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Sokolov B, Rossi MAC, García-Pérez G, Maniscalco S. Emergent entanglement structures and self-similarity in quantum spin chains. PHILOSOPHICAL TRANSACTIONS. SERIES A, MATHEMATICAL, PHYSICAL, AND ENGINEERING SCIENCES 2022; 380:20200421. [PMID: 35599560 DOI: 10.1098/rsta.2020.0421] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/07/2023]
Abstract
We introduce an experimentally accessible network representation for many-body quantum states based on entanglement between all pairs of its constituents. We illustrate the power of this representation by applying it to a paradigmatic spin chain model, the XX model, and showing that it brings to light new phenomena. The analysis of these entanglement networks reveals that the gradual establishment of quasi-long range order is accompanied by a symmetry regarding single-spin concurrence distributions, as well as by instabilities in the network topology. Moreover, we identify the existence of emergent entanglement structures, spatially localized communities enforced by the global symmetry of the system that can be revealed by model-agnostic community detection algorithms. The network representation further unveils the existence of structural classes and a cyclic self-similarity in the state, which we conjecture to be intimately linked to the community structure. Our results demonstrate that the use of tools and concepts from complex network theory enables the discovery, understanding and description of new physical phenomena even in models studied for decades. This article is part of the theme issue 'Emergent phenomena in complex physical and socio-technical systems: from cells to societies'.
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Affiliation(s)
- Boris Sokolov
- QTF Centre of Excellence, Department of Physics, Faculty of Science, University of Helsinki, Helsinki, Finland
- Algorithmiq Ltd, Kanavakatu 3C, Helsinki 00160, Finland
- InstituteQ - the Finnish Quantum Institute, University of Helsinki, Finland
| | - Matteo A C Rossi
- Algorithmiq Ltd, Kanavakatu 3C, Helsinki 00160, Finland
- QTF Centre of Excellence, Center for Quantum Engineering, Department of Applied Physics, Aalto University School of Science, Aalto 00076, Finland
- InstituteQ - the Finnish Quantum Institute, Aalto University, Finland
| | - Guillermo García-Pérez
- QTF Centre of Excellence, Department of Physics, Faculty of Science, University of Helsinki, Helsinki, Finland
- Algorithmiq Ltd, Kanavakatu 3C, Helsinki 00160, Finland
- InstituteQ - the Finnish Quantum Institute, University of Helsinki, Finland
- Complex Systems Research Group, Department of Mathematics and Statistics, University of Turku, Turun Yliopisto 20014, Finland
| | - Sabrina Maniscalco
- QTF Centre of Excellence, Department of Physics, Faculty of Science, University of Helsinki, Helsinki, Finland
- Algorithmiq Ltd, Kanavakatu 3C, Helsinki 00160, Finland
- InstituteQ - the Finnish Quantum Institute, University of Helsinki, Finland
- QTF Centre of Excellence, Center for Quantum Engineering, Department of Applied Physics, Aalto University School of Science, Aalto 00076, Finland
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Wang Y, Xue S, Wu J, Xu P. Continuous-time quantum walk based centrality testing on weighted graphs. Sci Rep 2022; 12:6001. [PMID: 35397632 PMCID: PMC8994786 DOI: 10.1038/s41598-022-09915-1] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/28/2022] [Accepted: 03/28/2022] [Indexed: 11/09/2022] Open
Abstract
Centrality measure is an essential tool in network analysis and widely used in the domain of computer science, biology and sociology. Taking advantage of the speedup offered by quantum computation, various quantum centrality measures have been proposed. However, few work of quantum centrality involves weighted graphs, while the weight of edges should be considered in certain real-world networks. In this work, we extend the centrality measure based on continuous-time quantum walk to weighted graphs. We testify the feasibility and reliability of this quantum centrality using an ensemble of 41,675 graphs with various topologies and comparing with the eigenvector centrality measure. The average Vigna’s correlation index of all the tested graphs with all edge weights in [1, 10] is as high as 0.967, indicating the pretty good consistency of rankings by the continuous-time quantum walk centrality and the eigenvector centrality. The intuitive consistency of the top-ranked vertices given by this quantum centrality measure and classical centrality measures is also demonstrated on large-scale weighted graphs. Moreover, the range of the continuous-time quantum walk centrality values is much bigger than that of classical centralities, which exhibits better distinguishing ability to pick the important vertices from the ones with less importance. All these results show that the centrality measure based on continuous-time quantum walk still works well on weighted graphs.
