  Quantum many-body systems are characterized by patterns of correlations that
define highly-non trivial manifolds when interpreted as data structures.
Physical properties of phases and phase transitions are typically retrieved via
simple correlation functions, that are related to observable response
functions. Recent experiments have demonstrated capabilities to fully
characterize quantum many-body systems via wave-function snapshots, opening new
possibilities to analyze quantum phenomena. Here, we introduce a method to data
mine the correlation structure of quantum partition functions via their path
integral (or equivalently, stochastic series expansion) manifold. We
characterize path-integral manifolds generated via state-of-the-art Quantum
Monte Carlo methods utilizing the intrinsic dimension (ID) and the variance of
distances from nearest neighbors (NN): the former is related to dataset
complexity, while the latter is able to diagnose connectivity features of
points in configuration space. We show how these properties feature universal
patterns in the vicinity of quantum criticality, that reveal how data
structures {\it simplify} systematically at quantum phase transitions. This is
further reflected by the fact that both ID and variance of NN-distances exhibit
universal scaling behavior in the vicinity of second-order and
Berezinskii-Kosterlitz-Thouless critical points. Finally, we show how
non-Abelian symmetries dramatically influence quantum data sets, due to the
nature of (non-commuting) conserved charges in the quantum case. Complementary
to neural network representations, our approach represents a first, elementary
step towards a systematic characterization of path integral manifolds before
any dimensional reduction is taken, that is informative about universal
behavior and complexity, and can find immediate application to both experiments
and Monte Carlo simulations.
