{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,17]],"date-time":"2026-02-17T11:31:08Z","timestamp":1771327868473,"version":"3.50.1"},"reference-count":49,"publisher":"Springer Science and Business Media LLC","issue":"1","license":[{"start":{"date-parts":[[2022,4,28]],"date-time":"2022-04-28T00:00:00Z","timestamp":1651104000000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2022,4,28]],"date-time":"2022-04-28T00:00:00Z","timestamp":1651104000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Commun Phys"],"abstract":"<jats:title>Abstract<\/jats:title><jats:p>Quantum networks allow us to harness networked quantum technologies and to develop a quantum internet. But how robust is a quantum network when its links and nodes start failing? We show that quantum complex networks based on typical noisy quantum-repeater nodes are prone to discontinuous phase transitions with respect to the random loss of operating links and nodes, abruptly compromising the connectivity of the network, and thus significantly limiting the reach of its operation. Furthermore, we determine the critical quantum-repeater efficiency necessary to avoid this catastrophic loss of connectivity as a function of the network topology, the network size, and the distribution of entanglement in the network. From all the network topologies tested, a scale-free network topology shows the best promise for a robust large-scale quantum internet.<\/jats:p>","DOI":"10.1038\/s42005-022-00866-7","type":"journal-article","created":{"date-parts":[[2022,4,28]],"date-time":"2022-04-28T10:04:20Z","timestamp":1651140260000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":29,"title":["Robustness of noisy quantum networks"],"prefix":"10.1038","volume":"5","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-9980-1857","authenticated-orcid":false,"given":"Bruno Coelho","family":"Coutinho","sequence":"first","affiliation":[]},{"given":"William John","family":"Munro","sequence":"additional","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0003-3938-0808","authenticated-orcid":false,"given":"Kae","family":"Nemoto","sequence":"additional","affiliation":[]},{"given":"Yasser","family":"Omar","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2022,4,28]]},"reference":[{"key":"866_CR1","doi-asserted-by":"publisher","first-page":"1023","DOI":"10.1038\/nature07127","volume":"453","author":"HJ Kimble","year":"2008","unstructured":"Kimble, H. J. The quantum internet. Nature 453, 1023 (2008).","journal-title":"Nature"},{"key":"866_CR2","doi-asserted-by":"crossref","unstructured":"Caleffi, M., Cacciapuoti, A. S. & Bianchi, G. Quantum internet: from communication to distributed computing! in Proc. 5th ACM International Conference on Nanoscale Computing and Communication, NANOCOM \u201918 (Association for Computing Machinery, 2018).","DOI":"10.1145\/3233188.3233224"},{"key":"866_CR3","doi-asserted-by":"crossref","unstructured":"Van Meter, R. Quantum Networking (Wiley, 2014).","DOI":"10.1002\/9781118648919"},{"key":"866_CR4","doi-asserted-by":"crossref","unstructured":"Perseguers, S., Lewenstein, M., Ac\u00edn, A. & Cirac, J. I. Quantum random networks. Nat. Phys. https:\/\/www.nature.com\/articles\/nphys1665 (2010).","DOI":"10.1038\/nphys1665"},{"key":"866_CR5","doi-asserted-by":"publisher","first-page":"250502","DOI":"10.1103\/PhysRevLett.118.250502","volume":"118","author":"IV Inlek","year":"2017","unstructured":"Inlek, I. V., Crocker, C., Lichtman, M., Sosnova, K. & Monroe, C. Multispecies trapped-ion node for quantum networking. Phys. Rev. Lett. 118, 250502 (2017).","journal-title":"Phys. Rev. Lett."