  Let X be a smooth projective toric surface, and H^d(X) the Hilbert scheme
parametrising the length d zero-dimensional subschemes of X. We compute the
rational Chow ring A^*(H^d(X))\_Q. More precisely, if T is the two-dimensional
torus contained in X, we compute the rational equivariant Chow ring
A\_T^*(H^d(X))\_Q and the usual Chow ring is an explicit quotient of the
equivariant Chow ring. The case of some quasi-projective toric surfaces such as
the affine plane are described by our method too.
