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  • Open Access

Self-Configuring Quantum Networks with Superposition of Trajectories

Albie Chan1,2,*, Zheng Shi1,2,*, Jorge Miguel-Ramiro3, Luca Dellantonio1,2,4, Christine A. Muschik1,2,5, and Wolfgang Dür3

  • *These authors contributed equally.
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PRX Quantum 7, 033038 – Published 25 August, 2026

DOI: https://doi.org/10.1103/rj5x-yqsj

Abstract

Quantum networks are a backbone of future quantum technologies thanks to their role in communication and scalable quantum computing. However, their performance is challenged by noise and decoherence. We propose a self-configuring approach that integrates superposed quantum paths with variational quantum optimization techniques. By estimating a cost function via repeated network uses, our approach dynamically optimizes the superposition of noisy paths across multiple nodes to establish high-fidelity connections between different parties. Our framework is in principle capable of adapting to unknown noise without assuming parametric noise models or performing full process tomography. We also discuss the role of vacuum coherence, a quantum effect central to path superposition that impacts protocol performance. Additionally, we demonstrate that our approach remains beneficial even in the presence of imperfections in the generation of path superposition.

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Popular Summary

Quantum networks are critical for future technologies such as secure communication, distributed quantum computing, and global quantum sensing. However, they are vulnerable to noise and decoherence. This work presents a “self-configuring” quantum network protocol that uses coherent superpositions of various noisy transmission paths between the sender and receiver. It learns how to find the best superposition of these paths to mitigate noise, without requiring detailed knowledge of the noise in each path. By sending entangled states in a superposition of paths and adjusting phases and amplitudes of the superposition coefficients in a quantum-classical variational feedback loop, the protocol improves end-to-end fidelity beyond what any single path or classical mixture offers. The authors identify the conditions that enable this improvement and also demonstrate that the protocol remains robust even if the generation of path superposition is itself noisy. The potentially all-photonic design of this protocol makes it promising for near-term demonstrations and deployment in quantum internet infrastructures.

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References (155)

