  Quantum simulations are quickly becoming an indispensable tool for studying
particle transport in correlated lattice models. One of the central topics in
the study of transport is the bad-metal behavior, characterized by the direct
current (dc) resistivity linear in temperature. In the fermionic Hubbard model,
optical conductivity has been studied extensively, and a recent optical lattice
experiment has demonstrated bad metal behavior in qualitative agreement with
theory. Far less is known about transport in the bosonic Hubbard model. We
investigate the conductivity in the Bose-Hubbard model, and focus on the regime
of strong interactions and high-temperatures. We use numerically exact
calculations for small lattice sizes. At weak tunneling, we find multiple peaks
in the optical conductivity that stem from the Hubbard bands present in the
many-body spectrum. This feature slowly washes out as the tunneling rate gets
stronger. At high temperature, we identify a regime of $T$-linear resistivity,
as expected. When the interactions are very strong, the leading
inverse-temperature coefficient in conductivity is proportional to the
tunneling amplitude. As the tunneling becomes stronger, this dependence takes
quadratic form. At very strong coupling and half filling, we identify a
separate linear resistivity regime at lower temperature, corresponding to the
hard-core boson regime. Additionally, we unexpectedly observe that at half
filling, in a big part of the phase diagram, conductivity is an increasing
function of the coupling constant before it saturates at the hard-core-boson
result. We explain this feature based on the analysis of the many-body energy
spectrum and the contributions to conductivity of individual eigenstates of the
system.
