  We study the XXZ spin-one quantum magnet on the kagome lattice as an example
where quantum fluctuations on highly degenerate classical ground states lead to
various exotic quantum ground states. Previous studies have predicted several
quantum phases, but different analytical approaches do not necessarily lead to
the same physical picture. In this work, we use Quantum Monte Carlo
computations to critically examine some of the predictions made in the
string-net mean-field theory and the degenerate perturbation theory combined
with duality analysis and effective field theory. It is found that the
resulting phase diagram differs from some of the previous predictions. Further
implications of our results to different analytical approaches are discussed.