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Affiliation(s)
- Yang Wang
- Institute for Quantum Information & State Key Laboratory of High Performance Computing, College of Computer Science and Technology, National University of Defense Technology, Changsha, 410073, China
| | - Shichuan Xue
- Institute for Quantum Information & State Key Laboratory of High Performance Computing, College of Computer Science and Technology, National University of Defense Technology, Changsha, 410073, China
| | - Junjie Wu
- Institute for Quantum Information & State Key Laboratory of High Performance Computing, College of Computer Science and Technology, National University of Defense Technology, Changsha, 410073, China.
| | - Ping Xu
- Institute for Quantum Information & State Key Laboratory of High Performance Computing, College of Computer Science and Technology, National University of Defense Technology, Changsha, 410073, China.
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Tang H, Shi R, He TS, Zhu YY, Wang TY, Lee M, Jin XM. TensorFlow solver for quantum PageRank in large-scale networks. Sci Bull (Beijing) 2021; 66:120-126. [PMID: 36654218 DOI: 10.1016/j.scib.2020.09.009] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/21/2020] [Revised: 07/24/2020] [Accepted: 08/31/2020] [Indexed: 01/20/2023]
Abstract
Google PageRank is a prevalent algorithm for ranking the significance of nodes or websites in a network, and a recent quantum counterpart for PageRank algorithm has been raised to suggest a higher accuracy of ranking comparing to Google PageRank. The quantum PageRank algorithm is essentially based on quantum stochastic walks and can be expressed using Lindblad master equation, which, however, needs to solve the Kronecker products of an O(N4) dimension and requires severely large memory and time when the number of nodes N in a network increases above 150. Here, we present an efficient solver for quantum PageRank by using the Runge-Kutta method to reduce the matrix dimension to O(N2) and employing TensorFlow to conduct GPU parallel computing. We demonstrate its performance in solving quantum stochastic walks on Erdös-Rényi graphs using an RTX 2060 GPU. The test on the graph of 6000 nodes requires a memory of 5.5 GB and time of 223 s, and that on the graph of 1000 nodes requires 226 MB and 3.6 s. Compared with QSWalk, a currently prevalent Mathematica solver, our solver for the same graph of 1000 nodes reduces the required memory and time to only 0.2% and 0.05%. We apply the solver to quantum PageRank for the USA major airline network with up to 922 nodes, and to quantum stochastic walk on a glued tree of 2186 nodes. This efficient solver for large-scale quantum PageRank and quantum stochastic walks would greatly facilitate studies of quantum information in real-life applications.
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Affiliation(s)
- Hao Tang
- Center for Integrated Quantum Information Technologies (IQIT), School of Physics and Astronomy and State Key Laboratory of Advanced Optical Communication Systems and Networks, Shanghai Jiao Tong University, Shanghai 200240, China; CAS Center for Excellence and Synergetic Innovation Center in Quantum Information and Quantum Physics, University of Science and Technology of China, Hefei 230026, China
| | - Ruoxi Shi
- Center for Integrated Quantum Information Technologies (IQIT), School of Physics and Astronomy and State Key Laboratory of Advanced Optical Communication Systems and Networks, Shanghai Jiao Tong University, Shanghai 200240, China
| | - Tian-Shen He
- Center for Integrated Quantum Information Technologies (IQIT), School of Physics and Astronomy and State Key Laboratory of Advanced Optical Communication Systems and Networks, Shanghai Jiao Tong University, Shanghai 200240, China
| | - Yan-Yan Zhu
- School of Physical Science, University of Chinese Academy of Sciences, Beijing 100049, China
| | - Tian-Yu Wang
- Center for Integrated Quantum Information Technologies (IQIT), School of Physics and Astronomy and State Key Laboratory of Advanced Optical Communication Systems and Networks, Shanghai Jiao Tong University, Shanghai 200240, China
| | - Marcus Lee
- Department of Physics, Cambridge University, Cambridge CB3 0HE, UK
| | - Xian-Min Jin
- Center for Integrated Quantum Information Technologies (IQIT), School of Physics and Astronomy and State Key Laboratory of Advanced Optical Communication Systems and Networks, Shanghai Jiao Tong University, Shanghai 200240, China; CAS Center for Excellence and Synergetic Innovation Center in Quantum Information and Quantum Physics, University of Science and Technology of China, Hefei 230026, China.
