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  • Open Access

Fusion for high-dimensional linear-optical quantum computing with improved success probability

Gözde Üstün1,2, Eleanor G. Rieffel3, Simon J. Devitt4,5, and Jason Saied3,*

  • *Contact author: jasonsaied@gmail.com
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Phys. Rev. Applied 24, 044024 – Published 8 October, 2025

DOI: https://doi.org/10.1103/l7bg-hc8c

Abstract

Type-II fusion is a probabilistic entangling measurement that is essential to measurement-based linear-optical quantum computing and can be used for quantum teleportation more broadly. However, it remains under-explored for high-dimensional qudits. Our main result gives a Type-II fusion protocol with proven success probability approximately 2/d2 for qudits of arbitrary dimension d. This generalizes a previous method that only applied to even-dimensional qudits. We believe this protocol to be the most efficient known protocol for Type-II fusion, with the d=5 case beating the previous record by a factor of approximately 723. We discuss the construction of the required (d−2)-qudit ancillary state using a silicon spin-qudit ancilla coupled to a microwave cavity through time-bin multiplexing. We then introduce a general framework of extra-dimensional corrections, a natural technique in linear optics that can be used to nondeterministically correct nonmaximally entangled projections into Bell measurements. We use this method to analyze and improve several different circuits for high-dimensional Type-II fusion and compare their benefits and drawbacks.

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References (58)

