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Generation of Frequency-Bin-Encoded Dual-Rail Cluster States via Time-Frequency Multiplexing of Microwave Photonic Qubits

Zhiling Wang1,*, Takeaki Miyamura2, Yoshiki Sunada3, Keika Sunada2, Jesper Ilves2, Kohei Matsuura2, and Yasunobu Nakamura1,2

  • *Contact author: zhiling.wang@riken.jp
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PRX Quantum 7, 010330 – Published 12 February, 2026

DOI: https://doi.org/10.1103/rrct-dpfv

Abstract

Cluster states are a class of multiqubit entangled states with broad applications such as quantum metrology and one-way quantum computing. Here, we present a protocol to generate frequency-bin-encoded dual-rail cluster states using a superconducting circuit consisting of a fixed-frequency transmon qubit, a resonator, and a Purcell filter. We implement time-frequency multiplexing by sequentially emitting copropagating microwave photons, where each time bin contains a dual-rail qubit encoded in two distinct frequency modes, and adjacent time bins are entangled to form the cluster state. The frequency-bin dual-rail encoding enables erasure detection based on photon occupancy. We characterize the state fidelity using quantum tomography and quantify the multipartite entanglement using localizable entanglement. Our implementation achieves a state fidelity exceeding 50% for a cluster state consisting of up to four logical qubits. The localizable entanglement remains across chains of up to seven logical qubits. After discarding the erasure errors, the fidelity exceeds 50% for states with up to eight logical qubits, and the entanglement persists across chains of up to 11 qubits. These results highlight the improved robustness of frequency-bin dual-rail encoding against photon loss compared to conventional single-rail schemes. This work provides a scalable pathway toward high-dimensional entangled state generation and photonic quantum information processing in the microwave domain.

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Popular Summary

Quantum entanglement is a cornerstone of quantum physics and an essential resource for quantum computing and quantum information processing. Among various entangled states, cluster states stand out because they have broad applications such as quantum metrology and one-way quantum computing. However, creating such entangled states in a way that is resilient to photon loss remains a major challenge.

In this work, we demonstrate the generation of microwave photonic cluster states using a superconducting circuit. Instead of encoding information in the simple presence or absence of a photon, we use pairs of photons of distinct frequencies, known as frequency-bin encoding. This encoding method supports dual-rail configuration, enabling photon-loss detection. Using this method, we demonstrate the creation of multiqubit cluster states with enhanced fidelities and show that entanglement persists over longer chains with this photon-loss detection.

Looking ahead, this technique offers a promising path toward building larger and more complex entangled states in the microwave domain. Moreover, because it is compatible with real-time error detection, it could become an essential ingredient for scalable fault-tolerant quantum computing and future quantum networks.

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References (75)