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5
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Wu T, Izaac JA, Li ZX, Wang K, Chen ZZ, Zhu S, Wang JB, Ma XS. Experimental Parity-Time Symmetric Quantum Walks for Centrality Ranking on Directed Graphs. PHYSICAL REVIEW LETTERS 2020; 125:240501. [PMID: 33412067 DOI: 10.1103/physrevlett.125.240501] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/17/2019] [Accepted: 10/28/2020] [Indexed: 06/12/2023]
Abstract
Using quantum walks (QWs) to rank the centrality of nodes in networks, represented by graphs, is advantageous compared to certain widely used classical algorithms. However, it is challenging to implement a directed graph via QW, since it corresponds to a non-Hermitian Hamiltonian and thus cannot be accomplished by conventional QW. Here we report the realizations of centrality rankings of a three-, a four-, and a nine-vertex directed graph with parity-time (PT) symmetric quantum walks by using high-dimensional photonic quantum states, multiple concatenated interferometers, and dimension dependent loss to achieve these. We demonstrate the advantage of the QW approach experimentally by breaking the vertex rank degeneracy in a four-vertex graph. Furthermore, we extend our experiment from single-photon to two-photon Fock states as inputs and realize the centrality ranking of a nine-vertex graph. Our work shows that a PT symmetric multiphoton quantum walk paves the way for realizing advanced algorithms.
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Affiliation(s)
- Tong Wu
- National Laboratory of Solid State Microstructures, School of Physics, Collaborative Innovation Center of Advanced Microstructures, Nanjing University, Nanjing 210093, China
| | - J A Izaac
- School of Physics, The University of Western Australia, Perth, Western Australia 6009, Australia
| | - Zi-Xi Li
- National Laboratory of Solid State Microstructures, School of Physics, Collaborative Innovation Center of Advanced Microstructures, Nanjing University, Nanjing 210093, China
| | - Kai Wang
- National Laboratory of Solid State Microstructures, School of Physics, Collaborative Innovation Center of Advanced Microstructures, Nanjing University, Nanjing 210093, China
| | - Zhao-Zhong Chen
- National Laboratory of Solid State Microstructures, School of Physics, Collaborative Innovation Center of Advanced Microstructures, Nanjing University, Nanjing 210093, China
| | - Shining Zhu
- National Laboratory of Solid State Microstructures, School of Physics, Collaborative Innovation Center of Advanced Microstructures, Nanjing University, Nanjing 210093, China
| | - J B Wang
- School of Physics, The University of Western Australia, Perth, Western Australia 6009, Australia
| | - Xiao-Song Ma
- National Laboratory of Solid State Microstructures, School of Physics, Collaborative Innovation Center of Advanced Microstructures, Nanjing University, Nanjing 210093, China
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Quantum walks and Dirac cellular automata on a programmable trapped-ion quantum computer. Nat Commun 2020; 11:3720. [PMID: 32709855 PMCID: PMC7381638 DOI: 10.1038/s41467-020-17519-4] [Citation(s) in RCA: 4] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/18/2020] [Accepted: 07/02/2020] [Indexed: 11/08/2022] Open
Abstract
The quantum walk formalism is a widely used and highly successful framework for modeling quantum systems, such as simulations of the Dirac equation, different dynamics in both the low and high energy regime, and for developing a wide range of quantum algorithms. Here we present the circuit-based implementation of a discrete-time quantum walk in position space on a five-qubit trapped-ion quantum processor. We encode the space of walker positions in particular multi-qubit states and program the system to operate with different quantum walk parameters, experimentally realizing a Dirac cellular automaton with tunable mass parameter. The quantum walk circuits and position state mapping scale favorably to a larger model and physical systems, allowing the implementation of any algorithm based on discrete-time quantum walks algorithm and the dynamics associated with the discretized version of the Dirac equation.
