Introducing complex networks into quantum regime 魏宗文 导师：汪秉宏 教授 中国科学技术大学 近代物理系 wbravo@mail.ustc.edu.cn

outline Brief introduction of the joint study of quantum information and complex networks Entanglement-based percolation Classical and quantum entanglement percolation Limited-path-length entanglement percolation Quantum random networks Small-world model of quantum repeater networks Summary and outlook 2019/4/13

Joint study of complex networks and quantum information Growth of graph states in quantum networks. PRA (2012) Bipartite quantum states and random complex networks. New J. Phys. (2012). Encoding graphs into quantum states: an axiomatic approach. PRA (2012). Synchronization, quantum correlations and entanglement in oscillator networks. Sci. Rep. (2013) Adiabatic quantum algorithm for search engine ranking. PRL (2012). Google in a quantum network. Sci. Rep. (2012) Quantum navigation and ranking in complex networks. Sci. Rep. (2012). featured progress 2019/4/13

Quantum networks provide access to exchange of quantum information. Entanglement percolation in quantum networks. Nature physics. (2007) Entanglement Percolation in Quantum Complex Networks. PRL (2009) Quantum random networks. Nature physics. (2010) Limited-path-length entanglement percolation in quantum complex networks. PRA (2011) Renormalization and small-world model of fractal quantum repeater networks. Sci. Rep. (2013) Quantum networks provide access to exchange of quantum information. The primary task of quantum networks is to distribute entanglement between remote nodes. Complex networks provide conceptual and mathematical preparation for design of quantum networks. 2019/4/13

Entanglement—assisted quantum communication Quantum Teleportation Teleportation Quantum cryptograph (Ekert 91) Dense coding Entangled link Entanglement Swapping 2019/4/13

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outline Brief introduction of the joint study of quantum information and complex networks Entanglement-based percolation Classical and quantum entanglement percolation Limited-path-length entanglement percolation Quantum random networks Small-world model of quantum repeater networks Summary and outlook 2019/4/13

From bond percolation to entanglement percolation Classical entanglement percolation Quantum entanglement percolation Singlet conversion probability 2019/4/13

Trick of QEP: change topology and decrease threshold Q-Swap: Trick of QEP: change topology and decrease threshold 2019/4/13

Advantage of scale-free networks Does QEP necessarily outperform CEP strategy? How do you like entanglement percolation ? Is entanglement percolation a new strategy ? Topology is essential for entanglement distribution. A trivial strategy Advantage of scale-free networks 2019/4/13

Significance of small-world for quantum networks 2019/4/13

outline Brief introduction of the joint study of quantum information and complex networks Entanglement-based percolation Classical and quantum entanglement percolation Limited-path-length entanglement percolation Quantum random networks Small-world model of quantum repeater networks Summary and outlook 2019/4/13

Beyond classical random networks model 2019/4/13

Threshold at which the giant connected cluster emerges? A surprising result that goes beyond classical model Collapse of critical exponents ～ Quantum: Classical: Threshold at which the giant connected cluster emerges? 2019/4/13

outline Brief introduction of the joint study of quantum information and complex networks Entanglement-based percolation Classical and quantum entanglement percolation Limited-path-length entanglement percolation Quantum random networks Small-world model of quantum repeater networks Summary and outlook 2019/4/13

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Possible topology of quantum repeater networks Principle of quantum repeater protocols Possible topology of quantum repeater networks Quantum repeater networks are fractal. Small-world effect could reinforce scalability. Scale-free networks are robust to channel noise. Generalize 1D quantum repeaters to high dimension: Coupled renormalization 2019/4/13

Box-covering technique: maximum-excluded-mass-burning algorithm Brief introduction to renormalization Nature, 433, 392 (2005). Box-covering technique: maximum-excluded-mass-burning algorithm 2019/4/13

Coupled Renormalization and high dimensional quantum repeaters 2019/4/13

Corresponding relationships Renormalization and its relationship with quantum repeaters Corresponding relationships 1D chain configuration Scale-free fractal network Length N Segmentation: grouped into units Box-covering : divided into boxes   2019/4/13

Application to a scale-free fractal network Nature Phys. 2, 275 (2006). Minimal Model Parameters: n, s, a,e 2019/4/13

Proofs(1)—renormalization flow Is there a critical nesting level n 2019/4/13

Proofs(2) definition Hierarchical routing method 2019/4/13

Criterion and direct evidence Recursive Derivation Critical Condition Critical Nesting Level Diameter Relevant comment on the simultaneous logarithmical behavior 2019/4/13

Statistical properties of coupled networks 2019/4/13

summary outlook Percolation phenomena in quantum networks. Quantum random networks. High fractal dimensional quantum repeaters: coupled renormalization. The first scalable small-world model of quantum networks. outlook Introducing complex networks into quantum regime and search for quantum effect . Does it make a difference? Is it complete? Design of quantum internet. Apart from quantum networks, there are a wide range of topics deserving pursuits for the joint study. 2019/4/13

Thanks for your attention! 2019/4/13