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Universal quantum computation via superposed orders of single-qubit gates

Kyrylo Simonov1,*, Marcello Caleffi2,†, Jessica Illiano2,‡, Jacquiline Romero3,§, and Angela Sara Cacciapuoti2,∥

  • *Contact author: kyrylo.simonov@univie.ac.at
  • †Contact author: marcello.caleffi@unina.it
  • ‡Contact author: jessica.illiano@unina.it
  • §Contact author: m.romero@uq.edu.au
  • ∥Contact author: angelasara.cacciapuoti@unina.it
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Phys. Rev. Research 8, 013239 – Published 5 March, 2026

DOI: https://doi.org/10.1103/6xwr-9b4y

Abstract

Superposed orders of quantum channels have already been proved—both theoretically and experimentally—to enable unparalleled opportunities in the quantum communication domain. Superposition of orders can be exploited within the quantum computing domain as well, by relaxing the (traditional) assumption underlying quantum computation about applying gates in a well-defined causal order. In this context, we address a fundamental question arising with quantum computing: Whether superposed orders of single-qubit gates can enable universal quantum computation. As shown in this paper, the answer to this key question is a definitive “yes.” Indeed, we prove that any two-qubit controlled quantum gate can be deterministically realized, including the so-called Barenco gate that alone enables universal quantum computation.

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References (85)

