)$ queries. The phase register returns to $|0^m⟩$, while the data register isAs a direct check, the same spectral promise implies $U^{†}=U^{2^m-1}$.