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Heralded high-dimensional photon–photon quantum gate

Nature Photonics (2026)Cite this article

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Abstract

High-dimensional encoding of quantum information holds the potential to greatly increase the computational power of existing devices by enlarging the accessible state space for a fixed register size and by reducing the number of required entangling gates. However, qudit-based quantum computation remains far less developed than conventional qubit-based approaches, particularly for photons, which represent natural multilevel information carriers that play a crucial role in the development of quantum networks. A major obstacle for realizing quantum gates between two individual photons is the restriction of direct interaction between photons in linear media. In particular, essential logic components for quantum operations such as native qudit–qudit entangling gates are still missing for optical quantum information processing. Here we address this challenge by presenting a protocol for realizing an entangling gate—the controlled phase-flip gate—for two photonic qudits in an arbitrary dimension. We experimentally demonstrate this protocol by realizing a four-dimensional qudit–qudit controlled phase-flip gate, whose decomposition would require at least 13 two-qubit entangling gates. Our photonic qudits are encoded in orbital angular momentum, and we have developed a new active high-precision phase-locking technology to construct a high-dimensional orbital angular momentum beamsplitter that increases the stability of the controlled phase-flip gate, resulting in a process fidelity within a range of [0.71 ± 0.01, 0.85 ± 0.01]. Our experiment represents an important advance for high-dimensional optical quantum information processing and has the potential for wider applications beyond optical system.

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Fig. 1: Protocol for realizing a two-qudit CPF gate.
Fig. 2: Structural diagram and operation principle of OAM HD beamsplitter with two input ports (A and B) and two output ports (C and D) for d = 4.
Fig. 3: Experimental setup for heralded four-dimensional OAM CPF gate.
Fig. 4: Experimental results to calculate the process fidelity of the realized heralded CPF gate.
Fig. 5: Operation of heralded CPF gate on two photons with seven input OAM superposition states.

Data availability

All data supporting the findings of this study are provided within the Article. Source data are provided with this paper. They are also available via Figshare at https://doi.org/10.6084/m9.figshare.30946265 (ref. 66).

References

  1. Vertesi, T., Pironio, S. & Brunner, N. Closing the detection loophole in Bell experiments using qudits. Phys. Rev. Lett. 104, 060401 (2010).

    Article  ADS  Google Scholar 

  2. Dada, A. C., Leach, J., Buller, G. S., Padgett, M. J. & Andersson, E. Experimental high-dimensional two-photon entanglement and violations of generalized Bell inequalities. Nat. Phys. 7, 677–680 (2011).

    Article  Google Scholar 

  3. Malik, M., Erhard, M., Huber, M., Krenn, M., Fickler, R. & Zeilinger, A. Multi-photon entanglement in high dimensions. Nat. Photon. 10, 248–252 (2016).

    Article  ADS  Google Scholar 

  4. Martin, A. et al. Quantifying photonic high-dimensional entanglement. Phys. Rev. Lett. 118, 110501 (2017).

    Article  ADS  Google Scholar 

  5. Erhard, M., Malik, M., Krenn, M. & Zeilinger, A. Experimental Greenberger-Horne-Zeilinger entanglement beyond qubits. Nat. Photon. 12, 759–764 (2018).

    Article  ADS  Google Scholar 

  6. Erhard, M., Krenn, M. & Zeilinger, A. Advances in high-dimensional quantum entanglement. Nat. Rev. Phys. 2, 365–381 (2020).

    Article  Google Scholar 

  7. Designolle, S. et al. Genuine high-dimensional quantum steering. Phys. Rev. Lett. 126, 200404 (2021).

    Article  ADS  MathSciNet  Google Scholar 

  8. Hiekkamäki, M. & Fickler, R. High-dimensional two-photon interference effects in spatial modes. Phys. Rev. Lett. 126, 123601 (2021).

    Article  ADS  Google Scholar 

  9. Chi, Y. et al. A programmable qudit-based quantum processor. Nat. Commun. 13, 1166 (2022).

    Article  ADS  Google Scholar 

  10. Bruß, D. & Macchiavello, C. Optimal eavesdropping in cryptography with three-dimensional quantum states. Phys. Rev. Lett. 88, 127901 (2002).

