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(N2) to
O(N) for variational calculations, where
N is the system size, and from
O(m2) 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
Ms of the two-dimensional Heisenberg model to high precision, using systems with up to $256\ifmmode\times\else\texttimes\fi{}256$ spins. Extrapolating the results to the thermodynamic limit gives
Ms=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.
We propose algorithms of the path-ingetral-based quantum Monte Carlo simulation, which is otherwise prohibitively slow. While the basic idea is the loop-cluster update, there are some important 'tricks' that are vital to make the simulation practical. In the present paper, we show two such techniques and their successful applications to the two-dimensional SU( N ) Heisenberg model and the three-dimensional Bose Hubbard model. In the former, we obtain a new type of the valence-bond-solid state for the two-boson representation, while in the latter we equillibrate a system of which the size is comparable to a typical experiment of optical lattices.