  Quantum information storage using charge-neutral quasiparticles are expected
to play a crucial role in the future of quantum computers. In this regard,
magnons or collective spin-wave excitations in solid-state materials are
promising candidates in the future of quantum computing. Here, we study the
quantum squeezing of Dirac and topological magnons in a bosonic honeycomb
optical lattice with spin-orbit interaction by utilizing the mapping to quantum
spin-$1/2$ XYZ Heisenberg model on the honeycomb lattice with discrete Z$_2$
symmetry and a Dzyaloshinskii-Moriya interaction. We show that the squeezed
magnons can be controlled by the Z$_2$ anisotropy and demonstrate how the noise
in the system is periodically modified in the ferromagnetic and
antiferromagnetic phases of the model. Our results also apply to solid-state
honeycomb (anti)ferromagnetic insulators.