    Article  ADS  Google Scholar 

  11. Pivoluska, M., Huber, M. & Malik, M. Layered quantum key distribution. Phys. Rev. A 97, 032312 (2018).

    Article  ADS  Google Scholar 

  12. Doda, M. et al. Quantum key distribution overcoming extreme noise: simultaneous subspace coding using high-dimensional entanglement. Phys. Rev. Appl. 15, 034003 (2021).

    Article  ADS  Google Scholar 

  13. Sit, A. et al. High-dimensional intracity quantum cryptography with structured photons. Optica 4, 1006–1010 (2017).

    Article  ADS  Google Scholar 

  14. Bulla, L. et al. Distribution of genuine high-dimensional entanglement over 10.2 km of noisy metropolitan atmosphere. Phys. Rev. A 107, L050402 (2023).

    Article  ADS  Google Scholar 

  15. Lanyon, B. P. et al. Simplifying quantum logic using higher-dimensional Hilbert spaces. Nat. Phys. 5, 134–140 (2009).

    Article  Google Scholar 

  16. Wang, Y., Hu, Z., Sanders, B. C. & Kais, S. Qudits and high-dimensional quantum computing. Front. Phys. 8, 479 (2020).

    Article  Google Scholar 

  17. Ringbauer, M. et al. A universal qudit quantum processor with trapped ions. Nat. Phys. 18, 1053–1057 (2022).

    Article  Google Scholar 

  18. Hrmo, P. et al. Native qudit entanglement in a trapped ion quantum processor. Nat. Commun. 14, 2242 (2023).

    Article  ADS  Google Scholar 

  19. Gao, X., Appel, P., Friis, N., Ringbauer, M. & Huber, M. On the role of entanglement in qudit–based circuit compression. Quantum 7, 1141 (2023).

    Article  Google Scholar 

  20. Kiktenko, E. O., Nikolaeva, A. S., Xu, P., Shlyapnikov, G. V. & Fedorov, A. K. Scalable quantum computing with qudits on a graph. Phys. Rev. A 101, 022304 (2020).

    Article  ADS  Google Scholar 

  21. Nikolaeva, A. S., Kiktenko, E. O. & Fedorov, A. K. Efficient realization of quantum algorithms with qudits. EPJ Quantum Technol. 11, 43 (2024).

    Article  ADS  Google Scholar 

  22. Gao, X., Erhard, M., Zeilinger, A. & Krenn, M. Computer-inspired concept for high-dimensional multipartite quantum gates. Phys. Rev. Lett. 125, 050501 (2020).

    Article  ADS  MathSciNet  Google Scholar 

  23. Babazadeh, A. et al. High-dimensional single-photon quantum gates: concepts and experiments. Phys. Rev. Lett. 119, 180510 (2017).

    Article  ADS  Google Scholar 

  24. Brandt, F. et al. High-dimensional quantum gates using full–field spatial modes of photons. Optica 7, 98–107 (2020).

    Article  ADS  Google Scholar 

  25. Nielsen, M. A. and Chuang, I. L. Quantum Computation and Quantum Information: 10th Anniversary Edition (Cambridge Univ. Press, 2010).

  26. Allen, L., Beijersbergen, M. W., Spreeuw, R. C. & Woerdman, J. P. Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes. Phys. Rev. A 45, 8185–8189 (1992).

    Article  ADS  Google Scholar 

  27. Krenn, M., Malik, M., Erhard, M. & Zeilinger, A. Orbital angular momentum of photons and the entanglement of Laguerre-Gaussian modes. Phil. Trans. R. Soc. A 375, 20150442 (2017).

    Article  ADS  MathSciNet  Google Scholar 

  28. Barenco, A. et al. Elementary gates for quantum computation. Phys. Rev. A 52, 3457–3467 (1995).

    Article  ADS  Google Scholar 

  29. O’Brien, J. L., Pryde, G. J., White, A. G., Ralph, T. C. & Branning, D. Demonstration of an all-optical quantum controlled-NOT gate. Nature 426, 264–267 (2003).

    Article  ADS  Google Scholar 

  30. Pittman, T. B., Jacobs, B. C. & Franson, J. D. Demonstration of nondeterministic quantum logic operations using linear optical elements. Phys. Rev. Lett. 88, 257902 (2002).

