Quantum Communication Networks Enhanced by Distributed Quantum Memories

Xiangyi Meng1,2,3, Nicolò Lo Piparo4, Kae Nemoto4, and István A. Kovács3,5,6

1Department of Physics, Applied Physics, and Astronomy, Rensselaer Polytechnic Institute, Troy, New York 12180, USA
2Network Science and Technology Center, Rensselaer Polytechnic Institute, Troy, New York 12180, USA
3Department of Physics and Astronomy, Northwestern University, Evanston, Illinois 60208, USA
4Okinawa Institute of Science and Technology Graduate University, 1919-1 Tancha, Onna-son, Okinawa 904-0495, Japan
5Northwestern Institute on Complex Systems, Northwestern University, Evanston, Illinois 60208, USA
6Department of Engineering Sciences and Applied Mathematics, Northwestern University, Evanston, Illinois 60208, USA

Find this paper interesting or want to discuss? Scite or leave a comment on SciRate.

Abstract

Building large-scale quantum communication networks has its unique challenges. Here, we demonstrate that a network-wide synergistic usage of quantum memories distributed in a quantum communication network offers a fundamental advantage. We first map the problem of quantum communication with local usage of memories into a classical continuum percolation model. Then, we show that this mapping can be improved through a cooperation of quantum distillation and relay protocols via remote access to distributed memories. This improved mapping, which we term $\alpha$-percolation, can be formulated in terms of graph-merging rules, analogous to the decimation rules of the renormalization group treatment of disordered quantum magnets. These rules can be performed in any order, yielding the same optimal result that is characterized by the emergence of a “positive feedback'' mechanism and the formation of spatially disconnected “hopping'' communication components – both marking significant improvements beyond the traditional point-to-point consideration of quantum communication in networked structures.

It is well established that efficient quantum communication relies on quantum memories. Here, however, we show that the collective advantage of using memories is far more profound than previously anticipated, particularly within a complex network topology. By leveraging network-wide synergy—specifically through remote distillation (remote gate teleportation) and relay protocols between interconnected nodes—communication efficiency is significantly enhanced. This results in a novel statistical physics model for large-scale quantum networks—which we term α-percolation—that reveals unique emergent statistical behaviors and calls for a distinct architectural design for such large-scale systems.

► BibTeX data

► References

[1] M. Razavi, M. Piani, and N. Lütkenhaus. ``Quantum repeaters with imperfect memories: Cost and scalability''. Phys. Rev. A 80, 032301 (2009).
https:/​/​doi.org/​10.1103/​PhysRevA.80.032301

[2] Mikael Afzelius, Nicolas Gisin, and Hugues de Riedmatten. ``Quantum memory for photons''. Phys. Today 68, 42–47 (2015).
https:/​/​doi.org/​10.1063/​PT.3.3021

[3] Yumang Jing and Mohsen Razavi. ``Quantum Repeaters with Encoding on Nitrogen-Vacancy-Center Platforms''. Phys. Rev. Appl. 18, 024041 (2022).
https:/​/​doi.org/​10.1103/​PhysRevApplied.18.024041

[4] Ofir Milul, Barkay Guttel, Uri Goldblatt, Sergey Hazanov, Lalit M. Joshi, Daniel Chausovsky, Nitzan Kahn, Engin Çiftyürek, Fabien Lafont, and Serge Rosenblum. ``Superconducting Cavity Qubit with Tens of Milliseconds Single-Photon Coherence Time''. PRX Quantum 4, 030336 (2023).
https:/​/​doi.org/​10.1103/​PRXQuantum.4.030336

[5] Sonali Gera, Chase Wallace, Mael Flament, Alessia Scriminich, Mehdi Namazi, Youngshin Kim, Steven Sagona-Stophel, Giuseppe Vallone, Paolo Villoresi, and Eden Figueroa. ``Hong-Ou-Mandel interference of single-photon-level pulses stored in independent room-temperature quantum memories''. npj Quantum Inf. 10, 1–8 (2024).
https:/​/​doi.org/​10.1038/​s41534-024-00803-2