  1. M. Caleffi and A. S. Cacciapuoti, Quantum internet architecture: Unlocking quantum-native routing via quantum addressing, IEEE Trans. Commun. 74 3577 (2026).
  2. G. Chiribella and H. Kristjánsson, Quantum Shannon theory with superpositions of trajectories, Proc. R. Soc. A 475, 20180903 (2019).
  3. A. A. Abbott, J. Wechs, D. Horsman, M. Mhalla, and C. Branciard, Communication through coherent control of quantum channels, Quantum 4, 333 (2020).
  4. H. Kristjánsson, G. Chiribella, S. Salek, D. Ebler, and M. Wilson, Resource theories of communication, New J. Phys. 22, 073014 (2020).
  5. 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).
  6. A. S. Cacciapuoti, C. Pellitteri, J. Illiano, L. d’Avossa, F. Mazza, S. Chen, and M. Caleffi, Quantum data centers: Why entanglement changes everything, Philos. Trans. R. Soc. A (2025).
  7. C. Pellitteri, M. Caleffi, and A. S. Cacciapuoti, Quantum paths: A quantum walk approach, in 2025 IEEE International Conference on Quantum Computing and Engineering (QCE), Albuquerque, NM, USA (IEEE, 2025), pp. 1248–1251.
  8. G. Chiribella, Perfect discrimination of no-signalling channels via quantum superposition of causal structures, Phys. Rev. A 86, 040301(R) (2012).
  9. O. Oreshkov, F. Costa, and Č. Brukner, Quantum correlations with no causal order, Nat. Commun. 3, 1092 (2012).
  10. J. Illiano, M. Caleffi, A. Manzalini, and A. S. Cacciapuoti, Quantum internet protocol stack: A comprehensive survey, Comput. Netw. 213, 109092 (2022).
  11. A. S. Cacciapuoti, J. Illiano, S. Koudia, K. Simonov, and M. Caleffi, The quantum internet: Enhancing classical internet services one qubit at a time, IEEE Netw. 36, 6 (2022).
  12. A. S. Cacciapuoti, J. Illiano, and M. Caleffi, Quantum internet addressing, IEEE Netw. 38, 104 (2023).
  13. L. M. Procopio, A. Moqanaki, M. Araújo, F. Costa, I. Alonso Calafell, E. G. Dowd, D. R. Hamel, L. A. Rozema, Č. Brukner, and P. Walther, Experimental superposition of orders of quantum gates, Nat. Commun. 6, 7913 (2015).
  14. G. Rubino, L. A. Rozema, A. Feix, M. Araújo, J. M. Zeuner, L. M. Procopio, Č. Brukner, and P. Walther, Experimental verification of an indefinite causal order, Sci. Adv. 3, e1602589 (2017).
  15. K. Goswami, C. Giarmatzi, M. Kewming, F. Costa, C. Branciard, J. Romero, and A. G. White, Indefinite causal order in a quantum switch, Phys. Rev. Lett. 121, 090503 (2018).
  16. 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).
  17. 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).
  18. D. Chandra, M. Caleffi, and A. S. Cacciapuoti, The entanglement-assisted communication capacity over quantum trajectories, IEEE Trans. Wireless Commun. 21, 3632 (2021).
  19. M. Caleffi, K. Simonov, and A. S. Cacciapuoti, Beyond Shannon limits: Quantum communications through quantum paths, IEEE J. Sel. Areas Commun. 41, 2707 (2023).
  20. 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).
  21. Y. Chen and Y. Hasegawa, Fundamental trade-off relation in probabilistic entanglement generation, arXiv:2112.03233.
  22. D. Felce and V. Vedral, Quantum refrigeration with indefinite causal order, Phys. Rev. Lett. 125, 070603 (2020).
  23. T. Guha, M. Alimuddin, and P. Parashar, Thermodynamic advancement in the causally inseparable occurrence of thermal maps, Phys. Rev. A 102, 032215 (2020).
  24. G. Chiribella, M. Wilson, and H. F. Chau, Quantum and classical data transmission through completely depolarizing channels in a superposition of cyclic orders, Phys. Rev. Lett. 127, 190502 (2021).
  25. K. Simonov, G. Francica, G. Guarnieri, and M. Paternostro, Work extraction from coherently activated maps via quantum switch, Phys. Rev. A 105, 032217 (2022).
  26. S. Milz, J. Bavaresco, and G. Chiribella, Resource theory of causal connection, Quantum 6, 788 (2022).
  27. A. G. Maity and S. Bhattacharya, Activating information backflow with the assistance of quantum switch, J. Phys. A: Math. Theor. 57, 215302 (2024).
  28. S. Mukherjee, B. Mallick, S. Yanamandra, S. Bhattacharya, and A. G. Maity, Interplay between the Hilbert-Space dimension of a control system and the memory induced by a quantum switch, Phys. Rev. A 110, 042624 (2024).
  29. K. Simonov, S. Roy, T. Guha, Z. Zimborás, and G. Chiribella, Activation of thermal states by coherently controlled thermalization processes, New J. Phys. 27, 074502 (2025).
  30. G. Chiribella, G. M. D'Ariano, P. Perinotti, and B. Valiron, Beyond quantum computers, arXiv:0912.0195v1.
