HHL controlled reciprocal rotation ($\sin {\theta }_{\lambda }=c/\lambda$)

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 $0<c\le {\lambda }_{\min }$:Uncomputation of the eigenvalue register followed by conditioning on flag one multiplies each input eigenvector amplitude by $c/\lambda$. If $|b⟩$ is normalized, the success probability is $c^2\Vert A^{-1}|b⟩{\Vert }^2$. Choosing $c={\lambda }_{\max }/\kappa$ from a valid condition number bound gives success probability at least $1/{\kappa }^2$.