[6] Jacob Biamonte, Peter Wittek, Nicola Pancotti, Patrick Rebentrost, Nathan Wiebe, and Seth Lloyd. ``Quantum machine learning''. Nature 549, 195–202 (2017).
https:/​/​doi.org/​10.1038/​nature23474

[7] Hsin-Yuan Huang, Richard Kueng, and John Preskill. ``Information-Theoretic Bounds on Quantum Advantage in Machine Learning''. Physical Review Letters 126, 190505 (2021).
https:/​/​doi.org/​10.1103/​PhysRevLett.126.190505

[8] Matthew S. Palsson, Mile Gu, Joseph Ho, Howard M. Wiseman, and Geoff J. Pryde. ``Experimentally modeling stochastic processes with less memory by the use of a quantum processor''. Science Advances 3, e1601302 (2017).
https:/​/​doi.org/​10.1126/​sciadv.1601302

[9] Thomas J. Elliott and Mile Gu. ``Superior memory efficiency of quantum devices for the simulation of continuous-time stochastic processes''. npj Quantum Information 4, 18 (2018).
https:/​/​doi.org/​10.1038/​s41534-018-0064-4

[10] Nicolas Gisin and Rob Thew. ``Quantum communication''. Nat. Photonics 1, 165–171 (2007).
https:/​/​doi.org/​10.1038/​nphoton.2007.22

[11] Zhen-Sheng Yuan, Xiao-Hui Bao, Chao-Yang Lu, Jun Zhang, Cheng-Zhi Peng, and Jian-Wei Pan. ``Entangled photons and quantum communication''. Phys. Rep. 497, 1–40 (2010).
https:/​/​doi.org/​10.1016/​j.physrep.2010.07.004

[12] Stephanie Wehner, David Elkouss, and Ronald Hanson. ``Quantum internet: A vision for the road ahead''. Science 362, eaam9288 (2018).
https:/​/​doi.org/​10.1126/​science.aam9288

[13] Paweł Horodecki and Ryszard Horodecki. ``Distillation and bound entanglement''. Quantum Inf. Comput. 1, 45–75 (2001).
https:/​/​doi.org/​10.26421/​QIC1.1-4

[14] W. Dür and H. J. Briegel. ``Entanglement purification and quantum error correction''. Rep. Prog. Phys. 70, 1381 (2007).
https:/​/​doi.org/​10.1088/​0034-4885/​70/​8/​R03

[15] Daniela Abdelkhalek, Mareike Syllwasschy, Nicolas J. Cerf, Jaromír Fiurášek, and Roman Schnabel. ``Efficient entanglement distillation without quantum memory''. Nat. Commun. 7, 11720 (2016).
https:/​/​doi.org/​10.1038/​ncomms11720

[16] Luiz F. Bittencourt, Alfredo Goldman, Edmundo R. M. Madeira, Nelson L. S. da Fonseca, and Rizos Sakellariou. ``Scheduling in distributed systems: A cloud computing perspective''. Comput. Sci. Rev. 30, 31–54 (2018).
https:/​/​doi.org/​10.1016/​j.cosrev.2018.08.002

[17] Shi-Hai Wei, Bo Jing, Xue-Ying Zhang, Jin-Yu Liao, Chen-Zhi Yuan, Bo-Yu Fan, Chen Lyu, Dian-Li Zhou, You Wang, Guang-Wei Deng, Hai-Zhi Song, Daniel Oblak, Guang-Can Guo, and Qiang Zhou. ``Towards Real-World Quantum Networks: A Review''. Laser Photonics Rev. 16, 2100219 (2022).
https:/​/​doi.org/​10.1002/​lpor.202100219

