The recently formulated quantum transition state theory (QTST) in which the quantum projection operator is approximated by its parabolic barrier limit and the symmetrized thermal flux is evaluated numerically exactly, is applied to the collinear hydrogen exchange reaction. The results are found to bound the exact results from above for temperatures ranging from T = 200 K to T = 1000 K. The QTST rate is almost exact at high temperature and is a factor of 3.7 greater than the exact rate at T = 200 K, where there is extensive tunneling. Contour plots of the quantum transition state theory reactive flux reveal that the theory accounts well for the "corner cutting" observed in the collinear hydrogen exchange reaction at low temperatures. These results demonstrate that one may estimate quantum rates of bimolecular reactions, using only thermodynamic information.