  1. T. Rudolph, Why I am optimistic about the silicon-photonic route to quantum computing, APL Photonics 2, 030901 (2017).
  2. K. Alexander et al., A manufacturable platform for photonic quantum computing, Nature 641, 876 (2025).
  3. P. Kok, W. J. Munro, K. Nemoto, T. C. Ralph, J. P. Dowling, and G. J. Milburn, Linear optical quantum computing with photonic qubits, Rev. Mod. Phys. 79, 135 (2007).
  4. G. Üstün, S. Elman, J. J. Pla, A. C. Doherty, A. Morello, and S. J. Devitt, Near-deterministic photon entanglement from a spin qudit in silicon using third quantisation, arXiv:2502.01096 [quant-ph].
  5. T. Rudolph, Perhaps they are everywhere? Undetectable distributed quantum computation and communication for alien civilizations can be established using thermal light from stars, arXiv:2107.13023 [quant-ph].
  6. M. Reck, A. Zeilinger, H. J. Bernstein, and P. Bertani, Experimental realization of any discrete unitary operator, Phys. Rev. Lett. 73, 58 (1994).
  7. Z. Raissi, E. Barnes, and S. E. Economou, Deterministic generation of qudit photonic graph states from quantum emitters, PRX Quantum 5, 020346 (2024).
  8. S. Paesani, J. F. F. Bulmer, A. E. Jones, R. Santagati, and A. Laing, Scheme for universal high-dimensional quantum computation with linear optics, Phys. Rev. Lett. 126, 230504 (2021).
  9. O. Lib and Y. Bromberg, Resource-efficient photonic quantum computation with high-dimensional cluster states, Nat. Photonics 18, 1218 (2024).
  10. M. Cabrejo-Ponce, A. L. M. Muniz, M. Huber, and F. Steinlechner, High-dimensional entanglement for quantum communication in the frequency domain, Laser Photonics Rev. 17, 2201010 (2023).
  11. M. Zahidy, D. Ribezzo, C. De Lazzari, I. Vagniluca, N. Biagi, R. Müller, T. Occhipinti, L. K. Oxenløwe, M. Galili, T. Hayashi, D. Cassioli, A. Mecozzi, C. Antonelli, A. Zavatta, and D. Bacco, Practical high-dimensional quantum key distribution protocol over deployed multicore fiber, Nat. Commun. 15, 1651 (2024).
  12. O. Lib, K. Sulimany, M. Araújo, M. Ben-Or, and Y. Bromberg, High-dimensional quantum key distribution using a multi-plane light converter, Opt. Quantum 3, 182 (2025).
  13. M. Shalaev, S. Saha, G. Toh, I. Goetting, A. Kalakuntla, H. B. Shi, J. O’Reilly, Y. Yu, and C. Monroe, Photonic networking of quantum memories in high-dimensions, arXiv:2505.11704 [quant-ph].
  14. T. Yamazaki and K. Azuma, Linear-optical fusion boosted by high-dimensional entanglement, Phys. Rev. Lett. 134, 200801 (2025).
  15. H. Chau, Five quantum register error correction code for higher spin systems, Phys. Rev. A 56, R1 (1997).
  16. S. S. Bullock and G. K. Brennen, Qudit surface codes and gauge theory with finite cyclic groups, J. Phys. A: Math. Theor. 40, 3481 (2007).
  17. S. Bartolucci, P. Birchall, H. Bombín, H. Cable, C. Dawson, M. Gimeno-Segovia, E. Johnston, K. Kieling, N. Nickerson, M. Pant, F. Pastawski, T. Rudolph, and C. Sparrow, Fusion-based quantum computation, Nat. Commun. 14, 912 (2023).
  18. H. Bombin, I. H. Kim, D. Litinski, N. Nickerson, M. Pant, F. Pastawski, S. Roberts, and T. Rudolph, Interleaving: Modular architectures for fault-tolerant photonic quantum computing, arXiv:2103.08612.
  19. D. Litinski and N. Nickerson, Active volume: An architecture for efficient fault-tolerant quantum computers with limited non-local connections, arXiv:2211.15465.
  20. H. Bombin, C. Dawson, T. Farrelly, Y. Liu, N. Nickerson, M. Pant, F. Pastawski, and S. Roberts, Fault-tolerant complexes, arXiv:2308.07844.
  21. D. E. Browne and T. Rudolph, Resource-efficient linear optical quantum computation, Phys. Rev. Lett. 95, 010501 (2005).
  22. S. Bartolucci, P. M. Birchall, M. Gimeno-Segovia, E. Johnston, K. Kieling, M. Pant, T. Rudolph, J. Smith, C. Sparrow, and M. D. Vidrighin, Creation of entangled photonic states using linear optics, arXiv:2106.13825 [quant-ph].
  23. J. Calsamiglia and N. Lütkenhaus, Maximum efficiency of a linear-optical Bell-state analyzer, Appl. Phys. B 72, 67 (2001).
  24. W. P. Grice, Arbitrarily complete Bell-state measurement using only linear optical elements, Phys. Rev. A 84, 042331 (2011).
  25. F. Ewert and P. van Loock, 3/4-efficient Bell measurement with passive linear optics and unentangled ancillae, Phys. Rev. Lett. 113, 140403 (2014).
  26. S. K. Goyal, P. E. Boukama-Dzoussi, S. Ghosh, F. S. Roux, and T. Konrad, Qudit-teleportation for photons with linear optics, Sci. Rep. 4, 4543 (2014).
  27. S. Goel, S. Leedumrongwatthanakun, N. H. Valencia, W. McCutcheon, A. Tavakoli, C. Conti, P. W. H. Pinkse, and M. Malik, Inverse design of high-dimensional quantum optical circuits in a complex medium, Nat. Phys. 20, 232 (2024).
  28. N. Bharos, L. Markovich, and J. Borregaard, Efficient high-dimensional entangled state analyzer with linear optics, Quantum 9, 1711 (2025).
  29. N. Bharos, Master thesis, High-dimensional entanglement generation with linear optics Delft University of Technology, 2023.
  30. Y.-H. Luo, H.-S. Zhong, M. Erhard, X.-L. Wang, L.-C. Peng, M. Krenn, X. Jiang, L. Li, N.-L. Liu, C.-Y. Lu, A. Zeilinger, and J.-W. Pan, Quantum teleportation in high dimensions, Phys. Rev. Lett. 123, 070505 (2019).
  31. M. Heinrich, Ph.D. thesis, On stabiliser techniques and their application to simulation and certification of quantum devices, Mathematisch-Naturwissenschaftlichen Fakultät der Universität zu Köln, Universität zu Köln, 2021.