[18] Xiangyi Meng, Xinqi Hu, Yu Tian, Gaogao Dong, Renaud Lambiotte, Jianxi Gao, and Shlomo Havlin. ``Percolation Theories for Quantum Networks''. Entropy 25, 1564 (2023).
https:/​/​doi.org/​10.3390/​e25111564

[19] Johannes Nokkala, Jyrki Piilo, and Ginestra Bianconi. ``Complex quantum networks: A topical review''. Journal of Physics A: Mathematical and Theoretical 57, 233001 (2024).
https:/​/​doi.org/​10.1088/​1751-8121/​ad41a6

[20] Samuraí Brito, Askery Canabarro, Rafael Chaves, and Daniel Cavalcanti. ``Statistical Properties of the Quantum Internet''. Phys. Rev. Lett. 124, 210501 (2020).
https:/​/​doi.org/​10.1103/​PhysRevLett.124.210501

[21] Samuraí Brito, Askery Canabarro, Daniel Cavalcanti, and Rafael Chaves. ``Satellite-Based Photonic Quantum Networks Are Small-World''. PRX Quantum 2, 010304 (2021).
https:/​/​doi.org/​10.1103/​PRXQuantum.2.010304

[22] Stefano Pirandola, Samuel L. Braunstein, Riccardo Laurenza, Carlo Ottaviani, Thomas P. W. Cope, Gaetana Spedalieri, and Leonardo Banchi. ``Theory of channel simulation and bounds for private communication''. Quantum Sci. Technol. 3, 035009 (2018).
https:/​/​doi.org/​10.1088/​2058-9565/​aac394

[23] Stefano Pirandola. ``End-to-end capacities of a quantum communication network''. Commun. Phys. 2, 1–10 (2019).
https:/​/​doi.org/​10.1038/​s42005-019-0147-3

[24] Xiangyi Meng, Jianxi Gao, and Shlomo Havlin. ``Concurrence Percolation in Quantum Networks''. Phys. Rev. Lett. 126, 170501 (2021).
https:/​/​doi.org/​10.1103/​PhysRevLett.126.170501

[25] Jianxiong Liang, Xiaoguang Chen, and Tianyi Wang. ``Percolation Distribution in Small-World Quantum Networks''. Appl. Sci. 12, 701 (2022).
https:/​/​doi.org/​10.3390/​app12020701

[26] Omar Malik, Xiangyi Meng, Shlomo Havlin, Gyorgy Korniss, Boleslaw Karol Szymanski, and Jianxi Gao. ``Concurrence percolation threshold of large-scale quantum networks''. Commun. Phys. 5, 1–11 (2022).
https:/​/​doi.org/​10.1038/​s42005-022-00958-4

[27] Xiangyi Meng, Yulong Cui, Jianxi Gao, Shlomo Havlin, and Andrei E. Ruckenstein. ``Deterministic entanglement distribution on series-parallel quantum networks''. Phys. Rev. Research 5, 013225 (2023).
https:/​/​doi.org/​10.1103/​PhysRevResearch.5.013225

[28] Ronald Meester and Rahul Roy. ``Continuum Percolation''. Cambridge Tracts in Mathematics. Cambridge University Press. Cambridge (1996). 1st edition.

[29] Liang Jiang, Jacob M. Taylor, Anders S. Sørensen, and Mikhail D. Lukin. ``Distributed quantum computation based on small quantum registers''. Physical Review A 76, 062323 (2007).
https:/​/​doi.org/​10.1103/​PhysRevA.76.062323

[30] B. C. Jacobs, T. B. Pittman, and J. D. Franson. ``Quantum relays and noise suppression using linear optics''. Phys. Rev. A 66, 052307 (2002).
https:/​/​doi.org/​10.1103/​PhysRevA.66.052307

[31] H. de Riedmatten, I. Marcikic, W. Tittel, H. Zbinden, D. Collins, and N. Gisin. ``Long Distance Quantum Teleportation in a Quantum Relay Configuration''. Phys. Rev. Lett. 92, 047904 (2004).
https:/​/​doi.org/​10.1103/​PhysRevLett.92.047904

