We quantify the emergent complexity of quantum states near quantum critical points on regular 1D lattices, via complex network measures based on quantum mutual information as the adjacency matrix, in direct analogy to quantifying the complexity of EEG/fMRI measurements of the brain. Using matrix product state methods, we show that network density, clustering, disparity, and Pearson's correlation obtain the critical point for both quantum Ising and Bose-Hubbard models to a high degree of accuracy in finite-size scaling for three classes of quantum phase transitions, Z2, mean field superfluid/Mott insulator, and a BKT crossover.Classical statistical physics has developed a powerful set of tools for analyzing complex systems, chief among them complex networks, in which connectivity and topology predominate over other system features [1]. Complex networks model systems as diverse as the brain and the internet; however, up till now they have been obtained in quantum systems by explicitly enforcing complex network structure in their quantum connections [2][3][4][5][6][7], e.g. entanglement percolation on a complex network [4]. In contrast, complexity measures on the brain observe emergent complexity arising out of, e.g., a regular array of EEG electrodes placed on the scalp, via an adjacency matrix formed from the classical mutual information calculated between them [8]. We apply the quantum generalization of this measure, an adjacency matrix of the quantum mutual information calculated on quantum states [9], to well known quantum many-body models on regular 1D lattices, and uncover emergent quantum complexity which clearly identifies quantum critical points (QCPs) [10,11]. Quantum mutual information bounds two-point correlations from above [12], measurable in a precise and tunable fashion in e.g. atom interferometry in 1D Bose gases [13], among many other quantum simulator architectures. Using matrix product state (MPS) computational methods [14,15], we demonstrate rapid finite size-scaling for both transverse Ising and Bose-Hubbard models, including Z 2 , mean field, and BKT quantum phase transitions.As we move toward more and more complex quantum systems in materials design and quantum simulators, involving a hierarchy of scales, diverse interacting components, and a structured environment, we expect to observe long-lived dynamical features, fat-tailed distributions, and other key identifiers of complexity [16][17][18]. Such systems include quantum simulator technologies based on ultracold atoms and molecules [19], trapped ions [20], and Rydberg gases [21], as well as superconducting Josephson-junction based nanoelectromechanical systems in which different quantum subsystems form compound quantum machines with both electrical and mechanical components [22]. A key area in which we have taken a first step beyond phase diagrams and ground state properties is non-equilibrium quantum dynamics, where critical exponents and renormalization group theory are only weakly applicable at best, e.g. in the KibbleZurek mechanism,...

We compute nodal centrality measures on the collaboration networks of students enrolled in three upper-division physics courses, usually taken sequentially, at the Colorado School of Mines. These are complex networks in which links between students indicate assistance with homework. The courses included in the study are intermediate Classical Mechanics, introductory Quantum Mechanics, and intermediate Electromagnetism. By correlating these nodal centrality measures with students' scores on homework and exams, we find four centrality measures that correlate significantly with students' homework scores in all three courses: in-strength, out-strength, closeness centrality, and harmonic centrality. These correlations suggest that students who not only collaborate often, but also collaborate significantly with many different people tend to achieve higher grades. Centrality measures between simultaneous collaboration networks (analytical vs. numerical homework collaboration) composed of the same students also correlate with each other, suggesting that students' collaboration strategies remain relatively stable when presented with homework assignments targeting different skills. Additionally, we correlate centrality measures between collaboration networks from different courses and find that the four centrality measures with the strongest relationship to students' homework scores are also the most stable measures across networks involving different courses. Correlations of centrality measures with exam scores were generally smaller than the correlations with homework scores, though this finding varied across courses.

Cellular automata are interacting classical bits that display diverse emergent behaviors, from fractals to random-number generators to Turing-complete computation. We discover that quantum cellular automata (QCA) can exhibit complexity in the sense of the complexity science that describes biology, sociology, and economics. QCA exhibit complexity when evolving under 'Goldilocks rules' that we define by balancing activity and stasis. Our Goldilocks rules generate robust dynamical features (entangled breathers), network structure and dynamics consistent with complexity, and persistent entropy fluctuations. Present-day experimental platforms-Rydberg arrays, trapped ions, and superconducting qubits-can implement our Goldilocks protocols, making testable the link between complexity science and quantum computation exposed by our QCA.

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While existing work has demonstrated that campaign donations can buy access to benefits such as favorable legislation and preferential contracting, we highlight another use of campaign contributions: buying reductions in regulatory enforcement. Specifically, we argue that in return for campaign contributions, Colombian mayors who rely on donor-funding (compared with those who do not) choose not to enforce sanctions against illegal deforestation activities. Using a regression discontinuity design, we show that deforestation is significantly higher in municipalities that elect donor-funded as opposed to self-funded politicians. Further analysis shows that only part of this effect can be explained by differences in contracting practices by donor-funded mayors. Instead, evidence of heterogeneity in the effects according to the presence of alternative formal and informal enforcement institutions, and analysis of fire clearance, support the interpretation that campaign contributions buy reductions in the enforcement of environmental regulations.