Magnetic domain walls (DWs) are topological defects that exhibit robust low-energy modes that can be harnessed for classical and neuromorphic computing. However, the quantum nature of these modes has been elusive thus far. Using the language of cavity optomechanics, we show how to exploit a geometric Berry-phase interaction between the localized DWs and the extended magnons in short ferromagnetic insulating wires to efficiently cool the DW to its quantum ground state or to prepare nonclassical states exhibiting a negative Wigner function that can be extracted from the power spectrum of the emitted magnons. Moreover, we demonstrate that magnons can mediate long-range entangling interactions between qubits stored in distant DWs, which could facilitate the implementation of a universal set of quantum gates. Our proposal relies only on the intrinsic degrees of freedom of the ferromagnet, and can be
naturally extended to explore the quantum dynamics of DWs in ferrimagnets and antiferromagnets, as well as quantum vortices or skyrmions confined in insulating magnetic nanodisks.