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Holonomic multicontrolled gates for single-photon states

Carlo Danieli1,2,*, Valentina Brosco1,3, Claudio Conti3,4, and Laura Pilozzi1,4

  • *Contact author: carlo.danieli@cnr.it
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Phys. Rev. A 113, 052606 – Published 8 May, 2026

DOI: https://doi.org/10.1103/hs1t-64zm

Abstract

Controlled and multicontrolled quantum gates whose action on a target qubit depends on the state of multiple control qubits, represent a fundamental logical building block for complex quantum algorithms. We propose a scheme for realizing this class of gates based on non-abelian holonomies in modulated photonic waveguide networks. Our approach relies on linear photonic cicuits formed by two star networks coupled via a two-path circuit. A star network with M peripheral waveguides coupled to a single central site, or M pod, naturally generalizes the tripod structure used in non-abelian Thouless pumping and stimulated Raman adiabatic passage. In the present work, we first analyze the minimal case of two connected tripods and design adiabatic driving loops that implement single-qubit, cnot, and swap gates. We then show how extending the approach to larger M-pod structures enables the realization of multiply controlled operations, which we exemplify by designing toffoli and the or gate on two coupled pentapods. Finally, we demonstrate that networks of connected tripods can implement the Deutsch quantum query algorithm.

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References (76)