},{"key":"866_CR6","doi-asserted-by":"publisher","first-page":"5932","DOI":"10.1103\/PhysRevLett.81.5932","volume":"81","author":"H-J Briegel","year":"1998","unstructured":"Briegel, H.-J., D\u00fcr, W., Cirac, J. I. & Zoller, P. Quantum repeaters: the role of imperfect local operations in quantum communication. Phys. Rev. Lett. 81, 5932\u20135935 (1998).","journal-title":"Phys. Rev. Lett."},{"key":"866_CR7","doi-asserted-by":"publisher","first-page":"259","DOI":"10.1038\/nphys2930","volume":"10","author":"\u010c Brukner","year":"2014","unstructured":"Brukner, \u010c. Quantum causality. Nat. Phys. 10, 259\u2013263 (2014).","journal-title":"Nat. Phys."},{"key":"866_CR8","doi-asserted-by":"publisher","first-page":"210501","DOI":"10.1103\/PhysRevLett.124.210501","volume":"124","author":"S Brito","year":"2020","unstructured":"Brito, S., Canabarro, A., Chaves, R. & Cavalcanti, D. Statistical properties of the quantum internet. Phys. Rev. Lett. 124, 210501 (2020).","journal-title":"Phys. Rev. Lett."},{"key":"866_CR9","doi-asserted-by":"publisher","first-page":"51","DOI":"10.1038\/s42005-019-0147-3","volume":"2","author":"S Pirandola","year":"2019","unstructured":"Pirandola, S. End-to-end capacities of a quantum communication network. Commun. Phys. 2, 51 (2019).","journal-title":"Commun. Phys."},{"key":"866_CR10","doi-asserted-by":"publisher","first-page":"22","DOI":"10.1049\/iet-qtc.2020.0006","volume":"1","author":"S Pirandola","year":"2020","unstructured":"Pirandola, S. General upper bound for conferencing keys in arbitrary quantum networks. IET Quantum Commun. 1, 22\u201325 (2020).","journal-title":"IET Quantum Commun."},{"key":"866_CR11","doi-asserted-by":"publisher","first-page":"052315","DOI":"10.1103\/PhysRevA.101.052315","volume":"101","author":"A-K Wu","year":"2020","unstructured":"Wu, A.-K., Tian, L., Coutinho, B. C., Omar, Y. & Liu, Y.-Y. Structural vulnerability of quantum networks. Phys. Rev. A 101, 052315 (2020).","journal-title":"Phys. Rev. A"},{"key":"866_CR12","doi-asserted-by":"publisher","first-page":"022308","DOI":"10.1103\/PhysRevA.77.022308","volume":"77","author":"S Perseguers","year":"2008","unstructured":"Perseguers, S., Cirac, J. I., Ac\u00edn, A., Lewenstein, M. & Wehr, J. Entanglement distribution in pure-state quantum networks. Phys. Rev. A 77, 022308 (2008).","journal-title":"Phys. Rev. A"},{"key":"866_CR13","doi-asserted-by":"publisher","first-page":"240503","DOI":"10.1103\/PhysRevLett.103.240503","volume":"103","author":"M Cuquet","year":"2009","unstructured":"Cuquet, M. & Calsamiglia, J. Entanglement percolation in quantum complex networks. Phys. Rev. Lett. 103, 240503 (2009).","journal-title":"Phys. Rev. Lett."},{"key":"866_CR14","doi-asserted-by":"publisher","first-page":"010304","DOI":"10.1103\/PRXQuantum.2.010304","volume":"2","author":"S Brito","year":"2021","unstructured":"Brito, S., Canabarro, A., Cavalcanti, D. & Chaves, R. Satellite-based photonic quantum networks are small-world. PRX Quantum 2, 010304 (2021).","journal-title":"PRX Quantum"},{"key":"866_CR15","doi-asserted-by":"publisher","first-page":"45007","DOI":"10.1088\/2058-9565\/ac1041","volume":"6","author":"B Zhang","year":"2021","unstructured":"Zhang, B. & Zhuang, Q. Quantum internet under random breakdowns and intentional attacks. Quantum Sci. Technol. 