  We report low-temperature thermal conductivity $\kappa$ of pyrochlore
Yb$_2$Ti$_2$O$_7$, which contains frustrated spin-ice correlations with
significant quantum fluctuations. In the disordered spin-liquid regime,
$\kappa(H)$ exhibits a nonmonotonic magnetic field dependence, which is well
explained by the strong spin-phonon scattering and quantum monopole
excitations. We show that the excitation energy of quantum monopoles is
strongly suppressed from that of dispersionless classical monopoles. Moreover,
in stark contrast to the diffusive classical monopoles, the quantum monopoles
have a very long mean free path. We infer that the quantum monopole is a novel
heavy particle, presumably boson, which is highly mobile in a three-dimensional
spin liquid.