[32] Sreraman Muralidharan, Linshu Li, Jungsang Kim, Norbert Lütkenhaus, Mikhail D. Lukin, and Liang Jiang. ``Optimal architectures for long distance quantum communication''. Sci. Rep. 6, 20463 (2016).
https:/​/​doi.org/​10.1038/​srep20463

[33] Peter C. Humphreys, Norbert Kalb, Jaco P. J. Morits, Raymond N. Schouten, Raymond F. L. Vermeulen, Daniel J. Twitchen, Matthew Markham, and Ronald Hanson. ``Deterministic delivery of remote entanglement on a quantum network''. Nature 558, 268–273 (2018).
https:/​/​doi.org/​10.1038/​s41586-018-0200-5

[34] Xiaotian Steve Yao. ``Devices for depolarizing polarized light''. U.S. Patent No. 6498869 (2002).

[35] C. King. ``The capacity of the quantum depolarizing channel''. IEEE Trans. Inf. Theor. 49, 221–229 (2003).
https:/​/​doi.org/​10.1109/​TIT.2002.806153

[36] Reinhard F. Werner. ``Quantum states with Einstein-Podolsky-Rosen correlations admitting a hidden-variable model''. Phys. Rev. A 40, 4277–4281 (1989).
https:/​/​doi.org/​10.1103/​PhysRevA.40.4277

[37] J. F. Dynes, H. Takesue, Z. L. Yuan, A. W. Sharpe, K. Harada, T. Honjo, H. Kamada, O. Tadanaga, Y. Nishida, M. Asobe, and A. J. Shields. ``Efficient entanglement distribution over 200 kilometers''. Opt. Express 17, 11440–11449 (2009).
https:/​/​doi.org/​10.1364/​OE.17.011440

[38] Manish Kumar Gupta. ``Minimizing Decoherence in Optical Fiber for Long Distance Quantum Communication''. PhD thesis. Louisiana State University. (2016).

[39] Simon J. Evered, Dolev Bluvstein, Marcin Kalinowski, Sepehr Ebadi, Tom Manovitz, Hengyun Zhou, Sophie H. Li, Alexandra A. Geim, Tout T. Wang, Nishad Maskara, Harry Levine, Giulia Semeghini, Markus Greiner, Vladan Vuletić, and Mikhail D. Lukin. ``High-fidelity parallel entangling gates on a neutral-atom quantum computer''. Nature 622, 268–272 (2023).
https:/​/​doi.org/​10.1038/​s41586-023-06481-y

[40] Charles H. Bennett, Gilles Brassard, Sandu Popescu, Benjamin Schumacher, John A. Smolin, and William K. Wootters. ``Purification of Noisy Entanglement and Faithful Teleportation via Noisy Channels''. Phys. Rev. Lett. 76, 722–725 (1996).
https:/​/​doi.org/​10.1103/​PhysRevLett.76.722

[41] S. F. Huelga, J. A. Vaccaro, A. Chefles, and M. B. Plenio. ``Quantum remote control: Teleportation of unitary operations''. Phys. Rev. A 63, 042303 (2001).
https:/​/​doi.org/​10.1103/​PhysRevA.63.042303

[42] M. Żukowski, A. Zeilinger, M. A. Horne, and A. K. Ekert. ````Event-Ready-Detectors'' Bell Experiment via Entanglement Swapping''. Phys. Rev. Lett. 71, 4287–4290 (1993).
https:/​/​doi.org/​10.1103/​PhysRevLett.71.4287

[43] Aditi Sen(De), Ujjwal Sen, Časlav Brukner, Vladimír Bužek, and Marek Żukowski. ``Entanglement swapping of noisy states: A kind of superadditivity in nonclassicality''. Phys. Rev. A 72, 042310 (2005).
https:/​/​doi.org/​10.1103/​PhysRevA.72.042310

[44] Béla Bollobás. ``Random graphs''. Volume 73 of Cambridge Studies in Advanced Mathematics. Academic Press. London, UK (1985). 1st edition.