  1. H. J. Kimble, The quantum internet, Nature (London) 453, 1023 (2008).
  2. S. Wehner, D. Elkouss, and R. Hanson, Quantum internet: A vision for the road ahead, Science 362, eaam9288 (2018).
  3. A. Acín, I. Bloch, H. Buhrman, T. Calarco, C. Eichler, J. Eisert, D. Esteve, N. Gisin, S. J. Glaser, F. Jelezko, S. Kuhr, M. Lewenstein, M. F. Riedel, P. O. Schmidt, R. Thew, A. Wallraff, I. Walmsley, and F. K. Wilhelm, The quantum technologies roadmap: a european community view, New J. Phys. 20, 080201 (2018).
  4. W. Kozlowski and S. Wehner, Towards large-scale quantum networks, in Proceedings of the Sixth Annual ACM International Conference on Nanoscale Computing and Communication, NANOCOM ’19 (Association for Computing Machinery, New York, NY, USA, 2019).
  5. K. Azuma, S. E. Economou, D. Elkouss, P. Hilaire, L. Jiang, H.-K. Lo, and I. Tzitrin, Quantum repeaters: From quantum networks to the quantum internet, Rev. Mod. Phys. 95, 045006 (2023).
  6. N. Gisin, Gégoire Ribordy, W. Tittel, and H. Zbinden, Quantum cryptography, Rev. Mod. Phys. 74, 145 (2002).
  7. S. Pirandola, U. L. Andersen, L. Banchi, M. Berta, D. Bunandar, R. Colbeck, D. Englund, T. Gehring, C. Lupo, C. Ottaviani, J. L. Pereira, M. Razavi, J. Shamsul Shaari, M. Tomamichel, V. C. Usenko, G. Vallone, P. Villoresi, and P. Wallden, Advances in quantum cryptography, Adv. Opt. Photonics 12, 1012 (2020).
  8. J. I. Cirac, A. K. Ekert, S. F. Huelga, and C. Macchiavello, Distributed quantum computation over noisy channels, Phys. Rev. A 59, 4249 (1999).
  9. M. Caleffi, M. Amoretti, D. Ferrari, J. Illiano, A. Manzalini, and A. S. Cacciapuoti, Distributed quantum computing: A survey, Comput. Netw. 254, 110672 (2024).
  10. C. Howe, M. Aziz, and A. Anwar, Towards scalable quantum networks, arXiv:2409.08416.
  11. E. M. Kessler, I. Lovchinsky, A. O. Sushkov, and M. D. Lukin, Quantum error correction for metrology, Phys. Rev. Lett. 112, 150802 (2014).
  12. P. Sekatski, S. Wölk, and W. Dür, Optimal distributed sensing in noisy environments, Phys. Rev. Res. 2, 023052 (2020).
  13. D.-H. Kim, S. Hong, Y.-S. Kim, Y. Kim, S.-W. Lee, R. C. Pooser, K. Oh, S.-Y. Lee, C. Lee, and H.-T. Lim, Distributed quantum sensing of multiple phases with fewer photons, Nat. Commun. 15, 266 (2024).
  14. N. Gisin and R. Thew, Quantum communication, Nat. Photonics 1, 165 (2007).
  15. T. E. Northup and R. Blatt, Quantum information transfer using photons, Nat. Photonics 8, 356 (2014).
  16. Y.-A. Chen et al., An integrated space-to-ground quantum communication network over 4, 600 kilometres, Nature (London) 589, 214 (2021).
  17. J. Preskill, Quantum computing in the NISQ era and beyond, Quantum 2, 79 (2018).
  18. K. Bharti, A. Cervera-Lierta, T. H. Kyaw, T. Haug, S. Alperin-Lea, A. Anand, M. Degroote, H. Heimonen, J. S. Kottmann, T. Menke, W.-K. Mok, S. Sim, L.-C. Kwek, and Aán Aspuru-Guzik, Noisy intermediate-scale quantum algorithms, Rev. Mod. Phys. 94, 015004 (2022).
  19. B. C. Coutinho, W. J. Munro, K. Nemoto, and Y. Omar, Robustness of noisy quantum networks, Commun. Phys. 5, 105 (2022).
  20. M. F. Mor-Ruiz and W. Dür, Influence of noise in entanglement-based quantum networks, IEEE J. Sel. Areas Commun. 42, 1793 (2024).
  21. E. Knill and R. Laflamme, Theory of quantum error-correcting codes, Phys. Rev. A 55, 900 (1997).
  22. W. Dür and H. J. Briegel, Entanglement purification and quantum error correction, Rep. Prog. Phys. 70, 1381 (2007).
  23. W. Dür, H.-J. Briegel, J. I. Cirac, and P. Zoller, Quantum repeaters based on entanglement purification, Phys. Rev. A 59, 169 (1999).
  24. J.-W. Pan, C. Simon, Časlav Brukner, and A. Zeilinger, Entanglement purification for quantum communication, Nature (London) 410, 1067 (2001).
  25. M. Cramer, M. B. Plenio, S. T. Flammia, R. Somma, D. Gross, S. D. Bartlett, O. Landon-Cardinal, D. Poulin, and Y.-K. Liu, Efficient quantum state tomography, Nat. Commun. 1, 1 (2010).
  26. J. Eisert, D. Hangleiter, N. Walk, I. Roth, D. Markham, R. Parekh, U. Chabaud, and E. Kashefi, Quantum certification and benchmarking, Nat. Rev. Phys. 2, 382 (2020).
