  We present a methodology to investigate phase-diagrams of quantum models
based on the principle of the reduced basis method (RBM). The RBM is built from
a few ground-state snapshots, i.e., lowest eigenvectors of the full system
Hamiltonian computed at well-chosen points in the parameter space of interest.
We put forward a greedy-strategy to assemble such small-dimensional basis,
i.e., to select where to spend the numerical effort needed for the snapshots.
Once the RBM is assembled, physical observables required for mapping out the
phase-diagram (e.g., structure factors) can be computed for any parameter value
with a modest computational complexity, considerably lower than the one
associated to the underlying Hilbert space dimension. We benchmark the method
in two test cases, a chain of excited Rydberg atoms and a geometrically
frustrated antiferromagnetic two-dimensional lattice model, and illustrate the
accuracy of the approach. In particular, we find that the ground-manifold can
be approximated to sufficient accuracy with a moderate number of basis
functions, which increases very mildly when the number of microscopic
constituents grows - in stark contrast to the exponential growth of the Hilbert
space needed to describe each of the few snapshots. A combination of the
presented RBM approach with other numerical techniques circumventing even the
latter big cost, e.g., Tensor Network methods, is a tantalising outlook of this
work.
