(color online) Two different representations used for the configuration space in PQMC simulation in the VB basis, exemplified with a Heisenberg chain of 10 sites. VBs of the sampled trial state are shown at the left and right edges. In panel (a), the linked vertex representation used for loop updates is shown, with solid and open circles indicating up and down spins. Diagonal and off-diagonal operators, and are denoted by open and solid vertical bars, respectively. An example of a loop is shown in red; flipping this loop changes all up spins to down, and vice versa, thereby also changing the operator type (diagonal or off-diagonal) on some of the vertices. The number of operators n and their locations on the lattice are updated separately in “diagonal updates”. In panel (b), the pure VB representation used for collecting operator expectation values is shown for the same configuration. Here there is no distinction between diagonal and off diagonal operators and the bars indicate the full singlet projectors P_ij. Propagated left and right VB configurations are shown at the center; they make up the transition graph on which correlation functions are evaluated.