  27. J. Helsen and S. Wehner, A benchmarking procedure for quantum networks, npj Quantum Inf. 9, 17 (2023).
  28. S. Das, S. Khatri, and J. P. Dowling, Robust quantum network architectures and topologies for entanglement distribution, Phys. Rev. A 97, 012335 (2018).
  29. M. Pant, H. Krovi, D. Towsley, L. Tassiulas, L. Jiang, P. Basu, D. Englund, and S. Guha, Routing entanglement in the quantum internet, npj Quantum Inf. 5, 25 (2019).
  30. L. D. Bugalho, B. C. Coutinho, F. A. Monteiro, and Y. Omar, Distributing multipartite entanglement over noisy quantum networks, Quantum 7, 920 (2023).
  31. G. Chiribella and Hér Kristjánsson, Quantum shannon theory with superpositions of trajectories, Proc. R. Soc. A. 475, 20180903 (2019).
  32. Hér Kristjánsson, G. Chiribella, S. Salek, D. Ebler, and M. Wilson, Resource theories of communication, New J. Phys. 22, 073014 (2020).
  33. A. A. Abbott, J. Wechs, D. Horsman, M. Mhalla, and C. Branciard, Communication through coherent control of quantum channels, Quantum 4, 333 (2020).
  34. J. Miguel-Ramiro, A. Pirker, and W. Dür, Genuine quantum networks with superposed tasks and addressing, npj Quantum Inf. 7, 135 (2021).
  35. G. Rubino, L. A. Rozema, D. Ebler, Hér Kristjánsson, S. Salek, P. Allard Guérin, A. A. Abbott, C. Branciard, Časlav Brukner, G. Chiribella, and P. Walther, Experimental quantum communication enhancement by superposing trajectories, Phys. Rev. Res. 3, 013093 (2021).
  36. J. Miguel-Ramiro, Z. Shi, L. Dellantonio, A. Chan, C. A. Muschik, and W. Dür, Superposed quantum error mitigation, Phys. Rev. Lett. 131, 230601 (2023).
  37. J. Miguel-Ramiro, Z. Shi, L. Dellantonio, A. Chan, C. A. Muschik, and W. Dür, Enhancing quantum computation via superposition of quantum gates, Phys. Rev. A 108, 062604 (2023).
  38. Hér Kristjánsson, Y. Zhong, A. Munson, and G. Chiribella, Quantum networks with coherent routing of information through multiple nodes, npj Quantum Inf. 10, 131 (2024).
  39. L. M. Procopio, A. Moqanaki, M. Araújo, F. Costa, I. Alonso Calafell, E. G. Dowd, D. R. Hamel, L. A. Rozema, Časlav Brukner, and P. Walther, Experimental superposition of orders of quantum gates, Nat. Commun. 6, 7913 (2015).
  40. D. Ebler, S. Salek, and G. Chiribella, Enhanced communication with the assistance of indefinite causal order, Phys. Rev. Lett. 120, 120502 (2018).
  41. P. A. Guérin, G. Rubino, and Časlav Brukner, Communication through quantum-controlled noise, Phys. Rev. A 99, 062317 (2019).
  42. M. Caleffi and A. S. Cacciapuoti, Quantum switch for the quantum internet: Noiseless communications through noisy channels, IEEE J. Sel. Areas Commun. 38, 575 (2020).
  43. Y. Guo, X.-M. Hu, Z.-B. Hou, H. Cao, J.-M. Cui, B.-H. Liu, Y.-F. Huang, C.-F. Li, G.-C. Guo, and G. Chiribella, Experimental transmission of quantum information using a superposition of causal orders, Phys. Rev. Lett. 124, 030502 (2020).
  44. G. Chiribella, M. Banik, S. S. Bhattacharya, T. Guha, M. Alimuddin, A. Roy, S. Saha, S. Agrawal, and G. Kar, Indefinite causal order enables perfect quantum communication with zero capacity channels, New J. Phys. 23, 033039 (2021).
  45. M. Araújo, F. Costa, and Časlav Brukner, Computational advantage from quantum-controlled ordering of gates, Phys. Rev. Lett. 113, 250402 (2014).
  46. G. Lee, C. T. Hann, S. Puri, S. M. Girvin, and L. Jiang, Error suppression for arbitrary-size black box quantum operations, Phys. Rev. Lett. 131, 190601 (2023).
  47. K. Simonov, M. Caleffi, J. Illiano, and A. S. Cacciapuoti, Universal quantum computation via superposed orders of single-qubit gates, arXiv:2311.13654.
  48. A. Ali, G. Scala, and C. Lupo, Optimal and robust error filtration for quantum information processing, Phys. Rev. A 112, 042418 (2025).
  49. D. Chandra, M. Caleffi, and A. S. Cacciapuoti, The entanglement-assisted communication capacity over quantum trajectories, IEEE Trans. Wirel. Commun. 21, 3632 (2021).
