  The development of quantum gas microscopy for two-dimensional optical
lattices has provided an unparalleled tool to study the Fermi-Hubbard model
(FHM) with ultracold atoms. Spin-resolved projective measurements, or
snapshots, have played a significant role in quantifying correlation functions
which uncover underlying physical phenomena such as antiferromagnetism at
commensurate filling on bipartite lattices, and other charge and spin
correlations, as well as dynamical properties at various densities. However, in
order to interpret these results and to establish temperature scales,
comparison against theory is generally needed. Here, we employ a recent
concept, the {\em multi-scale structural complexity}, and show that when
computed for the snapshots (of either single spin species, local moments, or
total density) it can provide a theory-free property, immediately accessible to
experiments. Specifically, after benchmarking results for Ising and XY models,
we study the structural complexity of snapshots of the repulsive FHM in the
two-dimensional square lattice as a function of doping and temperature. We
generate projective measurements using determinant quantum Monte Carlo and
compare their complexities against those from the experiment. We demonstrate
that these complexities are linked to relevant physical observables such as the
entropy and double occupancy. Their behaviors capture the development of
correlations and relevant length scales in the system. We provide an
open-source code in Python which can be implemented into data analysis routines
in experimental settings for the square lattice.
