  We study the $\bar b \bar b u d$ tetraquark with quantum numbers $I(J^P) =
0(1^+)$ as well as the $\bar b \bar b u s$ tetraquark with quantum numbers $J^P
= 1^+$ using lattice QCD. We improve on existing work by including both local
and scattering interpolating operators on both sides of the correlation
functions and use symmetric correlation matrices. This allows not only a
reliable determination of the energies of QCD-stable tetraquark ground states,
but also of low-lying excited states, which are meson-meson scattering states.
The latter is particularly important for future finite-volume scattering
analyses. Here, we perform chiral and continuum extrapolations of just the
ground-state energies, for which finite-volume effects are expected to be
small. Our resulting tetraquark binding energies, $-100 \pm
10\:^{+36}_{-43}\:\:{\rm MeV}$ for $\bar b \bar b u d$ and $-30 \pm
3\:^{+11}_{-31}\:\:{\rm MeV}$ for $\bar b \bar b u s$, are consistent with
other recent lattice-QCD predictions.