[45] Jonathan L. Gross and Jay Yellen. ``Graph theory and its applications''. CRC Press. Boca Raton, FL, USA (2005). 2nd edition.

[46] Raissa M. D'Souza and Jan Nagler. ``Anomalous critical and supercritical phenomena in explosive percolation''. Nat. Phys. 11, 531–538 (2015).
https:/​/​doi.org/​10.1038/​nphys3378

[47] Raissa M. D'Souza, Jesus Gómez-Gardeñes, Jan Nagler, and Alex Arenas. ``Explosive phenomena in complex networks''. Adv. Phys. 68, 123–223 (2019).
https:/​/​doi.org/​10.1080/​00018732.2019.1650450

[48] Sergey V. Buldyrev, Roni Parshani, Gerald Paul, H. Eugene Stanley, and Shlomo Havlin. ``Catastrophic cascade of failures in interdependent networks''. Nature 464, 1025–1028 (2010).
https:/​/​doi.org/​10.1038/​nature08932

[49] Jianxi Gao, Sergey V. Buldyrev, H. Eugene Stanley, and Shlomo Havlin. ``Networks Formed from Interdependent Networks''. Nat. Phys. 8, 40–48 (2012).
https:/​/​doi.org/​10.1038/​nphys2180

[50] Laurent Ménard and Arvind Singh. ``Percolation by cumulative merging and phase transition for the contact process on random graphs''. Annales scientifiques de l'Ecole Normale Supérieure 49, 1189–1238 (2016).

[51] Claudio Castellano and Romualdo Pastor-Satorras. ``Cumulative Merging Percolation and the Epidemic Transition of the Susceptible-Infected-Susceptible Model in Networks''. Physical Review X 10, 011070 (2020).
https:/​/​doi.org/​10.1103/​PhysRevX.10.011070

[52] Lorenzo Cirigliano, Giulio Cimini, Romualdo Pastor-Satorras, and Claudio Castellano. ``Cumulative merging percolation: A long-range percolation process in networks''. Physical Review E 105, 054310 (2022).
https:/​/​doi.org/​10.1103/​PhysRevE.105.054310

[53] Shawn Fostner, Richard Brown, James Carr, and Simon A. Brown. ``Continuum percolation with tunneling''. Phys. Rev. B 89, 075402 (2014).
https:/​/​doi.org/​10.1103/​PhysRevB.89.075402

[54] Lorenzo Cirigliano, Claudio Castellano, and Gábor Timár. ``Extended-range percolation in complex networks''. Phys. Rev. E 108, 044304 (2023).
https:/​/​doi.org/​10.1103/​PhysRevE.108.044304

[55] T. Yu and J. H. Eberly. ``Finite-Time Disentanglement via Spontaneous Emission''. Phys. Rev. Lett. 93, 140404 (2004).
https:/​/​doi.org/​10.1103/​PhysRevLett.93.140404

[56] T. Yu and J. H. Eberly. ``Sudden death of entanglement: Classical noise effects''. Opt. Commun. 264, 393–397 (2006).
https:/​/​doi.org/​10.1016/​j.optcom.2006.01.061

[57] Qing-Jun Tong, Jun-Hong An, Hong-Gang Luo, and C. H. Oh. ``Mechanism of Entanglement Preservation''. Phys. Rev. A 81, 052330 (2010).
https:/​/​doi.org/​10.1103/​PhysRevA.81.052330

[58] Sparkle. ``The Global Backbone Experience [http:/​/​www.globalbackbone.tisparkle.com]''.
http:/​/​www.globalbackbone.tisparkle.com

[59] Koji Azuma, Sophia E. Economou, David Elkouss, Paul Hilaire, Liang Jiang, Hoi-Kwong Lo, and Ilan Tzitrin. ``Quantum repeaters: From quantum networks to the quantum internet''. Rev. Mod. Phys. 95, 045006 (2023).
https:/​/​doi.org/​10.1103/​RevModPhys.95.045006

[60] Takahiro Inagaki, Nobuyuki Matsuda, Osamu Tadanaga, Masaki Asobe, and Hiroki Takesue. ``Entanglement distribution over 300 km of fiber''. Opt. Express 21, 23241–23249 (2013).
https:/​/​doi.org/​10.1364/​OE.21.023241

[61] Subir Sachdev. ``Quantum Phase Transitions''. Cambridge University Press. Cambridge (2001). reprint edition.

