Nearest-integer success bound for quantum phase estimation ($\mathrm{Pr}(\text{nearest phase label})\ge 4/{\pi }^2$)

For a unitary operator eigenphase $\varphi$, an $n$-qubit quantum phase estimation register of size $L=2^n$ has output probability amplitudeChoose the nearest integer label $m$ to $L\varphi /(2\pi )$, with labels understood modulo $L$, and put $d=L\varphi /(2\pi )-m$, taking the representative $|d|\le 1/2$. The finite geometric series gives $|a_m|=|\sin (\pi d)|/[L|\sin (\pi d/L)|]$. For $0<|d|\le 1/2$, concavity of sine gives $|\sin (\pi d)|\ge 2|d|$, while $|\sin (\pi d/L)|\le \pi |d|/L$. Thus the probability of the nearest label is at least $4/{\pi }^2$. At $d=0$ it is exactly one. If two labels tie for nearest, each satisfies the bound.