Complex networks defined on quantum states via quantum mutual information turn out to give a surprising level of new insight on physical problems ranging from quantum critical phenomena to far-from-equilibrium quantum dynamics. We first show how measures well-known to the complex network theory community, such as clustering and disparity, serve to rapidly and efficiently identify quantum critical points for workhorse many-body models such as the transverse quantum Ising model and Bose-Hubbard model, as studied in present quantum simulator experiments. Then we show how a small modification of such models allows one to produce entangled quantum cellular automata. Complex network-based averages and dynamics serve as key quantifiers for emergent complexity along with localized robust dynamical structures and entropy fluctuations. They show that a new class of highly entangled yet also highly structured quantum states arise out of dynamics just a short step away from present e xperimental protocols. They also identify a set of simple criteria, called Goldilocks Rules, which consistently produce complexity independent of the details of the protocol. Thus complex network theory not only presents a useful new tool we can apply to present quantum simulator experiments, but also suggests significant new directions and new physics to be discovered. References:

- Marc Andrew Valdez, Daniel Jaschke, David L. Vargas and Lincoln D. Carr, "Quantifying Complexity in Quantum Phase Transitions via Mutual Information Complex Networks," Phys. Rev. Lett., v. 119, p. 225301 (2017)
- Bhuvanesh Sundar, Marc Andrew Valdez, Lincoln D. Carr, and Kaden R. A. Hazzard, "A complex network description of thermal quantum states in the Ising spin chain," Phys. Rev. A, under review, arXiv:1803.00994 (2018)
- L. E. Hillberry, P. Rall, D. L. Vargas, H. L. North, N. Yunger Halpern, N. Bao, S. Montangero, and L. D. Carr “Entangled Quantum Cellular Automata, Physical Complexity, and Goldilocks Rules”, to be submitted (2018)

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