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Outeiral C, Strahm M, Shi J, Morris GM, Benjamin SC, Deane CM. The prospects of quantum computing in computational molecular biology. WILEY INTERDISCIPLINARY REVIEWS-COMPUTATIONAL MOLECULAR SCIENCE 2020. [DOI: 10.1002/wcms.1481] [Citation(s) in RCA: 39] [Impact Index Per Article: 9.8] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 11/11/2022]
Affiliation(s)
- Carlos Outeiral
- Department of Statistics University of Oxford Oxford UK
- Department of Materials University of Oxford Oxford UK
| | - Martin Strahm
- Pharma Research and Early Development F. Hoffmann‐La Roche Basel Switzerland
| | - Jiye Shi
- Computer‐Aided Drug Design UCB Pharma Slough UK
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8
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Xia F, Liu J, Nie H, Fu Y, Wan L, Kong X. Random Walks: A Review of Algorithms and Applications. IEEE TRANSACTIONS ON EMERGING TOPICS IN COMPUTATIONAL INTELLIGENCE 2020. [DOI: 10.1109/tetci.2019.2952908] [Citation(s) in RCA: 42] [Impact Index Per Article: 10.5] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/07/2022]
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9
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Liu Y, Yuan J, Duan B, Li D. Quantum walks on regular uniform hypergraphs. Sci Rep 2018; 8:9548. [PMID: 29934645 PMCID: PMC6015024 DOI: 10.1038/s41598-018-27825-z] [Citation(s) in RCA: 9] [Impact Index Per Article: 1.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/12/2017] [Accepted: 05/16/2018] [Indexed: 11/08/2022] Open
Abstract
Quantum walks on graphs have shown prioritized benefits and applications in wide areas. In some scenarios, however, it may be more natural and accurate to mandate high-order relationships for hypergraphs, due to the density of information stored inherently. Therefore, we can explore the potential of quantum walks on hypergraphs. In this paper, by presenting the one-to-one correspondence between regular uniform hypergraphs and bipartite graphs, we construct a model for quantum walks on bipartite graphs of regular uniform hypergraphs with Szegedy's quantum walks, which gives rise to a quadratic speed-up. Furthermore, we deliver spectral properties of the transition matrix, given that the cardinalities of the two disjoint sets are different in the bipartite graph. Our model provides the foundation for building quantum algorithms on the strength of quantum walks on hypergraphs, such as quantum walks search, quantized Google's PageRank, and quantum machine learning.
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Affiliation(s)
- Ying Liu
- Nanjing University of Aeronautics and Astronautics, College of Computer Science and Technology, Nanjing, 211106, China.
| | - Jiabin Yuan
- Nanjing University of Aeronautics and Astronautics, College of Computer Science and Technology, Nanjing, 211106, China.
| | - Bojia Duan
- Nanjing University of Aeronautics and Astronautics, College of Computer Science and Technology, Nanjing, 211106, China
| | - Dan Li
- Nanjing University of Aeronautics and Astronautics, College of Computer Science and Technology, Nanjing, 211106, China
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10
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Izaac JA, Wang JB. Systematic dimensionality reduction for continuous-time quantum walks of interacting fermions. Phys Rev E 2017; 96:032136. [PMID: 29346966 DOI: 10.1103/physreve.96.032136] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/27/2017] [Indexed: 06/07/2023]
Abstract
To extend the continuous-time quantum walk (CTQW) to simulate P distinguishable particles on a graph G composed of N vertices, the Hamiltonian of the system is expanded to act on an N^{P}-dimensional Hilbert space, in effect, simulating the multiparticle CTQW on graph G via a single-particle CTQW propagating on the Cartesian graph product G^{□P}. The properties of the Cartesian graph product have been well studied, and classical simulation of multiparticle CTQWs are common in the literature. However, the above approach is generally applied as is when simulating indistinguishable particles, with the particle statistics then applied to the propagated N^{P} state vector to determine walker probabilities. We address the following question: How can we modify the underlying graph structure G^{□P} in order to simulate multiple interacting fermionic CTQWs with a reduction in the size of the state space? In this paper, we present an algorithm for systematically removing "redundant" and forbidden quantum states from consideration, which provides a significant reduction in the effective dimension of the Hilbert space of the fermionic CTQW. As a result, as the number of interacting fermions in the system increases, the classical computational resources required no longer increases exponentially for fixed N.
