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All the references in this question refer to Quantum algorithm for solving linear systems of equations (Harrow, Hassidim & Lloyd, 2009).

The question I have is about the step where they apply controlled-rotations to transfer the eigenvalue encoded in a quantum register to the amplitudes of a state:

After the quantum phase estimation, the state of the quantum registers is (see page 3): $$ \sum_{j=1}^{N} \sum_{k=0}^{T-1} \alpha_{k|j}\beta_j \vert \tilde\lambda_k\rangle \vert u_j \rangle $$ Then, the HHL algorithm consists in applying rotations controlled by $\vert\tilde\lambda_k\rangle$ to produce the state $$ \sum_{j=1}^{N} \sum_{k=0}^{T-1} \alpha_{k|j}\beta_j \vert \tilde\lambda_k\rangle \vert u_j \rangle \left( \sqrt{1 - \frac{C^2}{\tilde\lambda_k^2}} \vert 0 \rangle + \frac{C}{\tilde\lambda_k}\vert 1 \rangle \right) $$ where "$C = O(1/\kappa)$" (still page 3).

My question: why do they introduce $C$? Isn't $C=1$ valid?

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If $\tilde{\lambda_{k}} < C$, the controlled rotation becomes non-physical since you have coeffecient greater than 1 on your $|1\rangle$ state.

As a result $C < \lambda_{min}$ is a safer choice, and that is $O(1/\kappa)$ according to the 4th paragraph in the intro.

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