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Within the nonrelativistic quantum chromodynamics framework, we make acomprehensive study on the exclusive production of excited charmonium andbottomonium in $e^+e^-\to \gamma^*/Z^0 \to|(Q\bar{Q})[n]\rangle +\gamma$ ($Q=c$or $b$ quarks) at future $Z$ factory, where the $[n]$ represents thecolor-singlet $n^1S_0,~n^3S_1,~n^1P_0$ and $n^3P_J$ ($n=1,2,3,4; J=0,1,2$) Fockstates. The "improved trace technology" is adopted to derive the analyticexpressions at the amplitude level, which is useful for calculating thecomplicated $nP$-wave channels. Total cross sections, differentialdistributions, and uncertainties are discussed in system. According to ourstudy, production rates of heavy quarkonia of high excited Fock states areconsiderable at future $Z$ factory. The cross sections of charmonium for $2S$,$3S$, $4S$, $1P$, $2P$, $3P$ and $4P$-wave states are about $53.5\%$, $30.4\%$,$23.7\%$, $13.7\%$, $6.8\%$, $9.2\%$, and $9.2\%$ of that of the $1S$ state,respectively. And cross sections of bottomonium for $2S$, $3S$, $4S$, $1P$,$2P$, $3P$ and $4P$-wave states are about $39.3\%$, $12.3\%$, $14.3\%$,$7.1\%$, $3.1\%$, $2.7\%$, and $3.1\%$ of that of the $1S$ state, respectively.The main uncertainties come from the radial wave functions at the origin andtheir derivatives at the origin under different potential models. Then, suchsuper $Z$ factory should be a good platform to study the properties of the highexcited charmonium and bottomonium states.

Production of excited heavy quarkonia in $e^+e^- \to \gamma^*/Z^0 \to |(Q\bar{Q})[n]\rangle +\gamma$ at super $Z$ factory