[62] Ferenc Iglói and Cécile Monthus. ``Strong disorder RG approach of random systems''. Phys. Rep. 412, 277–431 (2005).
https:/​/​doi.org/​10.1016/​j.physrep.2005.02.006

[63] Ferenc Iglói and Cécile Monthus. ``Strong disorder RG approach – a short review of recent developments''. Eur. Phys. J. B 91, 290 (2018).
https:/​/​doi.org/​10.1140/​epjb/​e2018-90434-8

[64] Barry M. McCoy and Tai Tsun Wu. ``Theory of a Two-Dimensional Ising Model with Random Impurities. I. Thermodynamics''. Phys. Rev. 176, 631–643 (1968).
https:/​/​doi.org/​10.1103/​PhysRev.176.631

[65] Daniel S. Fisher. ``Random transverse field Ising spin chains''. Phys. Rev. Lett. 69, 534–537 (1992).
https:/​/​doi.org/​10.1103/​PhysRevLett.69.534

[66] István A. Kovács and Ferenc Iglói. ``Renormalization group study of the two-dimensional random transverse-field Ising model''. Phys. Rev. B 82, 054437 (2010).
https:/​/​doi.org/​10.1103/​PhysRevB.82.054437

[67] Shang-Keng Ma, Chandan Dasgupta, and Chin-Kun Hu. ``Random Antiferromagnetic Chain''. Phys. Rev. Lett. 43, 1434–1437 (1979).
https:/​/​doi.org/​10.1103/​PhysRevLett.43.1434

[68] Ehud Altman, Yariv Kafri, Anatoli Polkovnikov, and Gil Refael. ``Phase Transition in a System of One-Dimensional Bosons with Strong Disorder''. Phys. Rev. Lett. 93, 150402 (2004).
https:/​/​doi.org/​10.1103/​PhysRevLett.93.150402

[69] Ehud Altman, Yariv Kafri, Anatoli Polkovnikov, and Gil Refael. ``Superfluid-insulator transition of disordered bosons in one dimension''. Phys. Rev. B 81, 174528 (2010).
https:/​/​doi.org/​10.1103/​PhysRevB.81.174528

[70] Fawaz Hrahsheh, José A. Hoyos, and Thomas Vojta. ``Rounding of a first-order quantum phase transition to a strong-coupling critical point''. Physical Review B 86, 214204 (2012).
https:/​/​doi.org/​10.1103/​PhysRevB.86.214204

[71] Filip Rozpedek, Thomas Schiet, Le Phuc Thinh, David Elkouss, Andrew C. Doherty, and Stephanie Wehner. ``Optimizing practical entanglement distillation''. Phys. Rev. A 97, 062333 (2018).
https:/​/​doi.org/​10.1103/​PhysRevA.97.062333

[72] Chenxu Liu, Meng Wang, Samuel A. Stein, Yufei Ding, and Ang Li. ``Quantum Memory: A Missing Piece in Quantum Computing Units''. Technical Report arXiv:2309.14432. arXiv (2023).
arXiv:2309.14432

[73] David Deutsch, Artur Ekert, Richard Jozsa, Chiara Macchiavello, Sandu Popescu, and Anna Sanpera. ``Quantum Privacy Amplification and the Security of Quantum Cryptography over Noisy Channels''. Phys. Rev. Lett. 77, 2818–2821 (1996).
https:/​/​doi.org/​10.1103/​PhysRevLett.77.2818

[74] István A. Kovács. ``Infinitely Disordered Critical Behavior in Higher Dimensional Quantum Systems''. PhD thesis. Eötvös Loránd University. (2012).