  1. F. Wilczek and A. Zee, Appearance of gauge structure in simple dynamical systems, Phys. Rev. Lett. 52, 2111 (1984).
  2. J. Anandan, Non-adiabatic non-Abelian geometric phase, Phys. Lett. A 133, 171 (1988).
  3. M. V. Berry, Quantal phase factors accompanying adiabatic changes, Proc. R. Soc. Lond. A 392, 45 (1984).
  4. Y.-K. Sun, X.-L. Zhang, F. Yu, Z.-N. Tian, Q.-D. Chen, and H.-B. Sun, Non-Abelian Thouless pumping in photonic waveguides, Nat. Phys. 18, 1080 (2022).
  5. Y. Chen, Y. Fan, G. Larsonneur, J. Xiang, A. He, G. Wang, X.-L. Zhang, G. Ma, Q. Zhou, G. Guo, Y. Su, and X. Guo, High-dimensional non-Abelian holonomy in integrated photonics, Nat. Commun. 16, 3650 (2025).
  6. O. You, S. Liang, B. Xie, W. Gao, W. Ye, J. Zhu, and S. Zhang, Observation of non-Abelian Thouless pump, Phys. Rev. Lett. 128, 244302 (2022).
  7. J. W. Zwanziger, M. Koenig, and A. Pines, Non-Abelian effects in a quadrupole system rotating around two axes, Phys. Rev. A 42, 3107 (1990).
  8. R. G. Unanyan, B. W. Shore, and K. Bergmann, Laser-driven population transfer in four-level atoms: Consequences of non-Abelian geometrical adiabatic phase factors, Phys. Rev. A 59, 2910 (1999).
  9. S. Sugawa, F. Salces-Carcoba, A. R. Perry, Y. Yue, and I. B. Spielman, Second chern number of a quantum-simulated non-Abelian yang monopole, Science 360, 1429 (2018).
  10. A. A. Abdumalikov, Jr., J. M. Fink, K. Juliusson, M. Pechal, S. Berger, A. Wallraff, and S. Filipp, Experimental realization of non-Abelian non-adiabatic geometric gates, Nature (London) 496, 482 (2013).
  11. S. Murakami, N. Nagaosa, and S.-C. Zhang, SU(2) non-Abelian holonomy and dissipationless spin current in semiconductors, Phys. Rev. B 69, 235206 (2004).
  12. Y. Chen, R.-Y. Zhang, Z. Xiong, Z. H. Hang, J. Li, J. Q. Shen, and C. Chan, non-Abelian gauge field optics, Nat. Commun. 10, 3125 (2019).
  13. V. Brosco, L. Pilozzi, R. Fazio, and C. Conti, non-Abelian Thouless pumping in a photonic lattice, Phys. Rev. A 103, 063518 (2021).
  14. L. Pilozzi and V. Brosco, Thouless pumping of light with a twist, Nat. Phys. 18, 968 (2022).
  15. L. Taddia, E. Cornfeld, D. Rossini, L. Mazza, E. Sela, and R. Fazio, Topological fractional pumping with alkaline-earth-like atoms in synthetic lattices, Phys. Rev. Lett. 118, 230402 (2017).
  16. C. Danieli, V. Brosco, L. Pilozzi, and R. Citro, non-Abelian thouless pumping in a rice–mele ladder, AVS Quantum Sci. 7, 022001 (2025).
  17. T. Iadecola, T. Schuster, and C. Chamon, non-Abelian braiding of light, Phys. Rev. Lett. 117, 073901 (2016).
  18. L.-M. Duan, J. I. Cirac, and P. Zoller, Geometric manipulation of trapped ions for quantum computation, Science 292, 1695 (2001).
  19. V. Brosco, R. Fazio, F. W. J. Hekking, and A. Joye, non-Abelian superconducting pumps, Phys. Rev. Lett. 100, 027002 (2008).
  20. C. Danieli, L. Pilozzi, C. Conti, and V. Brosco, Parity breaking in thouless quantum walks, arXiv:2412.02429.
  21. P. Zanardi and M. Rasetti, Holonomic quantum computation, Phys. Lett. A 264, 94 (1999).
  22. J. Zhang, T. H. Kyaw, S. Filipp, L.-C. Kwek, E. Sjöqvist, and D. Tong, Geometric and holonomic quantum computation, Phys. Rep. 1027, 1 (2023), Geometric and holonomic quantum computation.
  23. A. Y. Kitaev, Quantum computations: Algorithms and error correction, Russ. Math. Surv. 52, 1191 (1997).
  24. P. Solinas, P. Zanardi, and N. Zanghì, Robustness of non-Abelian holonomic quantum gates against parametric noise, Phys. Rev. A 70, 042316 (2004).
  25. S.-Y. Sun, L.-N. Ji, F.-F. Cui, Z.-Y. Xue, and T. Chen, Suppression of systematic errors in holonomic quantum gates via multipath correction, Phys. Rev. A 110, 062621 (2024).
  26. X. Y. Sun and P. Z. Zhao, Decoherence mitigation for geometric quantum computation, Phys. Rev. A 112, 032403 (2025).
  27. J. A. Jones, V. Vedral, A. Ekert, and G. Castagnoli, Geometric quantum computation using nuclear magnetic resonance, Nature (London) 403, 869 (2000).
