Recent experiments with quantum simulators and noisy intermediate-scale quantum devices have demonstrated unparalleled capabilities of probing many-body wave functions, via directly probing them at the single quantum level via projective measurements. However, very little is known about to interpret and analyse such huge datasets. This represent a fundamental challenge for theory to understand experimental data, that is also relevant to other fields where similarly large data sets are routinely explored - from classical simulations of gauge theories, to observatory studies of many-body ensembles.

In this talk, I will show how it is possible to provide such characterisation of quantum hardware via a direct and assumption-free data mining. The core idea of this programme is the fact that snapshots of many body systems can be construed as very high-dimensional manifolds. Such a manifolds can be characterised via basic topological concepts, in particular, by their intrinsic dimension, and by advanced theoretical tools from network theory and non-parametric, unsupervised learning.

This new approach to the many-body problem opens up a cornucopia of methods to connect physical properties to a stochastic sampling of the system wave function. I will focus here on two specific applications. Firstly, I will discuss theoretical results for both classical and quantum many-body spin systems that illustrate how data structures undergo structural transitions whenever the underlying physical system does, and display universal (critical) behavior in both classical and quantum mechanical cases. These results pave the way for a systematic understanding of field theory aspects in data space, a topic of current interesting in particle and statistical physics. Secondly, I will discuss how our methods allow to track Kolmogorov complexity in quantum simulators and quantum computers, providing novel insights onto the working of such systems, in terms of both practical and fundamental aspects - including cross-certification of quantum devices, a grand challenge in the field.