 Quantum phase estimation tail bound

Let $L=2^t$, let $j_0$ be the nearest label to $L\varphi$, and let $m\ge 2$. The exact quantum phase estimation amplitude is $L^{-1}{\sum }_{a=0}^{L-1}{e}^{2\pi ia(\varphi -j/L)}$. The sine bound on circular distances gives probability at most $1/(4d_j^2)$, where $d_j$ is the centered distance from $j$ to $L\varphi$. At label distance $\ell$ from $j_0$, $|d_j|\ge \ell -1/2$, with at most two labels for each $\ell$. Since $(\ell -1/2{)}^{-2}\le [(\ell -1)\ell {]}^{-1}$, the tail telescopes to the displayed bound. Adding $u$ control qubits beyond $b$ desired phase bits gives circular accuracy below $2^{-b}$ with failure probability at most $[2(2^u-1){]}^{-1}$.