6, 45007 (2021).","journal-title":"Quantum Sci. Technol"},{"key":"866_CR16","doi-asserted-by":"publisher","first-page":"022608","DOI":"10.1103\/PhysRevA.104.022608","volume":"104","author":"Q Zhuang","year":"2021","unstructured":"Zhuang, Q. & Zhang, B. Quantum communication capacity transition of complex quantum networks. Phys. Rev. A 104, 022608 (2021).","journal-title":"Phys. Rev. A"},{"key":"866_CR17","doi-asserted-by":"publisher","first-page":"165","DOI":"10.1038\/nphoton.2007.22","volume":"1","author":"N Gisin","year":"2007","unstructured":"Gisin, N. & Thew, R. Quantum communication. Nat. Photonics 1, 165\u2013171 (2007).","journal-title":"Nat. Photonics"},{"key":"866_CR18","doi-asserted-by":"publisher","first-page":"145","DOI":"10.1103\/RevModPhys.74.145","volume":"74","author":"N Gisin","year":"2002","unstructured":"Gisin, N., Ribordy, G., Tittel, W. & Zbinden, H. Quantum cryptography. Rev. Mod. Phys. 74, 145\u2013195 (2002).","journal-title":"Rev. Mod. Phys."},{"key":"866_CR19","doi-asserted-by":"publisher","first-page":"247","DOI":"10.1038\/35005001","volume":"404","author":"CH Bennett","year":"2000","unstructured":"Bennett, C. H. & DiVincenzo, D. P. Quantum information and computation. Nature 404, 247\u2013255 (2000).","journal-title":"Nature"},{"key":"866_CR20","doi-asserted-by":"publisher","first-page":"505","DOI":"10.1038\/s41586-019-1666-5","volume":"574","author":"F Arute","year":"2019","unstructured":"Arute, F. et al. Quantum supremacy using a programmable superconducting processor. Nature 574, 505\u2013510 (2019).","journal-title":"Nature"},{"key":"866_CR21","doi-asserted-by":"publisher","first-page":"1460","DOI":"10.1126\/science.abe8770","volume":"370","author":"H-S Zhong","year":"2020","unstructured":"Zhong, H.-S. et al. Quantum computational advantage using photons. Science 370, 1460\u20131463 (2020).","journal-title":"Science"},{"key":"866_CR22","doi-asserted-by":"publisher","first-page":"035002","DOI":"10.1103\/RevModPhys.89.035002","volume":"89","author":"CL Degen","year":"2017","unstructured":"Degen, C. L., Reinhard, F. & Cappellaro, P. Quantum sensing. Rev. Mod. Phys. 89, 035002 (2017).","journal-title":"Rev. Mod. Phys."},{"key":"866_CR23","doi-asserted-by":"publisher","first-page":"1817","DOI":"10.1103\/PhysRevD.26.1817","volume":"26","author":"CM Caves","year":"1982","unstructured":"Caves, C. M. Quantum limits on noise in linear amplifiers. Phys. Rev. D. 26, 1817\u20131839 (1982).","journal-title":"Phys. Rev. D."},{"key":"866_CR24","doi-asserted-by":"publisher","first-page":"1330","DOI":"10.1126\/science.1104149","volume":"306","author":"V Giovannetti","year":"2004","unstructured":"Giovannetti, V., Lloyd, S. & Maccone, L. Quantum-enhanced measurements: beating the standard quantum limit. Science 306, 1330\u20131336 (2004).","journal-title":"Science"},{"key":"866_CR25","doi-asserted-by":"crossref","unstructured":"Newman, M. Networks\u2014An Introduction (Oxford University Press, 2010).","DOI":"10.1093\/acprof:oso\/9780199206650.003.0001"},{"key":"866_CR26","unstructured":"Dorogovtsev, S. N. & Mendes, J. F. Evolution of networks: From biological nets to the Internet and WWW (OUP Oxford, 2013)."},{"key":"866_CR27","unstructured":"Barab\u00e1si, A.-L. et al. Network Science (Cambridge university press, 2016)."},{"key":"866_CR28","doi-asserted-by":"publisher","first-page":"4626","DOI":"10.1103\/PhysRevLett.85.4626","volume":"85","author":"R Cohen","year":"2000","unstructured":"Cohen, R., Erez, K., ben Avraham, D. & Havlin, S. Resilience of the internet to random breakdowns. Phys. Rev. Lett. 85, 4626\u20134628 (2000).","journal-title":"Phys. Rev. Lett."