The axion provides a solution for the strong CP problem and is one of the leading candidates for dark matter. This paper proposes an axion detection scheme based on quantum nondemolition detection of magnon, i.e., quanta of collective spin excitations in solid, which is expected to be excited by the axion-electron interaction predicted by the Dine-Fischer-Srednicki-Zhitnitsky (DFSZ) model. The prototype detector is composed of a ferromagnetic sphere as an electronic spin target and a superconducting qubit. Both of these are embedded inside a microwave cavity, which leads to a coherent effective interaction between the uniform magnetostatic mode in the ferromagnetic crystal and the qubit. An upper limit for the coupling constant between an axion and an electron is obtained as $g_{aee}<1.6times10^{-7}$ at the 95% confidence level for the axion mass of $33.117$$mu$eV$<m_{a}<33.130$$mu$eV.
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