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Affiliation(s)
- J A Izaac
- School of Physics, University of Western Australia, Crawley, Western Australia 6009, Australia
| | - J B Wang
- School of Physics, University of Western Australia, Crawley, Western Australia 6009, Australia
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11
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Chen T, Zhang X, Zhang X. Quantum sensing of noises in one and two dimensional quantum walks. Sci Rep 2017; 7:4962. [PMID: 28694443 PMCID: PMC5504040 DOI: 10.1038/s41598-017-04795-2] [Citation(s) in RCA: 8] [Impact Index Per Article: 1.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/16/2017] [Accepted: 05/19/2017] [Indexed: 11/22/2022] Open
Abstract
Quantum walk (QW) provides a versatile platform for the realization of quantum algorithms. Due to the existence of the inevitable noises in the walk, the different quantum algorithms accommodating to different noises are demanded. Thus, the success of the algorithms based on the QW requires us to sense different noises in the walk. Until now, the way to distinguish different noises in the walk has been discussed rarely. Here, we propose an efficient way to sense the noises in the one and two dimensional QWs. The populations of the coin in the walk with or without decoherence are presented. By only detecting the populations of the coin in the QW, we can determine whether there exists the decoherence in the total QW system. Moreover, the non-Markovianity of the coin in the one and two dimensional QWs is revealed, in which the coin is taken as an open quantum system, and the other components of the QW system is taken as the large environment. With the measured value of the non-Markovianity for the coin, we can conjecture which kinds of noise emerges in the one and two dimensional QWs.
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Affiliation(s)
- Tian Chen
- School of Physics, Beijing Institute of Technology, Beijing, 100081, China.
| | - Xiong Zhang
- School of Physics, Beijing Institute of Technology, Beijing, 100081, China
| | - Xiangdong Zhang
- School of Physics, Beijing Institute of Technology, Beijing, 100081, China
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12
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Abstract
Networks are widely used as a tool for describing diverse real complex systems and have been successfully applied to many fields. The distance between networks is one of the most fundamental concepts for properly classifying real networks, detecting temporal changes in network structures, and effectively predicting their temporal evolution. However, this distance has rarely been discussed in the theory of complex networks. Here, we propose a graph distance between networks based on a Laplacian matrix that reflects the structural and dynamical properties of networked dynamical systems. Our results indicate that the Laplacian-based graph distance effectively quantifies the structural difference between complex networks. We further show that our approach successfully elucidates the temporal properties underlying temporal networks observed in the context of face-to-face human interactions.
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13
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Chen T, Zhang X. The defect-induced localization in many positions of the quantum random walk. Sci Rep 2016; 6:25767. [PMID: 27216697 PMCID: PMC4877653 DOI: 10.1038/srep25767] [Citation(s) in RCA: 12] [Impact Index Per Article: 1.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/26/2016] [Accepted: 04/20/2016] [Indexed: 11/09/2022] Open
Abstract
We study the localization of probability distribution in a discrete quantum random walk on an infinite chain. With a phase defect introduced in any position of the quantum random walk (QRW), we have found that the localization of the probability distribution in the QRW emerges. Different localized behaviors of the probability distribution in the QRW are presented when the defect occupies different positions. Given that the coefficients of the localized stationary eigenstates relies on the coin operator, we reveal that when the defect occupies different positions, the amplitude of localized probability distribution in the QRW exhibits a non-trivial dependence on the coin operator.
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Affiliation(s)
- Tian Chen
- School of Physics, Beijing Institute of Technology, 100081, Beijing, China
| | - Xiangdong Zhang
- School of Physics, Beijing Institute of Technology, 100081, Beijing, China
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14
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Ma J, Shi Y, Wang Z, Yue J. On Wiener polarity index of bicyclic networks. Sci Rep 2016; 6:19066. [PMID: 26750820 PMCID: PMC4707490 DOI: 10.1038/srep19066] [Citation(s) in RCA: 63] [Impact Index Per Article: 7.9] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/16/2015] [Accepted: 12/02/2015] [Indexed: 11/09/2022] Open
Abstract
Complex networks are ubiquitous in biological, physical and social sciences. Network robustness research aims at finding a measure to quantify network robustness. A number of Wiener type indices have recently been incorporated as distance-based descriptors of complex networks. Wiener type indices are known to depend both on the network's number of nodes and topology. The Wiener polarity index is also related to the cluster coefficient of networks. In this paper, based on some graph transformations, we determine the sharp upper bound of the Wiener polarity index among all bicyclic networks. These bounds help to understand the underlying quantitative graph measures in depth.
