  We consider universal approximations of symmetric and anti-symmetric
functions, which are important for applications in quantum physics, as well as
other scientific and engineering computations. We give constructive
approximations with explicit bounds on the number of parameters with respect to
the dimension and the target accuracy $\epsilon$. While the approximation still
suffers from the curse of dimensionality, to the best of our knowledge, these
are the first results in the literature with explicit error bounds for
functions with symmetry or anti-symmetry constraints.