  28. J. Pachos and S. Chountasis, Optical holonomic quantum computer, Phys. Rev. A 62, 052318 (2000).
  29. L. Faoro, J. Siewert, and R. Fazio, non-Abelian holonomies, charge pumping, and quantum computation with Josephson junctions, Phys. Rev. Lett. 90, 028301 (2003).
  30. K. Nagata, K. Kuramitani, Y. Sekiguchi, and H. Kosaka, Universal holonomic quantum gates over geometric spin qubits with polarised microwaves, Nat. Commun. 9, 3227 (2018).
  31. S. Bhattacharyya and S. Bhattacharyya, Holonomic control of a three-qubits system in an NV center using a near-term quantum computer, arXiv:2202.08061.
  32. Y.-S. Chen, J. Gao, and L.-N. Ji, Realization of universal nonadiabatic holonomic quantum gates in trapped ion system, arXiv:2505.02137.
  33. D. David, Quantum theory, the Church-Turing principle and the universal quantum computer, Royal Soc. London: A 400, 97 (1985).
  34. P. Shor, Fault-tolerant quantum computation, in Proceedings of 37th Conference on Foundations of Computer Science (IEEE, New York, 1996), pp. 56–65.
  35. D. Gottesman, Theory of fault-tolerant quantum computation, Phys. Rev. A 57, 127 (1998).
  36. P. Boykin, T. Mor, M. Pulver, V. Roychowdhury, and F. Vatan, A new universal and fault-tolerant quantum basis, Inf. Process. Lett. 75, 101 (2000).
  37. A. Paetznick and B. W. Reichardt, Universal fault-tolerant quantum computation with only transversal gates and error correction, Phys. Rev. Lett. 111, 090505 (2013).
  38. E. Sjöqvist, D. M. Tong, L. Mauritz Andersson, B. Hessmo, M. Johansson, and K. Singh, Non-adiabatic holonomic quantum computation, New J. Phys. 14, 103035 (2012).
  39. G. F. Xu, J. Zhang, D. M. Tong, E. Sjoqvist, and L. C. Kwek, Nonadiabatic holonomic quantum computation in decoherence-free subspaces, Phys. Rev. Lett. 109, 170501 (2012).
  40. T. Yan, B.-J. Liu, K. Xu, C. Song, S. Liu, Z. Zhang, H. Deng, Z. Yan, H. Rong, K. Huang, M.-H. Yung, Y. Chen, and D. Yu, Experimental realization of nonadiabatic shortcut to non-Abelian geometric gates, Phys. Rev. Lett. 122, 080501 (2019).
  41. P. Z. Zhao and D. M. Tong, Nonadiabatic holonomic quantum computation based on a commutation relation, Phys. Rev. A 108, 012619 (2023).
  42. V. Neef, J. Pinske, T. A. W. Wolterink, K. Becker, M. Heinrich, S. Scheel, and A. Szameit, Non-adiabatic holonomies for photonic quantum computing, Optica Quantum 3, 93 (2025).
  43. F. Jin, J. Geng, and H. Zhou, Nonadiabatic holonomic quantum gates in a decoherence-insensitive space, Phys. Rev. A 111, 022620 (2025).
  44. W. Song, X. Liu, J. Sun, O. You, S. Wu, C. Chen, S. Zhu, T. Li, and S. Zhang, Shortcuts to adiabatic non-Abelian braiding on silicon photonic chips, Sci. Adv. 11, eadt7224 (2025).
  45. V. Neef, J. Pinske, F. Klauck, L. Teuber, M. Kremer, M. Ehrhardt, M. Heinrich, S. Scheel, and A. Szameit, Three-dimensional non-Abelian quantum holonomy, Nat. Phys. 19, 30 (2023).
  46. J. Pinske, L. Teuber, and S. Scheel, Highly degenerate photonic waveguide structures for holonomic computation, Phys. Rev. A 101, 062314 (2020).
  47. J. Pinske and S. Scheel, Symmetry-protected non-Abelian geometric phases in optical waveguides with nonorthogonal modes, Phys. Rev. A 105, 013507 (2022).
  48. A. Babazadeh, M. Erhard, F. Wang, M. Malik, R. Nouroozi, M. Krenn, and A. Zeilinger, High-dimensional single-photon quantum gates: Concepts and experiments, Phys. Rev. Lett. 119, 180510 (2017).
  49. R. Lapkiewicz, P. Li, C. Schaeff, N. K. Langford, S. Ramelow, M. Wieśniak, and A. Zeilinger, Experimental non-classicality of an indivisible quantum system, Nature (London) 474, 490 (2011).
  50. 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).
  51. A. Y. Vlasov, Clifford algebras and universal sets of quantum gates, Phys. Rev. A 63, 054302 (2001).
  52. J. Tolar, On clifford groups in quantum computing, J. Phys.: Conf. Ser. 1071, 012022 (2018).
  53. J. Helsen, J. J. Wallman, and S. Wehner, Representations of the multi-qubit Clifford group, J. Math. Phys. 59, 072201 (2018).
