  We make a detailed discussion on the one-loop QCD correction to the
bottomonium production associated with a photon, i.e. via the channel
$e^{+}e^{-} \to\gamma^*/Z^0 \to |H_{b\bar{b}}\rangle + \gamma$, where
$|H_{b\bar{b}}\rangle$ stands for the color-singlet bottomonium state as
$\eta_b$, $\Upsilon$, $h_b$ or $\chi_{bJ}$ ($J$=0, 1 or 2), respectively. At
the super $Z$ factory with the collision energy $E_{cm} \sim m_Z$, by summing
up the cross sections for all bottomonium states, we obtain a large one-loop
QCD correction, i.e. $|R|\sim 30\%$. This ensures the necessity and importance
of the one-loop QCD corrections for the present processes. Further more, for
the $\eta_b$, $h_b$ and $\chi_{bJ}$ production, their cross sections are
dominated by the $s$-channel diagrams and are enhanced by the $Z^0$ boson
resonance effect when $E_{cm}\sim m_Z$. While, for the $\Upsilon$ production,
such resonance effect shall be smeared by a large $t(u)$-channel contribution
that dominant over the $s$-channel one. Theoretical uncertainties caused by
slight change of $E_{cm}$, the $b$-quark mass, the renormalization scale and
etc. have been presented. At the super $Z$ factory with a high luminosity up to
${\cal L}=10^{36}{\rm cm}^{-2}{\rm s}^{-1}$, the bottomonium plus one photon
events are sizable, especially for $\eta_b$ and $\Upsilon$, which have large
signal significance. Summing up all bottomonium states, we shall totally have
$\sim 3.8\times10^{5}$ bottomonium events in one operation year. So, the super
$Z$ factory shall provide a good platform for studying the bottomonium
properties.