  50. S. Koudia, A. S. Cacciapuoti, K. Simonov, and M. Caleffi, How deep the theory of quantum communications goes: Superadditivity, superactivation and causal activation, IEEE Commun. Surv. Tutorials 24, 1926 (2022).
  51. S. Koudia, A. S. Cacciapuoti, and M. Caleffi, Deterministic generation of multipartite entanglement via causal activation in the quantum internet, IEEE Access 11, 73863 (2023).
  52. M. Caleffi, K. Simonov, and A. S. Cacciapuoti, Beyond shannon limits: Quantum communications through quantum paths, IEEE J. Sel. Areas Commun. 41, 2707 (2023).
  53. S. Kechrimparis, J. Moran, A. Karsa, C. Lee, and H. Kwon, Enhancing quantum state discrimination with indefinite causal order, New J. Phys. 26, 123030 (2024).
  54. H. Tang, Y. Guo, X.-M. Hu, Y.-F. Huang, B.-H. Liu, C.-F. Li, and G.-C. Guo, Demonstration of superior communication through thermodynamically free channels in an optical quantum switch, Phys. Rev. A 110, 032422 (2024).
  55. Z. Wu, J. Fullwood, Z. Ma, S.-Q. Zhou, Q. Zhao, and G. Chiribella, General communication enhancement via the quantum switch, Phys. Rev. A 111, 012605 (2025).
  56. S. Mondal, P. Ghosh, and U. Sen, Path superposition as resource for perfect quantum teleportation with separable states, arXiv:2505.11398.
  57. R. B. Mothe, C. Branciard, and A. A. Abbott, Reassessing the advantage of indefinite causal orders for quantum metrology, Phys. Rev. A 109, 062435 (2024).
  58. L. Chen, Y.-X. Yang, G.-C. Li, X.-S. Hong, S.-Q. Zhang, H.-Q. Xu, Y.-C. Liu, G. Chiribella, G. Chen, C.-F. Li et al., Nonlinear enhancement of measurement precision via a hybrid quantum switch, arXiv:2506.20632.
  59. Z. Huang and C. Lupo, Error filtration for quantum sensing via interferometry, Phys. Rev. A 110, 052431 (2024).
  60. J. Miguel-Ramiro, A. Pirker, and W. Dür, Improving entanglement purification through coherent superposition of roles, Quantum 9, 1702 (2025).
  61. J. Miguel-Ramiro, A. Pirker, and W. Dür, Coherent randomized benchmarking, Phys. Rev. Res. 3, 033038 (2021).
  62. P.-R. Lai, J.-D. Lin, Y.-T. Huang, H.-C. Jan, and Y.-N. Chen, Quick charging of a quantum battery with superposed trajectories, Phys. Rev. Res. 6, 023136 (2024).
  63. N. Moll, P. Barkoutsos, L. S. Bishop, J. M. Chow, A. Cross, D. J. Egger, S. Filipp, A. Fuhrer, J. M. Gambetta, M. Ganzhorn, A. Kandala, A. Mezzacapo, P. Müller, W. Riess, G. Salis, J. Smolin, I. Tavernelli, and K. Temme, Quantum optimization using variational algorithms on near-term quantum devices, Quantum Sci. Technol. 3, 030503 (2018).
  64. M. Cerezo, A. Arrasmith, R. Babbush, S. C. Benjamin, S. Endo, K. Fujii, J. R. McClean, K. Mitarai, X. Yuan, L. Cincio, and P. J. Coles, Variational quantum algorithms, Nat. Rev. Phys. 3, 625 (2021).
  65. M. A. Nielsen and I. L. Chuang, Distance measures to compare real and ideal quantum processes, Quantum Computation and Quantum Information: 10th Anniversary Edition (Cambridge University Press, Cambridge, England, 2010).
  66. R. Jozsa, Fidelity for mixed quantum states, J. Mod. Opt. 41, 2315 (1994).
  67. A. Gilchrist, N. K. Langford, and M. A. Nielsen, Distance measures to compare real and ideal quantum processes, Phys. Rev. A 71, 062310 (2005).
  68. W. Dür, M. Hein, J. I. Cirac, and H.-J. Briegel, Standard forms of noisy quantum operations via depolarization, Phys. Rev. A 72, 052326 (2005).
  69. Mł Horodecki, Pł Horodecki, and R. Horodecki, General teleportation channel, singlet fraction, and quasidistillation, Phys. Rev. A 60, 1888 (1999).
  70. M. A. Nielsen, A simple formula for the average gate fidelity of a quantum dynamical operation, Phys. Lett. A 303, 249 (2002).
  71. S. T. Flammia and Y.-K. Liu, Direct fidelity estimation from few Pauli measurements, Phys. Rev. Lett. 106, 230501 (2011).
  72. S. Attal and Y. Pautrat, From repeated to continuous quantum interactions, Annales Henri Poincaré (Springer, Basel, Switzerland, 2006), Vol. 7, pp. 59–104.
  73. D. Grimmer, D. Layden, R. B. Mann, and E. Martín-Martínez, Open dynamics under rapid repeated interaction, Phys. Rev. A 94, 032126 (2016).