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Affiliation(s)
- Jing Ma
- Center for Combinatorics and LPMC-TJKLC, Nankai University, Tianjin, 300071, China
| | - Yongtang Shi
- Center for Combinatorics and LPMC-TJKLC, Nankai University, Tianjin, 300071, China
| | - Zhen Wang
- Interdisciplinary Graduate School of Engineering Sciences, Kyushu University, Kasuga-koen, Kasugashi, Fukuoka, Japan
| | - Jun Yue
- School of Mathematical Sciences, Shandong Normal University, Jinan 250014, Shandong, China
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15
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Friis N, Melnikov AA, Kirchmair G, Briegel HJ. Coherent controlization using superconducting qubits. Sci Rep 2015; 5:18036. [PMID: 26667893 PMCID: PMC4678369 DOI: 10.1038/srep18036] [Citation(s) in RCA: 24] [Impact Index Per Article: 2.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/17/2015] [Accepted: 11/10/2015] [Indexed: 11/08/2022] Open
Abstract
Coherent controlization, i.e., coherent conditioning of arbitrary single- or multi-qubit operations on the state of one or more control qubits, is an important ingredient for the flexible implementation of many algorithms in quantum computation. This is of particular significance when certain subroutines are changing over time or when they are frequently modified, such as in decision-making algorithms for learning agents. We propose a scheme to realize coherent controlization for any number of superconducting qubits coupled to a microwave resonator. For two and three qubits, we present an explicit construction that is of high relevance for quantum learning agents. We demonstrate the feasibility of our proposal, taking into account loss, dephasing, and the cavity self-Kerr effect.
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Affiliation(s)
- Nicolai Friis
- Institute for Theoretical Physics, University of Innsbruck, Technikerstraße 21a, A-6020 Innsbruck, Austria
| | - Alexey A. Melnikov
- Institute for Theoretical Physics, University of Innsbruck, Technikerstraße 21a, A-6020 Innsbruck, Austria
- Institute for Quantum Optics and Quantum Information, Austrian Academy of Sciences, Technikerstraße 21a, A-6020 Innsbruck, Austria
| | - Gerhard Kirchmair
- Institute for Quantum Optics and Quantum Information, Austrian Academy of Sciences, Technikerstraße 21a, A-6020 Innsbruck, Austria
- Institute for Experimental Physics, University of Innsbruck, Technikerstraße 25, A-6020 Innsbruck, Austria
| | - Hans J. Briegel
- Institute for Theoretical Physics, University of Innsbruck, Technikerstraße 21a, A-6020 Innsbruck, Austria
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16
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Riascos AP, Mateos JL. Fractional quantum mechanics on networks: Long-range dynamics and quantum transport. Phys Rev E 2015; 92:052814. [PMID: 26651751 DOI: 10.1103/physreve.92.052814] [Citation(s) in RCA: 11] [Impact Index Per Article: 1.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/02/2015] [Indexed: 11/07/2022]
Abstract
In this paper we study the quantum transport on networks with a temporal evolution governed by the fractional Schrödinger equation. We generalize the dynamics based on continuous-time quantum walks, with transitions to nearest neighbors on the network, to the fractional case that allows long-range displacements. By using the fractional Laplacian matrix of a network, we establish a formalism that combines a long-range dynamics with the quantum superposition of states; this general approach applies to any type of connected undirected networks, including regular, random, and complex networks, and can be implemented from the spectral properties of the Laplacian matrix. We study the fractional dynamics and its capacity to explore the network by means of the transition probability, the average probability of return, and global quantities that characterize the efficiency of this quantum process. As a particular case, we explore analytically these quantities for circulant networks such as rings, interacting cycles, and complete graphs.
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Affiliation(s)
- A P Riascos
- Instituto de Física, Universidad Nacional Autónoma de México, Apartado Postal 20-364, 01000 México, D.F., México
| | - José L Mateos
- Instituto de Física, Universidad Nacional Autónoma de México, Apartado Postal 20-364, 01000 México, D.F., México
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17
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Luo MX, Xu G, Chen XB, Yang YX, Wang X. Efficient quantum transmission in multiple-source networks. Sci Rep 2014; 4:4571. [PMID: 24691590 PMCID: PMC3972689 DOI: 10.1038/srep04571] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/22/2013] [Accepted: 03/18/2014] [Indexed: 11/23/2022] Open
Abstract
A difficult problem in quantum network communications is how to efficiently transmit quantum information over large-scale networks with common channels. We propose a solution by developing a quantum encoding approach. Different quantum states are encoded into a coherent superposition state using quantum linear optics. The transmission congestion in the common channel may be avoided by transmitting the superposition state. For further decoding and continued transmission, special phase transformations are applied to incoming quantum states using phase shifters such that decoders can distinguish outgoing quantum states. These phase shifters may be precisely controlled using classical chaos synchronization via additional classical channels. Based on this design and the reduction of multiple-source network under the assumption of restricted maximum-flow, the optimal scheme is proposed for specially quantized multiple-source network. In comparison with previous schemes, our scheme can greatly increase the transmission efficiency.
