  We investigate a doubly-bottomed tetraquark state $T_{bb}$ $(bb
\bar{u}\bar{d})$ with quantum number $I(J^P)=0(1^+)$ in $(2+1)$-flavor lattice
QCD. Using the Non-Relativistic QCD (NRQCD) quark action for $b$ quarks, we
have extracted the coupled channel potential between $\bar{B}\bar{B}^*$ and
$\bar{B}^* \bar{B}^*$ in the HAL QCD method at $a \approx 0.09$ {fm} on
$32^3\times 64$ lattices. The potential predicts an existence of a bound
$T_{bb}$ below the $\bar{B}\bar{B}^*$ threshold. At the physical pion mass
$m_\pi\approx140$ {MeV} extrapolated from $m_\pi\approx 410,\, 570,\, 700$
{MeV}, a binding energy with its statistical error is given by $E_{\rm
binding}^{\rm (coupled)} = 83(10)$ MeV from a coupled channel analysis where
effects due to virtual $\bar{B}^* \bar{B}^*$ states are included through the
coupled channel potential, while we obtain $E_{\rm binding}^{\rm (single)} =
155(17)$ MeV only from a potential for a single $\bar{B}\bar{B}^*$ channel.
This difference indicates that the effect from virtual $\bar{B}^* \bar{B}^*$
states is sizable to the binding energy of $T_{bb}$. Adding $\pm 20$ MeV as
empirical systematic error caused by the NRQCD approximation for $b$ quarks,
our estimate of the $T_{bb}$ binding energy becomes $83(10)(20)$ MeV.
