  We systematically test the performance of several Monte Carlo update schemes
for the $(2+1)$d XY phase transition of quantum rotor model. By comparing the
local Metropolis (LM), LM plus over-relaxation (OR), Wolff-cluster (WC), hybrid
Monte Carlo (HM), hybrid Monte Carlo with Fourier acceleration (FA) scheme, it
is clear that among the five different update schemes, at the quantum critical
point, the WC and FA schemes acquire the smallest autocorrelation time and cost
the least amount of CPU hours in achieving the same level of relative error,
and FA enjoys a further advantage of easily implementable for more complicated
interactions such as the long-range ones. These results bestow one with the
necessary knowledge of extending the quantum rotor model, which plays the role
of ferromagnetic/antiferromagnetic critical bosons or Z$_2$ topological order,
to more realistic and yet challenging models such as Fermi surface
Yukawa-coupled to quantum rotor models.
