= Binary-angle implementation of a quantum variable rotation
Suppose a <reversible circuit> computes the angle bits $b_l(x)$ in $\theta_x=\sum_l b_l(x)\alpha_l$, where $\alpha_l$ are known binary place values. Apply a <controlled unitary gate> $R_y(2\alpha_l)$ to the target for each set angle bit. The <rotations about the y-axis> commute and their angles add, giving $R_y(2\theta_x)$. Finally perform <uncomputation> of the angle and arithmetic workspace. An $L$-bit angle needs $L$ controlled rotations and twice the reversible angle-computation cost. This preserves coherent superpositions of inputs and underlies the <HHL controlled reciprocal rotation>.
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