  1. O. Gühne and G. Tóth, Entanglement detection, Phys. Rep. 474, 1 (2009).
  2. J. Kempe, Multiparticle entanglement and its applications to cryptography, Phys. Rev. A 60, 910 (1999).
  3. R. Horodecki, P. Horodecki, M. Horodecki, and K. Horodecki, Quantum entanglement, Rev. Mod. Phys. 81, 865 (2009).
  4. H. J. Briegel and R. Raussendorf, Persistent entanglement in arrays of interacting particles, Phys. Rev. Lett. 86, 910 (2001).
  5. R. Raussendorf and H. J. Briegel, A one-way quantum computer, Phys. Rev. Lett. 86, 5188 (2001).
  6. R. Raussendorf, D. E. Browne, and H. J. Briegel, Measurement-based quantum computation on cluster states, Phys. Rev. A 68, 022312 (2003).
  7. H. J. Briegel, D. E. Browne, W. Dür, R. Raussendorf, and M. Van den Nest, Measurement-based quantum computation, Nat. Phys. 5, 19 (2009).
  8. D. Schlingemann and R. F. Werner, Quantum error-correcting codes associated with graphs, Phys. Rev. A 65, 012308 (2001).
  9. B. A. Bell, D. Herrera-Martí, M. Tame, D. Markham, W. Wadsworth, and J. Rarity, Experimental demonstration of a graph state quantum error-correction code, Nat. Commun. 5, 3658 (2014).
  10. N. Friis, D. Orsucci, M. Skotiniotis, P. Sekatski, V. Dunjko, H. J. Briegel, and W. Dür, Flexible resources for quantum metrology, New J. Phys. 19, 063044 (2017).
  11. N. Shettell and D. Markham, Graph states as a resource for quantum metrology, Phys. Rev. Lett. 124, 110502 (2020).
  12. D. Markham and B. C. Sanders, Graph states for quantum secret sharing, Phys. Rev. A 78, 042309 (2008).
  13. B. Bell, D. Markham, D. Herrera-Martí, A. Marin, W. Wadsworth, J. Rarity, and M. Tame, Experimental demonstration of graph-state quantum secret sharing, Nat. Commun. 5, 1 (2014).
  14. K. Azuma, K. Tamaki, and H.-K. Lo, All-photonic quantum repeaters, Nat. Commun. 6, 1 (2015).
  15. E. Kuznetsova, T. Bragdon, R. Côté, and S. F. Yelin, Cluster-state generation using van der waals and dipole-dipole interactions in optical lattices, Phys. Rev. A 85, 012328 (2012).
  16. I. Schwartz, D. Cogan, E. R. Schmidgall, Y. Don, L. Gantz, O. Kenneth, N. H. Lindner, and D. Gershoni, Deterministic generation of a cluster state of entangled photons, Science 354, 434 (2016).
  17. J.-C. Besse, K. Reuer, M. C. Collodo, A. Wulff, L. Wernli, A. Copetudo, D. Malz, P. Magnard, A. Akin, M. Gabureac, G. J. Norris, J. I. Cirac, A. Wallraff, and C. Eichler, Realizing a deterministic source of multipartite-entangled photonic qubits, Nat. Commun. 11, 4877 (2020).
  18. D. Istrati, Y. Pilnyak, J. Loredo, C. Antón, N. Somaschi, P. Hilaire, H. Ollivier, M. Esmann, L. Cohen, L. Vidro, C. Millet, A. Lemaître, I. Sagnes, A. Harouri, L. Lanco, P. Senellart, and H. Eisenberg, Sequential generation of linear cluster states from a single photon emitter, Nat. Commun. 11, 5501 (2020).
  19. P. Thomas, L. Ruscio, O. Morin, and G. Rempe, Efficient generation of entangled multiphoton graph states from a single atom, Nature 608, 677 (2022).
  20. D. Cogan, Z.-E. Su, O. Kenneth, and D. Gershoni, Deterministic generation of indistinguishable photons in a cluster state, Nat. Photonics 17, 324 (2023).
  21. M. Banic, V. Crescimanna, J. E. Bourassa, C. Gonzalez-Arciniegas, R. N. Alexander, and K. Heshami, Exact simulation of realistic Gottesman-Kitaev-Preskill cluster states, Phys. Rev. A 112, 052425 (2025).
  22. P. Kurpiers, M. Pechal, B. Royer, P. Magnard, T. Walter, J. Heinsoo, Y. Salathé, A. Akin, S. Storz, J. C. Besse, S. Gasparinetti, A. Blais, and A. Wallraff, Quantum communication with time-bin encoded microwave photons, Phys. Rev. Appl. 12, 044067 (2019).