Cited by

[1] Yaqi Zhao, Kan He, Yongtao Zhang, Jinchuan Hou, Jianxi Gao, Shlomo Havlin, and Xiangyi Meng, "Negativity percolation in continuous-variable quantum networks", npj Quantum Information 12 1, 77 (2026).

[2] Pedro Henrique Alvarez and Marcos César de Oliveira, "Coherence scaling in quantum communication protocols", Quantum Information Processing 25 8, 263 (2026).

[3] Alessandro Romancino, Jordi Romero-Pallejá, G Massimo Palma, and Anna Sanpera, "Entanglement percolation in random quantum networks", Quantum Science and Technology 11 2, 025044 (2026).

[4] Haigang Wang, Omar Malik, Jinchuan Hou, Yongtao Zhang, Kan He, and Xiangyi Meng, "A counter-intuitive low entanglement percolation threshold in mixed-state quantum networks", Communications Physics 9 1, 28 (2026).

[5] Xinqi Hu, Gaogao Dong, Kim Christensen, Hanlin Sun, Jingfang Fan, Zihao Tian, Jianxi Gao, Shlomo Havlin, Renaud Lambiotte, and Xiangyi Meng, "Unveiling the importance of nonshortest paths in quantum networks", Science Advances 11 9, eadt2404 (2025).

[6] Yaqi Zhao, Jinchuan Hou, Kan He, Nicolò Lo Piparo, and Xiangyi Meng, "Nonconvex entanglement monotone determining the characteristic length of entanglement distribution in continuous-variable quantum networks", Physical Review A 111 4, 042429 (2025).

[7] Xiangyi Meng, Bingjie Hao, Balázs Ráth, and István A. Kovács, "Path Percolation in Quantum Communication Networks", Physical Review Letters 134 3, 030803 (2025).

[8] Bethany Davies, Guus Avis, and Stephanie Wehner, "On the accuracy of twirled approximations in repeater chains", arXiv:2509.16689, (2025).

[9] Xiangyi Meng and Yu Tian, "Finding hypergraph immersion is fixed-parameter tractable", arXiv:2411.16017, (2024).

[10] Yu Tian, Yuefei Liu, and Xiangyi Meng, "Multipartite Entanglement Routing as a Hypergraph Immersion Problem", arXiv:2406.13452, (2024).

[11] Alessandro Romancino, "Random entanglement percolation on realistic quantum networks", arXiv:2604.21967, (2026).

[12] Yaqi Zhao, Kan He, Jinchuan Hou, and Xiangyi Meng, "Advantage of deterministic entanglement transmission on Gaussian-state quantum networks", Journal of Physics: Complexity 6 4, 045004 (2025).

[13] Yaqi Zhao, Kan He, Yongtao Zhang, Jinchuan Hou, Jianxi Gao, Shlomo Havlin, and Xiangyi Meng, "Negativity Percolation in Continuous-Variable Quantum Networks", arXiv:2507.16417, (2025).

[14] Otávio José R. Silveira, Nycolas B. da Silva, Saulo L. L. da Silva, and Angélica S. da Mata, "Characterizing quantum internet using complex network models", European Physical Journal Special Topics (2025).

[15] Yuan-Cheng Lai, Han Lin, Yen-Hung Chen, Yu-Ling Hsiao, and Liang-Chun Chen, "Selecting and Developing a Quantum Internet Simulator for the Needs of New Audience: Practices and Considerations", IEEE Access 12, 170965 (2024).

The above citations are from Crossref's cited-by service (last updated successfully 2026-08-13 17:32:00) and SAO/NASA ADS (last updated successfully 2026-08-13 17:32:01). The list may be incomplete as not all publishers provide suitable and complete citation data.