  There is a growing interest in the $U(1)$ Coulomb liquid in both quantum
materials in pyrochlore ice and cluster Mott insulators and cold atom systems.
We explore a paired hardcore boson model on a pyrochlore lattice. This model is
equivalent to the XYZ spin model that was proposed for rare-earth pyrochlores
with "dipole-octupole" doublets. Since this model has no sign problem for
quantum Monte Carlo (QMC) simulations in a large parameter regime, we carry out
both analytical and QMC calculations. We find that the $U(1)$ Coulomb liquid is
quite stable and spans a rather large portion of the phase diagram with boson
pairing. Moreover, we numerically find thermodynamic evidence that the boson
pairing could induce a possible $\mathbb{Z}_2$ liquid in the vicinity of the
phase boundary between Coulomb liquid and $\mathbb{Z}_2$ symmetry-broken phase.
Besides the materials' relevance with quantum spin ice, we point to quantum
simulation with cold atoms on optical lattices.