},{"key":"866_CR29","doi-asserted-by":"publisher","first-page":"115","DOI":"10.1109\/TCSII.2017.2705758","volume":"65","author":"Z Chen","year":"2017","unstructured":"Chen, Z., Wu, J., Xia, Y. & Zhang, X. Robustness of interdependent power grids and communication networks: a complex network perspective. IEEE Trans. Circuits Syst. II Express Briefs 65, 115\u2013119 (2017).","journal-title":"IEEE Trans. Circuits Syst. II Express Briefs"},{"key":"866_CR30","doi-asserted-by":"crossref","unstructured":"Mezard, M. & Montanari, A. Information, Physics, and Computation (Oxford University Press, 2009).","DOI":"10.1093\/acprof:oso\/9780198570837.001.0001"},{"key":"866_CR31","unstructured":"Newman, M. E. J. Symetrized snapshot of the structure of autonamous systems. http:\/\/www-personal.umich.edu\/m\u0303ejn\/netdata\/as-22july06.zip. (2006)."},{"key":"866_CR32","doi-asserted-by":"publisher","first-page":"722","DOI":"10.1103\/PhysRevLett.76.722","volume":"76","author":"CH Bennett","year":"1996","unstructured":"Bennett, C. H. et al. Purification of noisy entanglement and faithful teleportation via noisy channels. Phys. Rev. Lett. 76, 722\u2013725 (1996).","journal-title":"Phys. Rev. Lett."},{"key":"866_CR33","doi-asserted-by":"publisher","first-page":"4287","DOI":"10.1103\/PhysRevLett.71.4287","volume":"71","author":"M \u017bukowski","year":"1993","unstructured":"\u017bukowski, M., Zeilinger, A., Horne, M. A. & Ekert, A. K. \u201cevent-ready-detectors\u201d bell experiment via entanglement swapping. Phys. Rev. Lett. 71, 4287\u20134290 (1993).","journal-title":"Phys. Rev. Lett."},{"key":"866_CR34","doi-asserted-by":"publisher","first-page":"78","DOI":"10.1109\/JSTQE.2015.2392076","volume":"21","author":"WJ Munro","year":"2015","unstructured":"Munro, W. J., Azuma, K., Tamaki, K. & Nemoto, K. Inside quantum repeaters. IEEE J. Sel. Top. Quantum Electron. 21, 78\u201390 (2015).","journal-title":"IEEE J. Sel. Top. Quantum Electron."},{"key":"866_CR35","doi-asserted-by":"publisher","DOI":"10.1038\/srep20463","volume":"6","author":"S Muralidharan","year":"2016","unstructured":"Muralidharan, S. et al. Optimal architectures for long distance quantum communication. Sci. Rep. 6, 20463 \u2013 (2016).","journal-title":"Sci. Rep."},{"key":"866_CR36","doi-asserted-by":"crossref","unstructured":"Sahini, M. & Sahimi M. Applications of Percolation Theory (CRC Press, 1994).","DOI":"10.1201\/9781482272444"},{"key":"866_CR37","doi-asserted-by":"crossref","unstructured":"Karp, R. M. Reducibility among combinatorial problems. in Complexity of computer computations, 85\u2013103 (Springer, 1972).","DOI":"10.1007\/978-1-4684-2001-2_9"},{"key":"866_CR38","unstructured":"Kurose, J. F. & Ross, K. W. Computer networking: A top-down approach (Addison Wesley, 2010)."},{"key":"866_CR39","unstructured":"Tanenbaum, A. S. & Wetherall, D. Computer Networks, 5th edn. (Prentice Hall, 2011)."},{"key":"866_CR40","doi-asserted-by":"crossref","first-page":"290","DOI":"10.5486\/PMD.1959.6.3-4.12","volume":"6","author":"P Erd\u0151s","year":"1959","unstructured":"Erd\u0151s, P. & R\u00e9nyi, A. On random graphs I. Publ. Math. Debr. 6, 290\u2013297 (1959).","journal-title":"Publ. Math. Debr."},{"key":"866_CR41","doi-asserted-by":"publisher","first-page":"835","DOI":"10.1017\/S0963548310000325","volume":"19","author":"O Riordan","year":"2010","unstructured":"Riordan, O. & Wormald, N. The diameter of sparse random graphs. Comb. Probab. Comput. 19, 835\u2013926 (2010).","journal-title":"Comb. Probab. Comput."},{"key":"866_CR42","doi-asserted-by":"publisher","first-page":"378","DOI":"10.1038\/35019019","volume":"406","author":"R Albert","year":"2000","unstructured":"Albert, R., Jeong, H. & Barab\u00e1si, A.