  32. D. McNulty and S. Weigert, Mutually unbiased bases in composite dimensions—a review, arXiv:2410.23997.
  33. M. Rambach, M. Qaryan, M. Kewming, C. Ferrie, A. G. White, and J. Romero, Robust and efficient high-dimensional quantum state tomography, Phys. Rev. Lett. 126, 100402 (2021).
  34. G. Lima, L. Neves, R. Guzmán, E. S. Gómez, W. Nogueira, A. Delgado, A. Vargas, and C. Saavedra, Experimental quantum tomography of photonic qudits via mutually unbiased basis, Opt. Express 19, 3542 (2011).
  35. N. Bent, H. Qassim, A. Tahir, D. Sych, G. Leuchs, L. L. Sánchez-Soto, E. Karimi, and R. Boyd, Experimental realization of quantum tomography of photonic qudits via symmetric informationally complete positive operator-valued measures, Phys. Rev. X 5, 041006 (2015).
  36. F. C. Peres, Pauli-based model of quantum computation with higher-dimensional systems, Phys. Rev. A 108, 032606 (2023).
  37. G. Vidal, Entanglement of pure states for a single copy, Phys. Rev. Lett. 83, 1046 (1999).
  38. E. Knill, R. Laflamme, and G. J. Milburn, A scheme for efficient quantum computation with linear optics, Nature 409, 46 (2001).
  39. C. Zhang, J. Chen, C. Cui, J. P. Dowling, Z. Ou, and T. Byrnes, Quantum teleportation of photonic qudits using linear optics, Phys. Rev. A 100, 032330 (2019).
  40. X. Yu, B. Wilhelm, D. Holmes, A. Vaartjes, D. Schwienbacher, M. Nurizzo, A. Kringhøj, M. R. van Blankenstein, A. M. Jakob, P. Gupta, F. E. Hudson, K. M. Itoh, R. J. Murray, R. Blume-Kohout, T. D. Ladd, N. Anand, A. S. Dzurak, B. C. Sanders, D. N. Jamieson, and A. Morello, Schrödinger cat states of a nuclear spin qudit in silicon, Nat. Phys. 21, 362 (2025).
  41. I. Fernández de Fuentes, T. Botzem, M. A. I. Johnson, A. Vaartjes, S. Asaad, V. Mourik, F. E. Hudson, K. M. Itoh, B. C. Johnson, A. M. Jakob, J. C. McCallum, D. N. Jamieson, A. S. Dzurak, and A. Morello, Navigating the 16-dimensional Hilbert space of a high-spin donor qudit with electric and magnetic fields, Nat. Commun. 15, 1380 (2024).
  42. S. E. D’Aurelio, M. J. Bayerbach, and S. Barz, Boosted quantum teleportation, npj Quantum Inf. 11, 37 (2025).
  43. N. Hauser, M. J. Bayerbach, S. E. D’Aurelio, R. Weber, M. Santandrea, S. P. Kumar, I. Dhand, and S. Barz, Boosted Bell-state measurements for photonic quantum computation, npj Quantum Inf. 11, 41 (2025).
  44. R. Barak and Y. Ben-Aryeh, Quantum fast Fourier transform and quantum computation by linear optics, J. Opt. Soc. Am. B 24, 231 (2007).
  45. 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).
  46. D. Smith, C. Taballione, M. C. Anguita, M. De Goede, P. Venderbosch, B. Kassenberg, H. Snijders, J. P. Epping, R. van der Meer, P. W. Pinkse et al., A Universal 20-mode Quantum Photonic Processor in Silicon Nitride, in Quantum 2.0 (Optica Publishing Group, Washington DC, 2022), p. QW4B–2, ISBN: 978-1-957171-11-1.
  47. X.-L. Wang, L.-K. Chen, W. Li, H.-L. Huang, C. Liu, C. Chen, Y.-H. Luo, Z.-E. Su, D. Wu, Z.-D. Li et al., Experimental ten-photon entanglement, Phys. Rev. Lett. 117, 210502 (2016).
  48. P. G. Kwiat, K. Mattle, H. Weinfurter, A. Zeilinger, A. V. Sergienko, and Y. Shih, New high-intensity source of polarization-entangled photon pairs, Phys. Rev. Lett. 75, 4337 (1995).
  49. M. Michler, K. Mattle, H. Weinfurter, and A. Zeilinger, Interferometric Bell-state analysis, Phys. Rev. A 53, R1209 (1996).
  50. G. Üstün, E. G. Rieffel, S. J. Devitt, and J. Saied, Ancillary files for “Fusion for high-dimensional linear optical quantum computing with improved success probability”, 2025, available at https://arxiv.org/src/2505.16816v2/anc
  51. M. C. Tichy, M. Tiersch, F. de Melo, F. Mintert, and A. Buchleitner, Zero-transmission law for multiport beam splitters, Phys. Rev. Lett. 104, 220405 (2010).
  52. J. Saied, J. Marshall, N. Anand, and E. G. Rieffel, General protocols for the efficient distillation of indistinguishable photons, Phys. Rev. Appl. 23, 034079 (2025).
  53. W. Vine, A. Kringhøj, M. Savytskyi, D. Parker, T. Schenkel, B. C. Johnson, J. C. McCallum, A. Morello, and J. J. Pla, Latched detection of zeptojoule spin echoes with a kinetic inductance parametric oscillator, Sci. Adv. 10, eadm7624 (2024).
  54. G. Tosi, F. A. Mohiyaddin, V. Schmitt, S. Tenberg, R. Rahman, G. Klimeck, and A. Morello, Silicon quantum processor with robust long-distance qubit couplings, Nat. Commun. 8, 450 (2017).
  55. G. Tosi, F. A. Mohiyaddin, H. Huebl, and A. Morello, Circuit-quantum electrodynamics with direct magnetic coupling to single-atom spin qubits in isotopically enriched 28Si, AIP Adv. 4, 087122 (2014).
  56. S. Asaad, V. Mourik, B. Joecker, M. A. Johnson, A. D. Baczewski, H. R. Firgau, M. T. Madzik, V. Schmitt, J. J. Pla, F. E. Hudson et al., Coherent electrical control of a single high-spin nucleus in silicon, Nature 579, 205 (2020).
  57. B. Joecker, H. G. Stemp, I. Fernández de Fuentes, M. A. Johnson, and A. Morello, Error channels in quantum nondemolition measurements on spin systems, Phys. Rev. B 109, 085302 (2024).
  58. C. Dittel, G. Dufour, M. Walschaers, G. Weihs, A. Buchleitner, and R. Keil, Totally destructive interference for permutation-symmetric many-particle states, Phys. Rev. A 97, 062116 (2018).

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Phys. Rev. Applied 24, 044024– Published 8 October, 2025
  • Received 29 May 2025
  • Revised 26 August 2025
  • Accepted 28 August 2025
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