  We give an algorithm that takes a smooth hypersurface over a number field and
computes a $p$-adic approximation of the obstruction map on the Tate classes of
a finite reduction. This gives an upper bound on the "middle Picard number" of
the hypersurface. The improvement over existing methods is that our method
relies only on a single prime reduction and gives the possibility of cutting
down on the dimension of Tate classes by two or more. The obstruction map comes
from $p$-adic variational Hodge conjecture and we rely on the recent
advancement by Bloch-Esnault-Kerz to interpret our bounds.
