  We show how efficient loop updates, originally developed for Monte Carlo
simulations of quantum spin systems at finite temperature, can be combined with
a ground-state projector scheme and variational calculations in the valence
bond basis. The methods are formulated in a combined space of spin z-components
and valence bonds. Compared to schemes formulated purely in the valence bond
basis, the computational effort is reduced from up to O(N^2) to O(N) for
variational calculations, where N is the system size, and from O(m^2) to O(m)
for projector simulations, where m>> N is the projection power. These
improvements enable access to ground states of significantly larger lattices
than previously. We demonstrate the efficiency of the approach by calculating
the sublattice magnetization M_s of the two-dimensional Heisenberg model to
high precision, using systems with up to 256*256 spins. Extrapolating the
results to the thermodynamic limit gives M_s=0.30743(1). We also discuss
optimized variational amplitude-product states, which were used as trial states
in the projector simulations, and compare results of projecting different types
of trial states.
