Quantifying Complexity in Quantum Phase Transitions via Mutual Information Complex Networks

Lincoln D. Carr

Dept. of Physics, Colorado School of Mines in collaboration with Marc Valdez, Daniel Jaschke, and David Vargas (CSM) Bhuvanesh Sundar and Kaden Hazzard (Rice) Complex networks occur between usual quantum physics regimes

Clustered Uniform = Random = Lattices, Hard Nuclear Condensed Dense Disparate Theory, Matter Physics, Quantum Quantum Chaos information theory Disparate Clustered Social Dense Food Web Metabolic Network Network Where do complex networks appear in quantum systems? Near-term quantum devices Digital quantum computer • Long-range entangling gates key factor in quantum volume for quantum advantage Quantum internet • Distributed entanglement on classical internet backbone? Present quantum devices Adiabatic quantum computer: D-Wave • Chimera map ( 6) Analog quantum computer: Quantum simulators • At least 7 different platforms! Ø(see talk of Valentina Parigi) Complex networks in quantum simulators Whole Hamiltonian Quantum random walk on Encode complex network directly into the links Part of Hamiltonian E.g. single particle terms vs. interactions Environment in open quantum systems Arising spontaneously in the state Quantum critical points in quantum phase transitions Quantum cellular automata No complex network required in Hamiltonian! Quantum Phase Transitions: Quantum Matter in a Crystal of Light Density image Optical Lattices Superfluid after expansion

Mott Insulator

DnDf ³1 But what are quantum phase transitions?

Non-analyticity in correlator in thermodynamic limit Driven by quantum not thermal fluctuations In practice: mesoscopic systems ok Huge field Condensed matter, AMO, Nuclear, QCD, String theory, from quantum states (not operators, not “quantum networks” for a quantum internet) Density matrix � = |�⟩⟨�|

� = Tr � is a mixed state Describes quantum state of subsystem j Quantum entropy (von Neumann)

� = −Tr(� log�) Quantifies entanglement, non-additive Quantum mutual information = adjacency matrix � = (� + � − � ) Two point quantum measure = nonlocality Bounded from below by all two-point correlators • Captures most essential feature of quantum phase transitions Quantum Mutual Information Complex Network

� forms an undirected, weighted adjacency matrix which does not permit self connection

= , 0 ≤ ≤ 1, = 0

Network measures quantify the structure of connections

Network density: : Disparity:

1 Tr( ) 1 ∑ � = � = � = �(� − 1) ∑ ∑[ ] � ( ∑ ) , density transitivity uniformity of con- of of con- nections connections nections Many-Body Models and Methods Fruit Fly: Quantum Ising Model chain – quantum information Trapped ions, Rydberg gates, ultracold molecules

Zebra fish: Bose-Hubbard Model Discretized Bose Gas Cold atoms in optical lattices

Matrix Product States (https://sourceforge.net/projects/openmps/) Exact diagonalization, Jordan-Wigner transform Ising Symmetry and Phases Bose-Hubbard Symmetry and Phases Complex network measures identify critical phenomena

Left: Transverse quantum Ising Z2 Right: Bose-Hubbard BKT crossover D = density, C = clustering, Y = disparity, R = Pearson’s correlation Finite-size scaling studies for Bose- Hubbard and quantum Ising models

System Size L

(a) Bose-Hubbard phase diagram (b) BKT transition at tip of Mott lobe (c) Mean field superfluid/Mott transition (d) Ising Complex network measures Finite-size scaling convergence shows new features near BKT critical point ⁄ Power law scaling of critical points � � = � + � � in system size L: ⁄ �⁄� � = �⁄� + � �

All the same Highly Variable Temperature dependence via Jordan- Wigner transform

� adjacency matrix � # geodesic paths from j to k via I � geodesic What is Quantum Mutual Information Really Measuring? Bounded from below by all possible 2- point correlators Makes no assumptions on operators measured Sketch of Results not Shown…

Other Adjacency Matrices Quantum Renyi Entropies Negativity Concurrence any 2-index object will do…

Many (Most? All?) Hamiltonians display complexity for Goldilocks Quantum Elementary Cellular Automata Far-from-equilibrium dynamics Two-slide teaser… Complex On quantum Networks Conclusions states Introduced Mutual Information Quantum Complexity Measures Quantum Critical Points Most complex in critical region Converges faster to critical point in finite size scaling More Information in Quantum Critical Fan Future directions Quantum Elementary Cellular Automata (QECA) • A whole new class of highly entangled and highly structured states! Open system quantum games of life Quantum complex network robustness Papers Valdez, Jaschke, Vargas, and Carr, PRL 119, 225301 (2017) Sundar, Valdez, Carr, and Hazzard, PRA 97, 052320 (2018) “Entangled Quantum Cellular Automata, Physical Complexity, and Goldilocks Rules,” to be submitted (2019) Open Questions

What is the interplay between Hamiltonian complex networks and those arising spontaneously in the state? How much complexity in the model is required to produce complexity in the state? We know that classical complex networks display hyperbolic scaling under renormalization group. Might quantum complex networks show similar scaling? What is the relationship between lattice RG for quantum phase transitions and complex network RG? How can we mathematically generalize complex networks for quantum systems? Need notion of coherences in links. Are quantum complex networks more or less robust than classical ones? Notion of weak vs. strong measurement as “attack.” The End The End Quantum Macroscopicity

Nimmrichter and Hornberger PRL 2013

Hyunseok Jeong et al. Opt. Comm. 2015 Classical mutual information: structural and functional brain networks Complexity of usual Quantum Information States Min/Max Analytically derive averaged network quantities for GHZ, W, localized singlet, and singlet array states Bond Entropy, Negativity, Correlation Length, and Depletion Renyi Entropies Concurrence network Entropy, Negativity, Depletion, Correlation Length Apples to Apples Comparison of Compute Times All computed within MPS Correlations have to be summed to get quantum mutual information, puts back in factor of 10 (or more!) Quantum mutual information independent of knowledge of system Power Laws in Finite-Size Scaling Ising Quantum Mutual Information Trends in Temperature and Coupling