    Article  ADS  Google Scholar 

  31. Langford, N. K. et al. Demonstration of a simple entangling optical gate and its use in Bell-state analysis. Phys. Rev. Lett. 95, 210504 (2005).

    Article  ADS  Google Scholar 

  32. Okamoto, R., Hofmann, H. F., Takeuchi, S. & Sasaki, K. Demonstration of an optical quantum controlled-NOT gate without path interference. Phys. Rev. Lett. 95, 210506 (2005).

    Article  ADS  Google Scholar 

  33. Gasparoni, S., Pan, J.-W., Walther, P., Rudolph, T. & Zeilinger, A. Realization of a photonic controlled–NOT gate sufficient for quantum computation. Phys. Rev. Lett. 93, 020504 (2004).

    Article  ADS  Google Scholar 

  34. Zhao, Z. et al. Experimental demonstration of a nondestructive controlled-not quantum gate for two independent photon qubits. Phys. Rev. Lett. 94, 030501 (2005).

    Article  ADS  Google Scholar 

  35. Huang, Y.-F., Ren, X.-F., Zhang, Y.-S., Duan, L.-M. & Guo, G.-C. Experimental teleportation of a quantum controlled-NOT gate. Phys. Rev. Lett. 93, 240501 (2004).

    Article  ADS  Google Scholar 

  36. Zeuner, J. et al. Integrated-optics heralded controlled-NOT gate for polarization-encoded qubits. npj Quantum Inf. 4, 13 (2018).

    Article  ADS  Google Scholar 

  37. Bao, X.-H. et al. Optical nondestructive controlled-NOT gate without using entangled photons. Phys. Rev. Lett. 98, 170502 (2007).

    Article  ADS  Google Scholar 

  38. Li, J.-P. et al. Heralded nondestructive quantum entangling gate with single-photon sources. Phys. Rev. Lett. 126, 140501 (2021).

    Article  ADS  Google Scholar 

  39. Mair, A., Vaziri, A., Weihs, G. & Zeilinger, A. Entanglement of the orbital angular momentum states of photons. Nature 412, 313–316 (2001).

    Article  ADS  Google Scholar 

  40. Wang, X.-L. et al. 18-qubit entanglement with six photons’ three degrees of freedom. Phys. Rev. Lett. 120, 260502 (2018).

    Article  ADS  Google Scholar 

  41. Zhang, W., Qi, Q., Zhou, J. & Chen, L. Mimicking Faraday rotation to sort the orbital angular momentum of light. Phys. Rev. Lett. 112, 153601 (2014).

    Article  ADS  Google Scholar 

  42. Marrucci, L., Manzo, C. & Paparo, D. Optical spin-to-orbital angular momentum conversion in inhomogeneous anisotropic media. Phys. Rev. Lett. 96, 163905 (2006).

    Article  ADS  Google Scholar 

  43. Agnew, M., Leach, J., McLaren, M., Roux, F. S. & Boyd, R. W. Tomography of the quantum state of photons entangled in high dimensions. Phys. Rev. A 84, 062101 (2011).

    Article  ADS  Google Scholar 

  44. Wu, S., Huang, W., Yang, P., Liu, S. & Chen, L. Arbitrary phase-locking in Mach–Zehnder interferometer. Opt. Commun. 442, 148–151 (2019).

    Article  ADS  Google Scholar 

  45. Black, E. D. An introduction to Pound-Drever-Hall laser frequency stabilization. Am. J. Phys. 69, 79–87 (2001).

    Article  ADS  Google Scholar 

  46. Wan, P. et al. Postselection-free cavity enhanced narrow band orbital angular momentum entangled photon source. Phys. Rev. Lett. 134, 053801 (2025).

    Article  ADS  Google Scholar 

  47. Hofmann, H. F. Complementary classical fidelities as an efficient criterion for the evaluation of experimentally realized quantum operations. Phys. Rev. Lett. 94, 160504 (2005).

    Article  ADS  MathSciNet  Google Scholar 

  48. Chen, M. et al. High-dimensional two-photon quantum controlled phase-flip gate. Phys. Rev. Research 6, 033004 (2024).

