We quantify the emergent complexity of quantum states near quantum criticalpoints on regular 1D lattices, via complex network measures based on quantummutual information as the adjacency matrix, in direct analogy to quantifyingthe complexity of EEG/fMRI measurements of the brain. Using matrix productstate methods, we show that network density, clustering, disparity, andPearson's correlation obtain the critical point for both quantum Ising andBose-Hubbard models to a high degree of accuracy in finite-size scaling forthree classes of quantum phase transitions, $Z_2$, mean field superfluid/Mottinsulator, and a BKT crossover.
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