  Rydberg atom arrays have emerged as a powerful platform to simulate a number
of exotic quantum ground states and phase transitions. To verify these
capabilities numerically, we develop a versatile quantum Monte Carlo sampling
technique which operates in the reduced Hilbert space generated by enforcing
the constraint of a Rydberg blockade. We use the framework of stochastic series
expansion and show that in the restricted space, the configuration space of
operator strings can be understood as a hard rod gas in $d+1$ dimensions. We
use this mapping to develop cluster algorithms which can be visualized as
various non-local movements of rods. We study the efficiency of each of our
updates individually and collectively. To elucidate the utility of the
algorithm, we show that it can efficiently generate the phase diagram of a
Rydberg atom array, to temperatures much smaller than all energy scales
involved, on a Kagom\'e link lattice. This is of broad interest as the presence
of a $Z_2$ spin liquid has been hypothesized recently.