    Article  ADS  Google Scholar 

  49. Jeff Kimble, H. The quantum internet. Nature 453, 1023–1030 (2008).

    Article  Google Scholar 

  50. Vitelli, C. et al. Joining the quantum state of two photons into one. Nat. Photon. 7, 521–526 (2013).

    Article  ADS  Google Scholar 

  51. Gao, X., Krenn, M., Kysela, J. & Zeilinger, A. Arbitrary d-dimensional Pauli X gates of a flying qudit. Phys. Rev. A 99, 023825 (2019).

    Article  ADS  Google Scholar 

  52. Wang, X., Sanders, B. C. & Berry, D. W. Entangling power and operator entanglement in qudit systems. Phys. Rev. A 67, 042323 (2003).

    Article  ADS  Google Scholar 

  53. Barreiro, J. T., Wei, T.-C. & Kwiat, P. G. Beating the channel capacity limit for linear photonic superdense coding. Nat. Phys. 4, 282–286 (2008).

    Article  Google Scholar 

  54. Wang, X.-L. et al. Quantum teleportation of multiple degrees of freedom of a single photon. Nature 518, 516–519 (2015).

    Article  ADS  Google Scholar 

  55. Williams, B. P., Sadlier, R. J. & Humble, T. S. Superdense coding over optical fiber links with complete Bell-state measurements. Phys. Rev. Lett. 118, 050501 (2017).

    Article  ADS  Google Scholar 

  56. Luo, Y.-H. et al. Quantum teleportation in high dimensions. Phys. Rev. Lett. 123, 070505 (2019).

    Article  ADS  Google Scholar 

  57. Reimer, C. et al. High-dimensional one-way quantum processing implemented on d-level cluster states. Nat. Phys. 15, 148–153 (2019).

    Article  Google Scholar 

  58. Hu, X.-M. et al. Experimental high-dimensional quantum teleportation. Phys. Rev. Lett. 125, 230501 (2020).

    Article  ADS  Google Scholar 

  59. Terhal, B. M. Quantum error correction for quantum memories. Rev. Mod. Phys. 87, 307–346 (2015).

    Article  ADS  MathSciNet  Google Scholar 

  60. Bocharov, A., Roetteler, M. & Svore, K. M. Factoring with qutrits: Shor’s algorithm on ternary and metaplectic quantum architectures. Phys. Rev. A 96, 012306 (2017).

    Article  ADS  Google Scholar 

  61. Miao, K. C. et al. Overcoming leakage in quantum error correction. Nat. Phys. 19, 1780–1786 (2023).

    Article  Google Scholar 

  62. Muthukrishnan, A. & Stroud, C. R. Multi-valued logic gates for quantum computation. Phys. Rev. A 62, 052309 (2000).

    Article  ADS  MathSciNet  Google Scholar 

  63. Grover, L. K. Quantum mechanics helps in searching for a needle in a haystack. Phys. Rev. Lett. 79, 325–328 (1997).

    Article  ADS  Google Scholar 

  64. Wang, X.-L. et al. Experimental ten-photon entanglement. Phys. Rev. Lett. 117, 210502 (2016).

    Article  ADS  Google Scholar 

  65. Zhong, H.-S. et al. 12-photon entanglement and scalable scattershot boson sampling with optimal entangled-photon pairs from parametric down-conversion. Phys. Rev. Lett. 121, 250505 (2018).

    Article  ADS  Google Scholar 

  66. Liu, Z. et al. Supporting data for heralded high-dimensional photon-photon quantum gate. Figshare https://doi.org/10.6084/m9.figshare.30946265 (2025).

