Quantum complexity: quantum mutual information, complex networks, and emergent phenomena in quantum cellular automata

Abstract

Emerging quantum simulator technologies provide a new challenge to quantum many body theory. Quantifying the emergent order in and predicting the dynamics of such complex quantum systems requires a new approach. We develop such an approach based on complex network analysis of quantum mutual information. First, we establish the usefulness of quantum mutual information complex networks by reproducing the phase diagrams of transverse Ising and Bose-Hubbard models. By quantifying the complexity of quantum cellular automata we then demonstrate the applicability of complex network theory to non-equilibrium quantum dynamics. We conclude with a study of student collaboration networks, correlating a student's role in a collaboration network with their grades. This work thus initiates a quantitative theory of quantum complexity and provides a new tool for physics education research. We find that network density, clustering coefficient, disparity, and Pearson R correlation show systematic finite size scaling towards critical points of the transverse Ising model and the Bose-Hubbard model. Using matrix product state methods we are able to simulate lattices of hundreds of qubits, allowing us to verify the critical point of the transverse Ising model to within 0.001% of its known value. Furthermore, we find that complex network analysis identifies the Berezinskii-Kosterlitz-Thouless critical point of the Bose-Hubbard to within 3.6% of its accepted value. Finally, we identify the boundary separating the Mott Insulator phase from a superfluid phase in the Bose-Hubbard model by extremizing network density, clustering coefficient, and disparity. After studying the static properties of quantum many body systems, we study the entanglement and complexity generated by Hamiltonian based quantum cellular automata. In quantum cellular automata one defines a set of local rules that govern the evolution of the quantum state. A site in a quantum lattice evolves if the set of sites around it are in certain configurations. Configurations are defined in terms of the number of sites in the ``alive'' state about a site. We quantify entanglement in terms of the central bond entropy, and complexity in terms of persistent fluctuations of the central bond entropy, complex network measures of quantum mutual information networks far from their values for random/well known quantum states, and robust dynamical features. These Hamiltonians are a generalization of the Bleh, Calarco, Montangero Hamiltonian. Before beginning our study of the entanglement and complexity generated by these Hamiltonians we first perform a convergence analysis of the dynamics of the Bleh, Calarco, Montangero Hamiltonian using an open source matrix product state code, Open Source Matrix Product States. We find that the Bleh, Calarco, Montangero Hamiltonian rapidly saturates the entanglement cutoff of Open Source Matrix Product States for all initial conditions studied and is thus not a viable numerical method for studying the dynamics of quantum cellular automata. We conclude our convergence study with a case study of an emergent quantum blinker pattern also observed in exact simulation and a case study of a nearest neighbor quantum cellular automata. We conclude that while for generic initial conditions Open Source Matrix Product States is unable to meet its internal convergence criteria, for particular initial conditions and quantum cellular automata it is able to provide reliable estimates of entanglement and complexity measures. The failure of OpenMPS to provide reliably converged quantum states leads us to study our quantum cellular automata using a Trotter-based time evolution scheme. We quantify the entanglement and complexity generated by 13 next-nearest neighbor quantum cellular automata. We also define Goldilocks rules, rules that produce activity at a site if there are exactly the right number of alive sites in the neighborhood of a site, not too few, not too many. We identify a Goldilocks rule, rule 4, as the best complexity-generating rule out of the 13 rules tested, verifying our hypothesis that only Goldilocks rules are complexity generating. We also find that non-Goldilocks rules tend toward thermalization as quantified by reduced fluctuations in the central bond entropy. We find that both highly entangled quantum states and lowly entangled quantum states have complex structure in their quantum mutual information adjacency matrices. Finally, in keeping with the strong physics education research focus at the Colorado School of Mines, we apply complex network analysis to a key issue germane to the student experience, namely student collaboration networks. We compute nodal centrality measures on the collaboration networks of students enrolled in three upper-division physics courses at the Colorado School of Mines. These are 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. We find that almost all of the measures considered correlate with analytical homework grades. In contrast only net out-strength correlates with exam grade. The benefits of collaboration do not extend from homework to exams, and students who help more than they are helped perform well on exams. Centrality measures between simultaneous collaboration networks (analytical vs. numerical homework collaboration) composed of the same students correlate with each other. Students take on similar roles in response to analytical vs. numerical homework assignments. Changes in collaboration across semesters are also considered as students transition from classical mechanics in the fall to quantum mechanics and electromagnetism in the spring. We find the most frequent transition is that students that help many others and have high grades will continue to help many others and have high grades. Students that help few more frequently transition from low grades to high grades than students that help many.