  54. M. Bataille, Quantum circuits of cnot gates, arXiv:2009.13247.
  55. C. Monroe, D. M. Meekhof, B. E. King, W. M. Itano, and D. J. Wineland, Demonstration of a fundamental quantum logic gate, Phys. Rev. Lett. 75, 4714 (1995).
  56. A. Barenco, C. H. Bennett, R. Cleve, D. P. DiVincenzo, N. Margolus, P. Shor, T. Sleator, J. A. Smolin, and H. Weinfurter, Elementary gates for quantum computation, Phys. Rev. A 52, 3457 (1995).
  57. B. P. Lanyon, M. Barbieri, M. P. Almeida, T. Jennewein, T. C. Ralph, K. J. Resch, G. J. Pryde, J. L. O'brien, A. Gilchrist, and A. G. White, Simplifying quantum logic using higher-dimensional hilbert spaces, Nat. Phys. 5, 134 (2009).
  58. D. Qin, Y. Chen, and Y. Li, Error statistics and scalability of quantum error mitigation formulas, npj Quantum Inf. 9, 35 (2023).
  59. B. Zindorf and S. Bose, Efficient implementation of multicontrolled quantum gates, Phys. Rev. Appl. 24, 044030 (2025).
  60. D. Aharonov, A simple proof that Toffoli and Hadamard are quantum universal, arXiv:quant-ph/0301040.
  61. D. Deutsch, Quantum computational networks Proc, R. Soc. A 425, 90 (1989).
  62. R. Portugal, Basic quantum algorithms, arXiv:2201.10574.
  63. A. Ramachandran, A. Andreanov, and S. Flach, Chiral flat bands: Existence, engineering, and stability, Phys. Rev. B 96, 161104(R) (2017).
  64. R. Keil, C. Poli, M. Heinrich, J. Arkinstall, G. Weihs, H. Schomerus, and A. Szameit, Universal sign control of coupling in tight-binding lattices, Phys. Rev. Lett. 116, 213901 (2016).
  65. M. Kremer, I. Petrides, E. Meyer, M. Heinrich, O. Zilberberg, and A. Szameit, A square-root topological insulator with non-quantized indices realized with photonic Aharonov-Bohm cages, Nat. Commun. 11, 907 (2020).
  66. Y.-J. Chang, Y.-H. Lu, Y. Wang, X.-Y. Xu, W.-H. Zhou, W.-H. Cui, X.-W. Wang, J. Gao, L.-F. Qiao, and X.-M. Jin, Symmetry-induced error filtering in a photonic Lieb lattice, Phys. Rev. Lett. 126, 110501 (2021).
  67. D. Roman-Cortes, M. Mazanov, R. A. Vicencio, and M. A. Gorlach, Observation of invisibility angle and flat band physics in dipolar photonic lattices, Nano Lett. 25, 4291 (2025).
  68. R. A. Vicencio, Multi-orbital photonic lattices, APL Photon. 10, 071101 (2025).
  69. J. R. Morris and B. W. Shore, Reduction of degenerate two-level excitation to independent two-state systems, Phys. Rev. A 27, 906 (1983).
  70. For comparison, (i) each tripod used in Fig. 1 has M=3=21+1 which results in a single (m=1) control qubit, while (ii) each pentapod used in Fig. 4 has M=5=22+1 which results in two (m=2) control qubits.
  71. C. Figgatt, D. Maslov, K. A. Landsman, N. M. Linke, S. Debnath, and C. Monroe, Complete 3-qubit Grover search on a programmable quantum computer, Nat. Commun. 8, 1918 (2017).
  72. C. Gidney and M. Ekerå, How to factor 2048 bit RSA integers in 8 hours using 20 million noisy qubits, Quantum 5, 433 (2021).
  73. J. Chu, X. He, Y. Zhou, J. Yuan, L. Zhang, Q. Guo, Y. Hai, Z. Han, C.-K. Hu, W. Huang, H. Jia, D. Jiao, S. Li, Y. Liu, Z. Ni, L. Nie, X. Pan, J. Qiu, W. Wei, W. Nuerbolati, et al., Scalable algorithm simplification using quantum and logic, Nat. Phys. 19, 126 (2023).
  74. M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information (Cambridge University Press, Cambridge, England, 2010).
  75. Y. Yang, B. Yang, G. Ma, J. Li, S. Zhang, and C. T. Chan, non-Abelian physics in light and sound, Science 383, eadf9621 (2024).
  76. S. Sun, X. Wang, S. Li, D. Zhang, Q.-D. Chen, and X.-L. Zhang, Reconfigurable non-Abelian integrated photonics, Nat. Commun. 16, 7089 (2025).

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Phys. Rev. A 113, 052606– Published 8 May, 2026
  • Received 23 December 2025
  • Accepted 19 March 2026
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Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

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