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Acknowledgements

This work was supported by the National Key Research and Development Program of China (number 2020YFA0309500); the National Natural Science Foundation of China (numbers 12234009, 12274215, 12404382, 12574392, 12427808 and U25A20194); Quantum Science and Technology—National Science and Technology Major Project (number 2021ZD0301400); the Program for Innovative Talents and Entrepreneurs in Jiangsu; Key R&D Program of Jiangsu Province (BE2023002); the Natural Science Foundation of Jiangsu Province (numbers BK20233001, BK20220759 and BK20252118). M.H. acknowledges funding from the European Research Council (Consolidator grant ‘Cocoquest’ 101043705) and the Horizon-Europe research and innovation programme under grant agreement number 101070168 (HyperSpace). N.F. acknowledges financial support from the Austrian Science Fund (FWF) through the standalone project P 36478-N funded by the European Union—NextGenerationEU, as well as by the Austrian Federal Ministry of Education, Science and Research via the Austrian Research Promotion Agency (FFG) through the flagship project FO999897481 (HPQC) and the project FO999921407 (HDcode) funded by the European Union—NextGenerationEU. M.H. and N.F. acknowledge financial support from the Austrian Research Promotion Agency (FFG) through project FO999914030 (MUSIQ) and project FO999921415 (Vanessa-QC) funded by the European Union—NextGenerationEU. We would like to thank R. Hogan for valuable discussions.

Author information

Author notes

  1. These authors contributed equally: Zhi-Feng Liu, Zhi-Cheng Ren.

Authors and Affiliations

  1. National Laboratory of Solid State Microstructures, School of Physics, Nanjing University, Nanjing, China

    Zhi-Feng Liu, Zhi-Cheng Ren, Pei Wan, Wen-Zheng Zhu, Zi-Mo Cheng, Jing Wang, Yu-Peng Shi, Han-Bing Xi, Xiaoqin Gao, Xi-Lin Wang & Hui-Tian Wang

  2. Collaborative Innovation Center of Advanced Microstructures, Nanjing University, Nanjing, China

    Zhi-Feng Liu, Zhi-Cheng Ren, Pei Wan, Wen-Zheng Zhu, Zi-Mo Cheng, Jing Wang, Yu-Peng Shi, Han-Bing Xi, Xiaoqin Gao, Xi-Lin Wang & Hui-Tian Wang

  3. Collaborative Innovation Center of Extreme Optics, Shanxi University, Taiyuan, China

    Zhi-Cheng Ren & Hui-Tian Wang

  4. Atominstitut, Technische Universität Wien, Vienna, Austria

    Marcus Huber & Nicolai Friis

  5. Institute for Quantum Optics and Quantum Information—IQOQI Vienna, Austrian Academy of Sciences, Vienna, Austria

    Marcus Huber & Nicolai Friis

  6. National Research Council of Canada, Ottawa, Ontario, Canada

    Xiaoqin Gao

  7. University of Ottawa, Ottawa, Ontario, Canada

    Xiaoqin Gao

  8. Jiangsu Physical Science Research Center, Nanjing, China

    Xi-Lin Wang

  9. Jiangsu Key Laboratory of Quantum Information Science and Technology, Nanjing University, Nanjing, China

    Xi-Lin Wang

  10. Hefei National Laboratory, Hefei, China

    Xi-Lin Wang

  11. Synergetic Innovation Center of Quantum Information and Quantum Physics, University of Science and Technology of China, Hefei, China

    Xi-Lin Wang

Authors
  1. Zhi-Feng Liu
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  2. Zhi-Cheng Ren
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Contributions

H.-T.W., X.-L.W. and X.G. conceived the idea, designed the research methodology and supervised the project. Z.-F.L. and Z.-C.R. presented the experimental scheme and performed the experiment. Z.-C.R. performed the revision experiment. P.W., W.-Z.Z., Z.-M.C., J.W., Y.-P.S. and H.-B.X. assisted in the experiment. X.G., M.H. and N.F. contributed to the theoretical proposal. All authors analysed and discussed the results and reviewed the manuscript. Z.-F.L., Z.-C.R., N.F., M.H., X.G., X.-L.W. and H.-T.W. wrote the manuscript.

Corresponding authors

Correspondence to Xiaoqin Gao, Xi-Lin Wang or Hui-Tian Wang.

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Supplementary Information

Supplementary Sections 1–5, Figs. 1–6 and Tables 1–3.

Source data

Source Data Fig. 4

Experimental raw data.

Source Data Fig. 5

Experimental raw data.

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Liu, ZF., Ren, ZC., Wan, P. et al. Heralded high-dimensional photon–photon quantum gate. Nat. Photon. (2026). https://doi.org/10.1038/s41566-026-01846-x

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