  Long-range interactions are relevant for a large variety of quantum systems
in quantum optics and condensed matter physics. In particular, the control of
quantum-optical platforms promises to gain deep insights in quantum-critical
properties induced by the long-range nature of interactions. From a theoretical
perspective, long-range interactions are notoriously complicated to treat.
Here, we give an overview of recent advancements to investigate quantum magnets
with long-range interactions focusing on two techniques based on Monte Carlo
integration. First, the method of perturbative continuous unitary
transformations where classical Monte Carlo integration is applied within the
embedding scheme of white graphs. This linked-cluster expansion allows to
extract high-order series expansions of energies and observables in the
thermodynamic limit. Second, stochastic series expansion quantum Monte Carlo
which enables calculations on large finite systems. Finite-size scaling can
then be used to determine physical properties of the infinite system. In recent
years, both techniques have been applied successfully to one- and
two-dimensional quantum magnets involving long-range Ising, XY, and Heisenberg
interactions on various bipartite and non-bipartite lattices. Here, we
summarise the obtained quantum-critical properties including critical exponents
for all these systems in a coherent way. Further, we review how long-range
interactions are used to study quantum phase transitions above the upper
critical dimension and the scaling techniques to extract these quantum critical
properties from the numerical calculations.
