= HHL controlled reciprocal rotation
{c}
{title2=$\sin\theta_\lambda=c/\lambda$}
On a positive <eigenvalue> label $\lambda$ and a clean <quantum ancilla>, the <HHL algorithm> applies a <quantum variable rotation> with angle $\theta_\lambda=\arcsin(c/\lambda)$, where $0<c\leq\lambda_{\min}$:
$$
|\lambda\rangle|0\rangle\longmapsto|\lambda\rangle\left(\sqrt{1-c^2/\lambda^2}|0\rangle+\frac c\lambda|1\rangle\right).
$$
<Uncomputation> of the eigenvalue register followed by conditioning on flag one multiplies each input <eigenvector> amplitude by $c/\lambda$. If $|b\rangle$ is normalized, the success probability is $c^2\|A^{-1}|b\rangle\|^2$. Choosing $c=\lambda_{\max}/\kappa$ from a valid <condition number> bound gives success probability at least $1/\kappa^2$.
Back to article page