  74. The local unitaries are applied to maximize the output CJ fidelity FCJ, see Step 7. FCJ is upper bounded by the largest eigenvalue of the output density matrix, i.e., λmax(ρabout), and a unitary rotation can maximize the overlap of |Φ+⟩ with an eigenvector corresponding to λmax.

  75. A. Shlosberg, A. J. Jena, P. Mukhopadhyay, J. F. Haase, F. Leditzky, and L. Dellantonio, Adaptive estimation of quantum observables, Quantum 7, 906 (2023).
  76. G. Iannelli and K. Jansen, Noisy Bayesian optimization for variational quantum eigensolvers, in Proceedings of the 38th International Symposium on Lattice Field Theory—PoS(LATTICE2021) (2022),Vol. 396, p. 251.
  77. K. Nicoli, C. J. Anders, L. Funcke, T. Hartung, K. Jansen, S. Kühn, K.-R. Müller, P. Stornati, P. Kessel, and S. Nakajima, Physics-informed bayesian optimization of variational quantum circuits, Adv. Neural Inf. Process. Syst. 36, 18341 (2023).
  78. J. R. Johansson, P. D. Nation, and F. Nori, Qutip: An open-source python framework for the dynamics of open quantum systems, Comput. Phys. Commun. 183, 1760 (2012).
  79. A. Chan, Z. Shi, W. Dür, and C. A. Muschik, in preparation (2026).
  80. G. Chen, D. Church, B. Englert, C. Henkel, B. Rohwedder, M. Scully, and M. Zubairy, Experimental realization of any discrete unitary operator, Quantum Computing Devices: Principles, Designs, and Analysis, Chapman & Hall/CRC Applied Mathematics & Nonlinear Science (CRC Press, Boca Raton, FL, 2006).
  81. Y. Ikeda, yuzie007/mpltern: 1.0.4 (2024).
  82. M. J. Powell, A direct search optimization method that models the objective and constraint functions by linear interpolation, Advances in Optimization and Numerical Analysis (Springer, 1994), pp. 51–67.
  83. Z. Zhang, PRIMA: Reference implementation for Powell’s methods with modernization and amelioration, available at http://www.libprima.net (2023), 10.5281/zenodo.8052654.
  84. P. Virtanen et al. (SciPy 1.0 Contributors), SciPy 1.0: Fundamental Algorithms for Scientific Computing in Python, Nat. Methods 17, 261 (2020).