  23. J. Ilves, S. Kono, Y. Sunada, S. Yamazaki, M. Kim, K. Koshino, and Y. Nakamura, On-demand generation and characterization of a microwave time-bin qubit, Npj Quantum Inf. 6, 34 (2020).
  24. G. F. Peñas, R. Puebla, and J. J. García-Ripoll, Multiplexed quantum state transfer in waveguides, Phys. Rev. Res. 6, 033294 (2024).
  25. B. Kannan, D. L. Campbell, F. Vasconcelos, R. Winik, D. K. Kim, M. Kjaergaard, P. Krantz, A. Melville, B. M. Niedzielski, J. L. Yoder, T. P. Orlando, S. Gustavsson, and W. D. Oliver, Generating spatially entangled itinerant photons with waveguide quantum electrodynamics, Sci. Adv. 6, eabb8780 (2020).
  26. J. Yang, M. Khanahmadi, I. Strandberg, A. Gaikwad, C. Castillo-Moreno, A. F. Kockum, M. A. Ullah, G. Johansson, A. M. Eriksson, and S. Gasparinetti, Deterministic generation of frequency-bin-encoded microwave photons, Phys. Rev. Lett. 134, 240803 (2025).
  27. M. Grassl, T. Beth, and T. Pellizzari, Codes for the quantum erasure channel, Phys. Rev. A 56, 33 (1997).
  28. E. Knill, Scalable quantum computing in the presence of large detected-error rates, Phys. Rev. A 71, 042322 (2005).
  29. J. D. Teoh, P. Winkel, H. K. Babla, B. J. Chapman, J. Claes, S. J. de Graaf, J. W. Garmon, W. D. Kalfus, Y. Lu, A. Maiti et al., Dual-rail encoding with superconducting cavities, Proc. Natl. Acad. Sci. U.S.A. 120, e2221736120 (2023).
  30. N. Mehta, J. D. Teoh, T. Noh, A. Agrawal, A. Anderson, B. Birdsall, A. Brahmbhatt, W. Byrd, M. Cacioppo, A. Cabrera et al., Bias-preserving and error-detectable entangling operations in a superconducting dual-rail system, arXiv:2503.10935.
  31. W. Huang, X. Sun, J. Zhang, Z. Guo, P. Huang, Y. Liang, Y. Liu, D. Sun, Z. Wang, Y. Xiong et al., Logical multi-qubit entanglement with dual-rail superconducting qubits, arXiv:2504.12099.
  32. K. S. Chou, T. Shemma, H. McCarrick, T.-C. Chien, J. D. Teoh, P. Winkel, A. Anderson, J. Chen, J. C. Curtis, S. J. de Graaf et al., A superconducting dual-rail cavity qubit with erasure-detected logical measurements, Nat. Phys. 20, 1454 (2024).
  33. H. Levine, A. Haim, J. S. Hung, N. Alidoust, M. Kalaee, L. DeLorenzo, E. A. Wollack, P. Arrangoiz-Arriola, A. Khalajhedayati, R. Sanil et al., Demonstrating a long-coherence dual-rail erasure qubit using tunable transmons, Phys. Rev. X 14, 011051 (2024).
  34. A. Koottandavida, I. Tsioutsios, A. Kargioti, C. R. Smith, V. R. Joshi, W. Dai, J. D. Teoh, J. C. Curtis, L. Frunzio, R. J. Schoelkopf, and M. H. Devoret, Erasure detection of a dual-rail qubit encoded in a double-post superconducting cavity, Phys. Rev. Lett. 132, 180601 (2024).
  35. M. Pechal, L. Huthmacher, C. Eichler, S. Zeytinoğlu, A. A. Abdumalikov, S. Berger, A. Wallraff, and S. Filipp, Microwave-controlled generation of shaped single photons in circuit quantum electrodynamics, Phys. Rev. X 4, 041010 (2014).
  36. S. Zeytinoğlu, M. Pechal, S. Berger, A. A. Abdumalikov, A. Wallraff, and S. Filipp, Microwave-induced amplitude- and phase-tunable qubit-resonator coupling in circuit quantum electrodynamics, Phys. Rev. A 91, 043846 (2015).
  37. T. Miyamura, Y. Sunada, Z. Wang, J. Ilves, K. Matsuura, and Y. Nakamura, Generation of frequency-tunable shaped single microwave photons using a fixed-frequency superconducting qubit, PRX Quantum 6, 020347 (2025).
  38. N. H. Lindner and T. Rudolph, Proposal for pulsed on-demand sources of photonic cluster state strings, Phys. Rev. Lett. 103, 113602 (2009).
