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Quantum many-body systems are characterized by patterns of correlations thatdefine highly-non trivial manifolds when interpreted as data structures.Physical properties of phases and phase transitions are typically retrieved viasimple correlation functions, that are related to observable responsefunctions. Recent experiments have demonstrated capabilities to fullycharacterize quantum many-body systems via wave-function snapshots, opening newpossibilities to analyze quantum phenomena. Here, we introduce a method to datamine the correlation structure of quantum partition functions via their pathintegral (or equivalently, stochastic series expansion) manifold. Wecharacterize path-integral manifolds generated via state-of-the-art QuantumMonte Carlo methods utilizing the intrinsic dimension (ID) and the variance ofdistances from nearest neighbors (NN): the former is related to datasetcomplexity, while the latter is able to diagnose connectivity features ofpoints in configuration space. We show how these properties feature universalpatterns in the vicinity of quantum criticality, that reveal how datastructures {\it simplify} systematically at quantum phase transitions. This isfurther reflected by the fact that both ID and variance of NN-distances exhibituniversal scaling behavior in the vicinity of second-order andBerezinskii-Kosterlitz-Thouless critical points. Finally, we show hownon-Abelian symmetries dramatically influence quantum data sets, due to thenature of (non-commuting) conserved charges in the quantum case. Complementaryto neural network representations, our approach represents a first, elementarystep towards a systematic characterization of path integral manifolds beforeany dimensional reduction is taken, that is informative about universalbehavior and complexity, and can find immediate application to both experimentsand Monte Carlo simulations.

Intrinsic dimension of path integrals: data mining quantum criticality and emergent simplicity