= Quantum Fourier sampling bound for order finding
{title2=$\Pr(|j/Q-s/r|\leq1/(2Q)\mid s)\geq4/\pi^2$}
In <quantum order finding>, a phase $s/r$ produces amplitude $Q^{-1}\sum_{x=0}^{Q-1}e^{2\pi ix(s/r-j/Q)}$ after an inverse <quantum Fourier transform>. Its nearest grid point has probability at least $4/\pi^2$, by bounding the sine ratio of this <finite geometric Fourier amplitude>. The initial modular orbit state is an equal-weight superposition of the $r$ phase eigenstates, so the phase label is uniform.
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