  39. J. Koch, T. M. Yu, J. Gambetta, A. A. Houck, D. I. Schuster, J. Majer, A. Blais, M. H. Devoret, S. M. Girvin, and R. J. Schoelkopf, Charge-insensitive qubit design derived from the cooper pair box, Phys. Rev. A 76, 042319 (2007).
  40. Y. Sunada, S. Kono, J. Ilves, S. Tamate, T. Sugiyama, Y. Tabuchi, and Y. Nakamura, Fast readout and reset of a superconducting qubit coupled to a resonator with an intrinsic purcell filter, Phys. Rev. Appl. 17, 044016 (2022).
  41. C. Eichler, D. Bozyigit, C. Lang, L. Steffen, J. Fink, and A. Wallraff, Experimental state tomography of itinerant single microwave photons, Phys. Rev. Lett. 106, 220503 (2011).
  42. C. Eichler, C. Lang, J. M. Fink, J. Govenius, S. Filipp, and A. Wallraff, Observation of entanglement between itinerant microwave photons and a superconducting qubit, Phys. Rev. Lett. 109, 240501 (2012).
  43. T. Baumgratz, Efficient system identification and characterization for quantum many-body systems, Ph.D. thesis, Universität Ulm (2014).
  44. F. Verstraete, M. Popp, and J. I. Cirac, Entanglement versus correlations in spin systems, Phys. Rev. Lett. 92, 027901 (2004).
  45. M. Popp, F. Verstraete, M. A. Martín-Delgado, and J. I. Cirac, Localizable entanglement, Phys. Rev. A 71, 042306 (2005).
  46. T. Miyamura, Generation of frequency-bin microwave photons enabled by a broadband resonator, Oral presentation at the SQA Conference, Delft, The Netherlands (2025), available at https://www.youtube.com/watch?v=am26Q7Yi60c.
  47. K. Reuer, J.-C. Besse, L. Wernli, P. Magnard, P. Kurpiers, G. J. Norris, A. Wallraff, and C. Eichler, Realization of a universal quantum gate set for itinerant microwave photons, Phys. Rev. X 12, 011008 (2022).
  48. M. Jerger, P. Macha, A. R. Hamann, Y. Reshitnyk, K. Juliusson, and A. Fedorov, Realization of a binary-outcome projection measurement of a three-level superconducting quantum system, Phys. Rev. Appl. 6, 014014 (2016).
  49. R. Raussendorf, J. Harrington, and K. Goyal, A fault-tolerant one-way quantum computer, Ann. Phys. 321, 2242 (2006).
  50. M. P. Bland, F. Bahrami, J. G. Martinez, P. H. Prestegaard, B. M. Smitham, A. Joshi, E. Hedrick, S. Kumar, A. Yang, A. C. Pakpour-Tabrizi et al., Millisecond lifetimes and coherence times in 2d transmon qubits, Nature 647, 343 (2025).
  51. M. Tuokkola, Y. Sunada, H. Kivijärvi, J. Albanese, L. Grönberg, J.-P. Kaikkonen, V. Vesterinen, J. Govenius, and M. Möttönen, Methods to achieve near-millisecond energy relaxation and dephasing times for a superconducting transmon qubit, Nat. Commun. 16, 5421 (2025).
  52. M. Bal, A. A. Murthy, S. Zhu, F. Crisa, X. You, Z. Huang, T. Roy, J. Lee, D. v. Zanten, R. Pilipenko et al., Systematic improvements in transmon qubit coherence enabled by niobium surface encapsulation, Npj Quantum Inf. 10, 43 (2024).
  53. C. Wang, Y. Y. Gao, I. M. Pop, U. Vool, C. Axline, T. Brecht, R. W. Heeres, L. Frunzio, M. H. Devoret, G. Catelani et al., Measurement and control of quasiparticle dynamics in a superconducting qubit, Nat. Commun. 5, 5836 (2014).
  54. M. McEwen, K. C. Miao, J. Atalaya, A. Bilmes, A. Crook, J. Bovaird, J. M. Kreikebaum, N. Zobrist, E. Jeffrey, B. Ying et al., Resisting high-energy impact events through gap engineering in superconducting qubit arrays, Phys. Rev. Lett. 133, 240601 (2024).
  55. T. Connolly, P. D. Kurilovich, S. Diamond, H. Nho, Charlotte G. L. Bøttcher, L. I. Glazman, V. Fatemi, and M. H. Devoret, Coexistence of nonequilibrium density and equilibrium energy distribution of quasiparticles in a superconducting qubit, Phys. Rev. Lett. 132, 217001 (2024).
