Algorithm 6 Conjugate Pair Oracle Recursive Fourier Sampling

Level:  $1\le k\le l$ Conjugate pair oracles: ${\mathcal{F}}_{k+1}$ and ${\mathcal{G}}_k$ Controls: conjugate pairs ${\mathcal{X}}_j$, $1\le j\le k$ Bias/Target: conjugate pair $\mathcal{Y}$ 1:create ancillas ${\mathcal{X}}_{k+1}=[x_{k+1},{\chi }_{k+1}]$ with ${\chi }_{k+1}:=0$ and ${\mathcal{Y}}^{\prime }=[y^{\prime },{\upsilon }^{\prime }]$ with ${\upsilon }^{\prime }:=1$ 2:apply ${\mathcal{F}}_{k+1}$ to $({\mathcal{X}}_1,\dots ,{\mathcal{X}}_{k+1},{\mathcal{Y}}^{\prime })$ 3:apply ${\mathcal{H}}^{n_{k+1}}$ to ${\mathcal{X}}_{k+1}$ 4:apply ${\mathcal{G}}_k$ to $({\mathcal{X}}_1,\dots ,{\mathcal{X}}_{k+1},\mathcal{Y})$ 5:apply ${\mathcal{H}}^{n_{k+1}}$ to ${\mathcal{X}}_{k+1}$ 6:apply ${\mathcal{F}}_{k+1}$ to $({\mathcal{X}}_1,\dots ,{\mathcal{X}}_{k+1},{\mathcal{Y}}^{\prime })$ 7:discard ancillas  $({\mathcal{X}}_{k+1},{\mathcal{Y}}^{\prime })$