  85. In our examples the number of cost function evaluations can be approximated as a linear function of the number of iterations with a slope of 2, but this can vary depending on both the cost function and the optimization algorithm.

  86. M. Reck, A. Zeilinger, H. J. Bernstein, and P. Bertani, Experimental realization of any discrete unitary operator, Phys. Rev. Lett. 73, 58 (1994).
  87. W. R. Clements, P. C. Humphreys, B. J. Metcalf, W. S. Kolthammer, and I. A. Walmsley, Optimal design for universal multiport interferometers, Optica 3, 1460 (2016).

  88. A superposition of more than two paths can be realized through successive pairwise splitting in a tree-like structure, as in Fig. 7. Alternatively, we can directly embed higher-dimensional parameterized unitary matrices in relevant subspaces.

  89. E. W. Dijkstra, A note on two problems in connexion with graphs, Numer. Math. 50, 269 (1959).
  90. M. Mičuda, E. Doláková, I. Straka, M. Miková, M. Dušek, J. Fiurášek, and M. Ježek, Highly stable polarization independent Mach-Zehnder interferometer, Rev. Sci. Instrum. 85 (2014).

  91. Here, the transit nodes are physically meaningful and no longer “virtual” when separating different noise types.

  92. Owing to statistical noise, the best-case Rprob at convergence may be worse than that attained during the optimization, as occurs for Nshots=103 and 105.
  93. R. Myerson, Game Theory (Harvard University Press, 2013).
  94. T. Baumgratz, M. Cramer, and M. B. Plenio, Quantifying coherence, Phys. Rev. Lett. 113, 140401 (2014).

  95. Nevertheless, when the noise level is extremely low, the optimization loop cannot proceed without a very large Nshots and consequently a prohibitive measurement budget, even though path superposition may remain useful as a noise reduction mechanism.

