Near-grid quantum phase estimation ($P(n)={\sin }^2(Nϵ/2)/[N^2{\sin }^2(ϵ/2)]$)

For register size $N=2^t$ and eigenphase $\varphi =2\pi n/N+ϵ$, the amplitude of decoded label $m$ is $N^{-1}{\sum }_{k=0}^{N-1}{e}^{ik(\varphi -2\pi m/N)}$. The finite geometric series gives $P(n)={\sin }^2(Nϵ/2)/[N^2{\sin }^2(ϵ/2)]$. If $N|ϵ|\ll 1$, this is $1-(N^2-1){ϵ}^2/12+O(N^4{ϵ}^4)$. Thus the nearest exactly representable phase is returned with probability near one, not with certainty unless the offset vanishes.

// Target: quantum-error-correction.bigb