  The detection of quantum and classical phase transitions in the absence of an
order parameter is possible using the Fisher information metric (FIM), also
known as fidelity susceptibility. Here, we propose and investigate an
unsupervised machine learning (ML) task: estimating the FIM given limited
samples from a multivariate probability distribution of measurements made
throughout the phase diagram. We utilize an unsupervised ML method called
ClassiFIM (developed in a companion paper) to solve this task and demonstrate
its empirical effectiveness in detecting both quantum and classical phase
transitions using a variety of spin and fermionic models, for which we generate
several publicly available datasets with accompanying ground-truth FIM. We find
that ClassiFIM reliably detects both topological (e.g., XXZ chain) and
dynamical (e.g., metal-insulator transition in Hubbard model) quantum phase
transitions. We perform a detailed quantitative comparison with prior
unsupervised ML methods for detecting quantum phase transitions. We demonstrate
that ClassiFIM is competitive with these prior methods in terms of appropriate
accuracy metrics while requiring significantly less resource-intensive training
data compared to the original formulation of the prior methods. In particular,
ClassiFIM only requires classical (single-basis) measurements. As part of our
methodology development, we prove several theorems connecting the classical and
quantum fidelity susceptibilities through equalities or bounds. We also
significantly expand the existence conditions of the fidelity susceptibility,
e.g., by relaxing standard differentiability conditions. These results may be
of independent interest to the mathematical physics community.