-L. Error and attack tolerance of complex networks. Nature 406, 378\u2013382 (2000).","journal-title":"Nature"},{"key":"866_CR43","doi-asserted-by":"publisher","first-page":"5468","DOI":"10.1103\/PhysRevLett.85.5468","volume":"85","author":"DS Callaway","year":"2000","unstructured":"Callaway, D. S., Newman, M. E. J., Strogatz, S. H. & Watts, D. J. Network robustness and fragility: Percolation on random graphs. Phys. Rev. Lett. 85, 5468\u20135471 (2000).","journal-title":"Phys. Rev. Lett."},{"key":"866_CR44","doi-asserted-by":"publisher","first-page":"509","DOI":"10.1126\/science.286.5439.509","volume":"286","author":"A-L Barab\u00e1si","year":"1999","unstructured":"Barab\u00e1si, A.-L. & Albert, R. Emergence of scaling in random networks. Science 286, 509\u2013512 (1999).","journal-title":"Science"},{"key":"866_CR45","doi-asserted-by":"publisher","first-page":"016121","DOI":"10.1103\/PhysRevE.66.016121","volume":"66","author":"J Dall","year":"2002","unstructured":"Dall, J. & Christensen, M. Random geometric graphs. Phys. Rev. E 66, 016121 (2002).","journal-title":"Phys. Rev. E"},{"key":"866_CR46","doi-asserted-by":"publisher","first-page":"1617","DOI":"10.1109\/49.12889","volume":"6","author":"B Waxman","year":"1988","unstructured":"Waxman, B. Routing of multipoint connections. IEEE J. Sel. Areas Commun. 6, 1617\u20131622 (1988).","journal-title":"IEEE J. Sel. Areas Commun."},{"key":"866_CR47","doi-asserted-by":"publisher","first-page":"250501","DOI":"10.1103\/PhysRevLett.112.250501","volume":"112","author":"S Muralidharan","year":"2014","unstructured":"Muralidharan, S., Kim, J., L\u00fctkenhaus, N., Lukin, M. D. & Jiang, L. Ultrafast and fault-tolerant quantum communication across long distances. Phys. Rev. Lett. 112, 250501 (2014).","journal-title":"Phys. Rev. Lett."},{"key":"866_CR48","doi-asserted-by":"publisher","first-page":"eaam9288","DOI":"10.1126\/science.aam9288","volume":"3622","author":"S Wehner","year":"2018","unstructured":"Wehner, S., Elkouss, D. & Hanson, R. Quantum internet: a vision for the road ahead. Science 3622, eaam9288 (2018).","journal-title":"Science"},{"key":"866_CR49","doi-asserted-by":"crossref","unstructured":"Bollob\u00e1s, B. & Riordan, O. M. Mathematical results on scale-free random graphs. In Handbook of Graphs and Networks: From the Genome to the Internet (Wiley, 2003).","DOI":"10.1002\/3527602755.ch1"}],"container-title":["Communications Physics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.nature.com\/articles\/s42005-022-00866-7.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.nature.com\/articles\/s42005-022-00866-7","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.nature.com\/articles\/s42005-022-00866-7.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,11,20]],"date-time":"2023-11-20T19:56:26Z","timestamp":1700510186000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.nature.com\/articles\/s42005-022-00866-7"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,4,28]]},"references-count":49,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2022,12]]}},"alternative-id":["866"],"URL":"https:\/\/doi.org\/10.1038\/s42005-022-00866-7","relation":{},"ISSN":["2399-3650"],"issn-type":[{"value":"2399-3650","type":"electronic"}],"subject":[],"published":{"date-parts":[[2022,4,28]]},"assertion":[{"value":"1 September 2021","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"23 March 2022","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"28 April 2022","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"The authors declare no competing interests.","order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Competing interests"}}],"article-number":"105"}}