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Affiliation(s)
- Ming-Xing Luo
- 1] Information Security and National Computing Grid Laboratory, Southwest Jiaotong University, Chengdu 610031, China [2] State Key Laboratory of Networking and Switching Technology, Beijing University of Posts and Telecommunications, Beijing 100876, China [3] State Key Laboratory of Information Security (Graduate University of Chinese Academy of Sciences), Beijing 100049, China
| | - Gang Xu
- 1] State Key Laboratory of Networking and Switching Technology, Beijing University of Posts and Telecommunications, Beijing 100876, China [2] School of Software Engineering, Beijing University of Posts and Telecommunications, Beijing 100876, China
| | - Xiu-Bo Chen
- State Key Laboratory of Networking and Switching Technology, Beijing University of Posts and Telecommunications, Beijing 100876, China
| | - Yi-Xian Yang
- State Key Laboratory of Networking and Switching Technology, Beijing University of Posts and Telecommunications, Beijing 100876, China
| | - Xiaojun Wang
- School of Electronic Engineering, Dublin City University, Dublin 9, Ireland
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18
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Abstract
We investigate the behaviour of the recently proposed Quantum PageRank algorithm, in large complex networks. We find that the algorithm is able to univocally reveal the underlying topology of the network and to identify and order the most relevant nodes. Furthermore, it is capable to clearly highlight the structure of secondary hubs and to resolve the degeneracy in importance of the low lying part of the list of rankings. The quantum algorithm displays an increased stability with respect to a variation of the damping parameter, present in the Google algorithm, and a more clearly pronounced power-law behaviour in the distribution of importance, as compared to the classical algorithm. We test the performance and confirm the listed features by applying it to real world examples from the WWW. Finally, we raise and partially address whether the increased sensitivity of the quantum algorithm persists under coordinated attacks in scale-free and random networks.
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19
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Renormalization and small-world model of fractal quantum repeater networks. Sci Rep 2013; 3:1222. [PMID: 23386977 PMCID: PMC3564040 DOI: 10.1038/srep01222] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/20/2012] [Accepted: 01/15/2013] [Indexed: 11/25/2022] Open
Abstract
Quantum networks provide access to exchange of quantum information. The primary task of quantum networks is to distribute entanglement between remote nodes. Although quantum repeater protocol enables long distance entanglement distribution, it has been restricted to one-dimensional linear network. Here we develop a general framework that allows application of quantum repeater protocol to arbitrary quantum repeater networks with fractal structure. Entanglement distribution across such networks is mapped to renormalization. Furthermore, we demonstrate that logarithmical times of recursive such renormalization transformations can trigger fractal to small-world transition, where a scalable quantum small-world network is achieved. Our result provides new insight into quantum repeater theory towards realistic construction of large-scale quantum networks.
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Sánchez-Burillo E, Duch J, Gómez-Gardeñes J, Zueco D. Quantum navigation and ranking in complex networks. Sci Rep 2012; 2:605. [PMID: 22930671 PMCID: PMC3428603 DOI: 10.1038/srep00605] [Citation(s) in RCA: 69] [Impact Index Per Article: 5.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/04/2012] [Accepted: 08/07/2012] [Indexed: 11/09/2022] Open
Abstract
Complex networks are formal frameworks capturing the interdependencies between the elements of large systems and databases. This formalism allows to use network navigation methods to rank the importance that each constituent has on the global organization of the system. A key example is Pagerank navigation which is at the core of the most used search engine of the World Wide Web. Inspired in this classical algorithm, we define a quantum navigation method providing a unique ranking of the elements of a network. We analyze the convergence of quantum navigation to the stationary rank of networks and show that quantumness decreases the number of navigation steps before convergence. In addition, we show that quantum navigation allows to solve degeneracies found in classical ranks. By implementing the quantum algorithm in real networks, we confirm these improvements and show that quantum coherence unveils new hierarchical features about the global organization of complex systems.
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Affiliation(s)
- Eduardo Sánchez-Burillo
- Instituto de Ciencia de Materiales de Aragón (ICMA), CSIC-Universidad de Zaragoza, E-50012 Zaragoza, Spain
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