  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.
