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Quantum Complexity in Ultracold Quantum Systems

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Ultracold quantum simulators have proven a tremendous success. These analog quantum computers have allowed us to explore diverse quantum many-body phenomena from quantum phase transitions to the Kibble-Zurek mechanism to many-body localization. We propose a new direction for analog quantum computations, quantum complexity. Despite hundreds of thousands of empirical examples of complexity ranging from complex networks like the internet to diverse mixed geometry microbial communities like the microbiome found in the human gut, we have no first principles theory of complexity – we don’t know why nature seems to prefer complexity. Moreover, unlike the other senses of the word “macroscopicity” we don’t have a good sense of how classical complexity, associated with macroscopic classical systems, results from the underlying quantum dynamics – so we don’t know where this preference first appears at the quantum level. In this talk, we principally explore two topics in this untested regime of quantum mechanics. First, we propose a new set of quantum measures for complex quantum dynamics, namely quantum mutual information complex networks. As a proof of principle, we show that complex network measures like clustering reproduce and improve upon traditional correlation-based measures for the critical point of quantum phase transitions in transverse Ising and Bose-Hubbard models. Second, we identify new concrete regimes of quantum complexity in quantum simulators implementing molecular Hubbard Hamiltonians. For singlet sigma molecules we calculate specific parameter sets for five kinds of molecules either already quantum degenerate and at unit filling in optical lattices in lab experiments, or close to quantum degeneracy: LiCs, NaK, RbCs, KRb, RbCs, and LiNa. Finally, we briefly outline progress in two other areas of quantum complexity: new regimes accessible in many-body symmetric top molecular systems, and quantum games of life

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