We present energy spectra of various tetraquark states with one or more heavy quarks using lattice quantum chromodynamics. These calculations are performed on
$N_f=2+1+1$ MILC ensembles at lattice spacings ~ 0.12, 0.09 and 0.06
fm. A relativistic action with overlap fermions is employed for the
light and charm quarks while a non-relativistic action with
non-perturbatively improved coefficients is used for bottom quarks.
Our results provide a clear indication of the presence of
energy levels below the relevant thresholds of different tetraquark
states. While in double charm sector we find very shallow bound levels, our
results suggest deeply bound energy levels with double bottom tetraquarks.