  Using a soft-wall AdS/QCD approach we derive the Schr\"odinger-type equation
of motion for the tetraquark wave function, which is dual to the dimension-4
AdS bulk profile. The latter coincides with the number of constituents in the
leading Fock state of the tetraquark. The obtained equation of motion is solved
analytically, providing predictions for both the tetraquark wave function and
its mass. A low mass limit for possible tetraquark states is given by $M \ge 2
\kappa = 1$ GeV, where $\kappa = 0.5$ GeV is the typical value of the scale
parameter in soft-wall AdS/QCD. We confirm results of the COMPASS Collaboration
recently reported on the discovery of the $a_1(1414)$ state, interpreted as a
tetraquark state composed of light quarks and having $J^{PC} = 1^{++}$. Our
prediction for the mass of this state, $M_{a_1} = \sqrt{2}$ GeV $\simeq 1.414$
GeV, is in good agreement with the COMPASS result $M_{a_1} =
1.414^{+0.015}_{-0.013}$ GeV. Next we included finite quark mass effects, which
are essential for the tetraquark states involving heavy quarks.