  56. D. Colao Zanuz, Q. Ficheux, L. Michaud, A. Orekhov, K. Hanke, A. Flasby, M. Bahrami Panah, G. J. Norris, M. Kerschbaum, A. Remm et al., Mitigating losses of superconducting qubits strongly coupled to defect modes, Phys. Rev. Appl. 23, 044054 (2025).
  57. M. Chen, J. C. Owens, H. Putterman, M. Schäfer, and O. Painter, Phonon engineering of atomic-scale defects in superconducting quantum circuits, Sci. Adv. 10, eado6240 (2024).
  58. J. Lisenfeld, A. Bilmes, and A. V. Ustinov, Enhancing the coherence of superconducting quantum bits with electric fields, Npj Quantum Inf. 9, 8 (2023).
  59. M. V. Larsen, X. Guo, C. R. Breum, J. S. Neergaard-Nielsen, and U. L. Andersen, Deterministic generation of a two-dimensional cluster state, Science 366, 369 (2019).
  60. W. Asavanant, Y. Shiozawa, S. Yokoyama, B. Charoensombutamon, H. Emura, R. N. Alexander, S. Takeda, J. ichi Yoshikawa, N. C. Menicucci, H. Yonezawa, and A. Furusawa, Generation of time-domain-multiplexed two-dimensional cluster state, Science 366, 373 (2019).
  61. C. Roh, G. Gwak, Y.-D. Yoon, and Y.-S. Ra, Generation of three-dimensional cluster entangled state, Nat. Photonics 19, 526 (2025).
  62. V. S. Ferreira, G. Kim, A. Butler, H. Pichler, and O. Painter, Deterministic generation of multidimensional photonic cluster states with a single quantum emitter, Nat. Phys. 20, 865 (2024).
  63. J. O’Sullivan, K. Reuer, A. Grigorev, X. Dai, A. Hernández-Antón, M. H. Muñoz-Arias, C. Hellings, A. Flasby, D. Colao Zanuz, J.-C. Besse, A. Blais, D. Malz, C. Eichler, and A. Wallraff, Deterministic generation of a 20-qubit two-dimensional photonic cluster state, Nat. Commun. 16, 5505 (2025).
  64. R. Raussendorf, J. Harrington, and K. Goyal, Topological fault-tolerance in cluster state quantum computation, New J. Phys. 9, 199 (2007).
  65. M. Pechal, Microwave photonics in superconducting circuits, Ph.D. thesis, ETH Zurich (2016).
  66. L. Cohen, in Proceedings of the 1998 IEEE International Conference on Acoustics, Speech and Signal Processing, ICASSP’98 (Cat. No. 98CH36181) (IEEE, Seattle, WA, USA, 1998), Vol. 3, p. 1577.
  67. Y. Sunada, S. Kono, J. Ilves, T. Sugiyama, Y. Suzuki, T. Okubo, S. Tamate, Y. Tabuchi, and Y. Nakamura, Efficient tomography of microwave photonic cluster states, arXiv:2410.03345.
  68. M. Hein, J. Eisert, and H. J. Briegel, Multiparty entanglement in graph states, Phys. Rev. A 69, 062311 (2004).
  69. S. Diamond, E. Chu, and S. Boyd, CVXPY: A Python-embedded modeling language for convex optimization, version 0.2 (2014), available at http://cvxpy.org/.
  70. M. P. Da Silva, D. Bozyigit, A. Wallraff, and A. Blais, Schemes for the observation of photon correlation functions in circuit QED with linear detectors, Phys. Rev. A 82, 043804 (2010).
  71. C. Eichler, Experimental characterization of quantum microwave radiation and its entanglement with a superconducting qubit, Ph.D. thesis, ETH Zurich (2013).
  72. C. Schön, E. Solano, F. Verstraete, J. I. Cirac, and M. M. Wolf, Sequential generation of entangled multiqubit states, Phys. Rev. Lett. 95, 110503 (2005).
  73. T. Baumgratz, D. Gross, M. Cramer, and M. B. Plenio, Scalable reconstruction of density matrices, Phys. Rev. Lett. 111, 020401 (2013).
  74. J. Biamonte, Lectures on quantum tensor networks, arXiv:1912.10049.
  75. A. H. Kiilerich and K. Mølmer, Quantum interactions with pulses of radiation, Phys. Rev. A 102, 023717 (2020).

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PRX Quantum 7, 010330– Published 12 February, 2026
  • Received 21 August 2025
  • Revised 18 November 2025
  • Accepted 6 January 2026
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