  96. N. V. Nardelli, D. V. Reddy, M. Grayson, D. Sorensen, M. J. Stevens, M. D. Mazurek, L. K. Shalm, and T. M. Fortier, Phase-stable optical fiber links for quantum network protocols, Opt. Quantum 4, 138 (2025).
  97. S. Grebien, J. Gurs, R. Schnabel, and M. Korobko, Hyperloss from coherent spatial-mode mixing in quantum-correlated networks, arXiv:2603.21982.
  98. S. Rojas-Rojas, D. Martínez, K. Sawada, L. Pereira, S. P. Walborn, E. S. Gómez, N. K. Bernardes, and G. Lima, Non-Markovianity in high-dimensional open quantum systems using next-generation multicore optical fibers, Quantum 8, 1436 (2024).
  99. B. Kar and P. Kumar, Routing protocols for quantum networks: overview and challenges, arXiv:2305.00708.
  100. C. Li, T. Li, Y.-X. Liu, and P. Cappellaro, Effective routing design for remote entanglement generation on quantum networks, arXiv:2001.02204.
  101. E. O. Kiktenko, A. Tayduganov, and A. K. Fedorov, Routing algorithm within the multiple non-overlapping paths’ approach for quantum key distribution networks, Entropy 26, 1102 (2024).
  102. S. Simmons, Scalable fault-tolerant quantum technologies with silicon color centers, PRX Quantum 5, 010102 (2024).
  103. R. Bedington, J. M. Arrazola, and A. Ling, Progress in satellite quantum key distribution, npj Quantum Inf. 3, 30 (2017).
  104. A. Shabani, Building a global quantum internet using a satellite constellation with inter-satellite links, arXiv:2505.08075.
  105. J. Yin et al., Satellite-based entanglement distribution over 1200 kilometers, Science 356, 1140 (2017).
  106. S.-K. Liao et al., Satellite-to-ground quantum key distribution, Nature (London) 549, 43 (2017).
  107. J. Yin et al., Satellite-to-ground entanglement-based quantum key distribution, Phys. Rev. Lett. 119, 200501 (2017).
  108. H. Zhang, Z. Sun, R. Qi, L. Yin, G.-L. Long, and J. Lu, Realization of quantum secure direct communication over 100 km fiber with time-bin and phase quantum states, Light Sci. Appl. 11, 83 (2022).
  109. T. Roger, R. Singh, C. Perumangatt, D. G. Marangon, M. Sanzaro, P. R. Smith, R. I. Woodward, and A. J. Shields, Real-time gigahertz free-space quantum key distribution within an emulated satellite overpass, Sci. Adv. 9, eadj5873 (2023).
  110. H. Peng, S. Koudia, and S. Chatzinotas, Security analysis of MDI-QKD in turbulent free-space polarization channels: a composite channel framework, arXiv:2509.02087.
  111. T. Gonzalez-Raya, S. Pirandola, and M. Sanz, Satellite-based entanglement distribution and quantum teleportation with continuous variables, Commun. Phys. 7, 126 (2024).
  112. T. Jaouni, L. Scarfe, F. Bouchard, M. Krenn, K. Heshami, F. Di Colandrea, and E. Karimi, Predicting atmospheric turbulence for secure quantum communications in free space, Opt. Express 33, 10759 (2025).
  113. N. Maharjan, N. Devkota, and B. W. Kim, Atmospheric effects on satellite–œground free space uplink and downlink optical transmissions, Appl. Sci. 12, 10944 (2022).
  114. Y. Shao, S. Guha, and A. E. Motter, Hybrid satellite-fiber quantum network, Phys. Rev. Appl. 24, 024033 (2025).
  115. G. Vallone, D. Dequal, M. Tomasin, F. Vedovato, M. Schiavon, V. Luceri, G. Bianco, and P. Villoresi, Satellite quantum communications exploiting SLR at MLRO, 2015 ILRS Technical Workshop (International Laser Ranging Service, Matera, Italy, 2015).
  116. Z. Wang, R. Malaney, and R. Aguinaldo, Temporal modes of light in satellite-to-earth quantum communications, IEEE Commun. Lett. 26, 311 (2022).
  117. S.-K. Liao et al., Long-distance free-space quantum key distribution in daylight towards inter-satellite communication, Nat. Photonics 11, 509 (2017).
  118. K. Azuma, K. Tamaki, and H.-K. Lo, All-photonic quantum repeaters, Nat. Commun. 6, 6787 (2015).
  119. Y. Hasegawa, R. Ikuta, N. Matsuda, K. Tamaki, H.-K. Lo, T. Yamamoto, K. Azuma, and N. Imoto, Experimental time-reversed adaptive Bell measurement towards all-photonic quantum repeaters, Nat. Commun. 10, 378 (2019).
  120. Z.-D. Li, R. Zhang, X.-F. Yin, L.-Z. Liu, Y. Hu, Y.-Q. Fang, Y.-Y. Fei, X. Jiang, J. Zhang, L. Li, N.-L. Liu, F. Xu, Y.-A. Chen, and J.-W. Pan, Experimental quantum repeater without quantum memory, Nat. Photonics 13, 644 (2019).
  121. S. Takeuchi, Photonic quantum information: Science and technology, Proc. Jpn. Acad., Ser. B 92, 29 (2016).
  122. J. Wang, S. Paesani, Y. Ding, R. Santagati, P. Skrzypczyk, A. Salavrakos, J. Tura, R. Augusiak, L. Mančinska, D. Bacco, D. Bonneau, J. W. Silverstone, Q. Gong, A. Acín, K. Rottwitt, L. K. Oxenløwe, J. L. O’Brien, A. Laing, and M. G. Thompson, Multidimensional quantum entanglement with large-scale integrated optics, Science 360, 285 (2018).
  123. D. Llewellyn, Y. Ding, I. I. Faruque, S. Paesani, D. Bacco, R. Santagati, Y.-J. Qian, Y. Li, Y.-F. Xiao, M. Huber et al., Chip-to-chip quantum teleportation and multi-photon entanglement in silicon, Nat. Phys. 16, 148 (2020).
  124. Y. Ding, D. Bacco, K. Dalgaard, X. Cai, X. Zhou, K. Rottwitt, and L. K. Oxenløwe, High-dimensional quantum key distribution based on multicore fiber using silicon photonic integrated circuits, npj Quantum Inf. 3, 25 (2017).
  125. W. Liu, M. Li, R. S. Guzzon, E. J. Norberg, J. S. Parker, M. Lu, L. A. Coldren, and J. Yao, A fully reconfigurable photonic integrated signal processor, Nat. Photonics 10, 190 (2016).
  126. J.-Z. Zhang, X.-B. Xu, Y. Gong, Z.-B. Wang, X.-Z. Qi, X.-J. Liu, Y.-H. Yang, Z.-H. Tian, J.-Q. Wang, Y.-L. Zhang, M. Li, Y. Guo, Y. Yan, C.-H. Dong, X.-F. Ren, Y. Zhang, C. Zhang, G.-C. Guo, Y. Che, and C.-L. Zou, Optically-driven organic nano-step actuator for reconfigurable photonic circuits, Nat. Commun. 16, 8213 (2025).
