Algorithm 4 Quantum Recursive Fourier Sampling

Level:  $1\le k\le l$ Quantum oracles: $U_{f_{k+1}}$ and $U_{g_k}$ Controls: qubit strings $|{\psi }_{x_j}\rangle$, $1\le j\le k$ Bias/Target: single qubit $|{\psi }_y\rangle$ 1:create ancilla qubit string $|{\psi }_{x_{k+1}}\rangle :=|0\rangle$ and qubit $|{\psi }_{y^{\prime }}\rangle :=|1\rangle$ 2:apply $H^{\otimes n_{k+1}+1}$ to $|{\psi }_{x_{k+1}},{\psi }_{y^{\prime }}\rangle$ 3:apply $U_{f_{k+1}}$ to $|{\psi }_{x_1},\dots ,{\psi }_{x_{k+1}},{\psi }_{y^{\prime }}\rangle$ 4:apply $H^{\otimes n_{k+1}}$ to $|{\psi }_{x_{k+1}}\rangle$ 5:apply $U_{g_k}$ to $|{\psi }_{x_1},\dots ,{\psi }_{x_{k+1}},{\psi }_y\rangle$ 6:apply $H^{\otimes n_{k+1}}$ to $|{\psi }_{x_{k+1}}\rangle$ 7:apply $U_{f_{k+1}}$ to $|{\psi }_{x_1},\dots ,{\psi }_{x_{k+1}},{\psi }_{y^{\prime }}\rangle$ 8:apply $H^{\otimes n_{k+1}+1}$ to $|{\psi }_{x_{k+1}},{\psi }_{y^{\prime }}\rangle$ 9:discard ancillas  $|{\psi }_{x_{k+1}},{\psi }_{y^{\prime }}\rangle$