The circuit for Kitaev phase estimation is given as:

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By varying $\theta$, we are able to determine $\sin(2 \pi M \phi_k)$ and $\cos (2 \pi M \phi_k)$ from sampling the circuit and calculating the probabilities of different outputs of the first qubit ($\phi_k$ is a real number, with $e^{2 \pi i \phi_k}$ being the eigenvalue corresponding to $U$ and $|\xi_k\rangle$). This makes sense, except, the paper I'm reading says we have to take measurements with different values of $M$ to get maximum precision for our algorithm's output. Why is this the case?


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