  127. D. Pérez-López and L. Torrijos-Morán, Large-scale photonic processors and their applications, npj Nanophotonics 2, 32 (2025).
  128. G. J. Mendoza, R. Santagati, J. Munns, E. Hemsley, M. Piekarek, E. Martín-López, G. D. Marshall, D. Bonneau, M. G. Thompson, and J. L. O’Brien, Active temporal and spatial multiplexing of photons, Optica 3, 127 (2016).
  129. R. Quan, Y. Zhai, M. Wang, F. Hou, S. Wang, X. Xiang, T. Liu, S. Zhang, and R. Dong, Demonstration of quantum synchronization based on second-order quantum coherence of entangled photons, Sci. Rep. 6, 30453 (2016).
  130. J.-H. Kim, J.-W. Chae, Y.-C. Jeong, and Y.-H. Kim, Quantum communication with time-bin entanglement over a wavelength-multiplexed fiber network, APL Photonics 7, 016106 (2022).
  131. N. Lal, I. A. Burenkov, Y.-S. Li-Baboud, M. V. Jabir, P. S. Kuo, T. Gerrits, O. Slattery, and S. V. Polyakov, Synchronized source of indistinguishable photons for quantum networks, Opt. Express 32, 18257 (2024).
  132. Mš Rančić, M. P. Hedges, R. L. Ahlefeldt, and M. J. Sellars, Coherence time of over a second in a telecom-compatible quantum memory storage material, Nat. Phys. 14, 50 (2018).
  133. Y. Ma, Y.-Z. Ma, Z.-Q. Zhou, C.-F. Li, and G.-C. Guo, One-hour coherent optical storage in an atomic frequency comb memory, Nat. Commun. 12, 2381 (2021).
  134. Y.-P. Liu, Z.-W. Ou, T.-X. Zhu, M.-X. Su, C. Liu, Y.-J. Han, Z.-Q. Zhou, C.-F. Li, and G.-C. Guo, A millisecond integrated quantum memory for photonic qubits, Sci. Adv. 11, adu5264 (2025).
  135. D. Cozzolino, B. Da Lio, D. Bacco, and L. K. Oxenløwe, High-dimensional quantum communication: Benefits, progress, and future challenges, Adv. Quantum Technol. 2, 1900038 (2019).
  136. Y. Wang, Z. Hu, B. C. Sanders, and S. Kais, Qudits and high-dimensional quantum computing, Front. Phys. 8, 589504 (2020).
  137. R. T. Thew, K. Nemoto, A. G. White, and W. J. Munro, Qudit quantum-state tomography, Phys. Rev. A 66, 012303 (2002).
  138. D. Martínez, L. Pereira, K. Sawada, P. González, J. Cariñe, M. Muñoz, A. Delgado, E. S. Gómez, S. P. Walborn, and G. Lima, Experimental qudit state estimation via point tomography, Phys. Rev. Res. 7, 033119 (2025).
  139. G. Garberoglio, M. Dapor, D. Maragnano, M. Liscidini, and D. Binosi, Enhanced compressive threshold quantum state tomography for qudit systems, Phys. Rev. A 111, 032436 (2025).
  140. Y. Kakihara, D. Yamamoto, and G. Marmorini, Compressed sensing quantum state tomography for qudits: A comparison of Gell-Mann and Heisenberg-Weyl observable bases, arXiv:2505.10462.
  141. J. C. Garcia-Escartin and P. Chamorro-Posada, swap test and Hong-Ou-Mandel effect are equivalent, Phys. Rev. A 87, 052330 (2013).
  142. J. Kadlec, A. Barasiński, and K. Lemr, Resource-efficient entanglement detection in high-dimensional states via two-qubit witnesses, arXiv:2509.13061.
  143. T. Kraft, C. Ritz, N. Brunner, M. Huber, and O. Gühne, Characterizing genuine multilevel entanglement, Phys. Rev. Lett. 120, 060502 (2018).
  144. W. Huggins, P. Patil, B. Mitchell, K. B. Whaley, and E. M. Stoudenmire, Towards quantum machine learning with tensor networks, Quantum Sci. Technol. 4, 024001 (2019).
  145. A. A. Mele, G. B. Mbeng, G. E. Santoro, M. Collura, and P. Torta, Avoiding barren plateaus via transferability of smooth solutions in a Hamiltonian variational ansatz, Phys. Rev. A 106, L060401 (2022).
  146. J. Dborin, F. Barratt, V. Wimalaweera, L. Wright, and A. G. Green, Matrix product state pre-training for quantum machine learning, Quantum Sci. Technol. 7, 035014 (2022).
  147. H. R. Grimsley, G. S. Barron, E. Barnes, S. E. Economou, and N. J. Mayhall, Adaptive, problem-tailored variational quantum eigensolver mitigates rough parameter landscapes and barren plateaus, npj Quantum Inf. 9, 19 (2023).
  148. M. S. Rudolph, J. Miller, D. Motlagh, J. Chen, A. Acharya, and A. Perdomo-Ortiz, Synergistic pretraining of parametrized quantum circuits via tensor networks, Nat. Commun. 14, 8367 (2023).
  149. A. Chan, Z. Shi, L. Dellantonio, W. Dür, and C. A. Muschik, Measurement-based infused circuits for variational quantum eigensolvers, Phys. Rev. Lett. 132, 240601 (2024).
  150. Q. Miao and T. Barthel, Isometric tensor network optimization for extensive Hamiltonians is free of barren plateaus, Phys. Rev. A 109, L050402 (2024).
  151. R. Puig, M. Drudis, S. Thanasilp, and Z. B. Holmes, Variational quantum simulation: A case study for understanding warm starts, PRX Quantum 6, 010317 (2025).
  152. H.-J. Briegel, W. Dür, J. I. Cirac, and P. Zoller, Quantum repeaters: The role of imperfect local operations in quantum communication, Phys. Rev. Lett. 81, 5932 (1998).
  153. B. Hensen, H. Bernien, A. E. Dréau, A. Reiserer, N. Kalb, M. S. Blok, J. Ruitenberg, R. F. Vermeulen, R. N. Schouten, C. Abellán et al., Loophole-free bell inequality violation using electron spins separated by 1.3 kilometres, Nature (London) 526, 682 (2015).
  154. D. Niu, Y. Zhang, A. Shabani, and H. Shapourian, All-photonic one-way quantum repeaters with measurement-based error correction, npj Quantum Inf. 9, 106 (2023).
  155. N. Benchasattabuse, M. Hajdušek, and R. Van Meter, Engineering challenges in all-photonic quantum repeaters, IEEE Network 39, 132 (2025).

Outline

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PRX Quantum 7, 033038– Published 25 August, 2026
  • Received 21 October 2025
  • Revised 5 May 2026
  • Accepted 26 June 2026
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