  31. G. Chiribella, G. M. D’Ariano, P. Perinotti, and B. Valiron, Quantum computations without definite causal structure, Phys. Rev. A 88, 022318 (2013).
  32. J. Wechs, H. Dourdent, A. A. Abbott, and C. Branciard, Quantum circuits with classical versus quantum control of causal order, PRX Quantum 2, 030335 (2021).
  33. T. Colnaghi, G. M. D’Ariano, S. Facchini, and P. Perinotti, Quantum computation with programmable connections between gates, Phys. Lett. A 376, 2940 (2012).
  34. M. Araújo, F. Costa, and Č. Brukner, Computational advantage from quantum-controlled ordering of gates, Phys. Rev. Lett. 113, 250402 (2014).
  35. M. M. Taddei, R. V. Nery, and L. Aolita, Quantum superpositions of causal orders as an operational resource, Phys. Rev. Res. 1, 033174 (2019).
  36. M. J. Renner and Č. Brukner, Computational advantage from a quantum superposition of qubit gate orders, Phys. Rev. Lett. 128, 230503 (2022).
  37. J. Escandón-Monardes, A. Delgado, and S. P. Walborn, Practical computational advantage from the quantum switch on a generalized family of promise problems, Quantum 7, 945 (2023).
  38. W.-Q. Liu, Z. Meng, B.-W. Song, J. Li, Q.-Y. Wu, X.-X. Chen, J.-Y. Hong, A.-N. Zhang, and Z.-Q. Yin, Experimentally demonstrating indefinite causal order algorithms to solve the generalized Deutsch's problem, Adv. Quantum Technol. 7, 2400181 (2024).
  39. M. M. Taddei, J. Cariñe, D. Martínez, T. García, N. Guerrero, A. A. Abbott, M. Araújo, C. Branciard, E. S. Gómez, S. P. Walborn, L. Aolita, and G. Lima, Computational advantage from quantum superposition of multiple temporal orders of photonic gates, PRX Quantum 2, 010320 (2021).
  40. A. Barenco, A universal two-bit gate for quantum computation, Proc. R. Soc. Lond. A 449, 679 (1995).
  41. V. Vilasini and R. Renner, Fundamental limits for realizing quantum processes in spacetime, Phys. Rev. Lett. 133, 080201 (2024).
  42. O. Oreshkov, Time-delocalized quantum subsystems and operations: On the existence of processes with indefinite causal structure in quantum mechanics, Quantum 3, 206 (2019).
  43. L. M. Procopio, F. Delgado, M. Enríquez, N. Belabas, and J. A. Levenson, Sending classical information via three noisy channels in superposition of causal orders, Phys. Rev. A 101, 012346 (2020).
  44. G. Rubino, L. A. Rozema, F. Massa, M. Araújo, M. Zych, Č. Brukner, and P. Walther, Experimental entanglement of temporal orders, Quantum 6, 621 (2022).
  45. T. Strömberg, P. Schiansky, R. W. Peterson, M. T. Quintino, and P. Walther, Demonstration of a quantum switch in a Sagnac configuration, Phys. Rev. Lett. 131, 060803 (2023).
  46. K. Goswami, Y. Cao, G. A. Paz-Silva, J. Romero, and A. G. White, Increasing communication capacity via superposition of order, Phys. Rev. Res. 2, 033292 (2020).
  47. A. Vanrietvelde and G. Chiribella, Universal control of quantum processes using sector-preserving channels, Quantum Inf. Comput. 21, 1320 (2021).
  48. M. Reck, A. Zeilinger, H. J. Bernstein, and P. Bertani, Experimental realization of any discrete unitary operator, Phys. Rev. Lett. 73, 58 (1994).
  49. J. Wang, F. Sciarrino, A. Laing, and M. G. Thompson, Integrated photonic quantum technologies, Nat. Photon. 14, 273 (2020).
  50. E. Knill, R. Laflamme, and G. J. Milburn, A scheme for efficient quantum computation with linear optics, Nature (London) 409, 46 (2001).
  51. X.-H. Bao, T.-Y. Chen, Q. Zhang, J. Yang, H. Zhang, T. Yang, and J.-W. Pan, Optical nondestructive controlled-not gate without using entangled photons, Phys. Rev. Lett. 98, 170502 (2007).
  52. J. Zeuner, A. N. Sharma, M. Tillmann, R. Heilmann, M. Gräfe, A. Moqanaki, A. Szameit, and P. Walther, Integrated-optics heralded controlled-not gate for polarization-encoded qubits, npj Quantum Inf. 4, 13 (2018).
  53. W.-Q. Liu, H.-R. Wei, and L.-C. Kwek, Low-cost Fredkin gate with auxiliary space, Phys. Rev. Appl. 14, 054057 (2020).
  54. J.-P. Li, X. Gu, J. Qin, D. Wu, X. You, H. Wang, C. Schneider, S. Höfling, Y.-H. Huo, C.-Y. Lu, N.-L. Liu, L. Li, and J.-W. Pan, Heralded nondestructive quantum entangling gate with single-photon sources, Phys. Rev. Lett. 126, 140501 (2021).
  55. W.-Q. Liu and H.-R. Wei, Linear optical universal quantum gates with higher success probabilities, Adv. Quantum Technol. 6, 2300009 (2023).
  56. F. Pegoraro, P. Held, J. Lammers, B. Brecht, and C. Silberhorn, Demonstration of a photonic time-multiplexed C-NOT gate, arXiv:2412.02478.
  57. R. Okamoto, J. L. O’Brien, H. F. Hofmann, and S. Takeuchi, Realization of a Knill-Laflamme-Milburn controlled-not photonic quantum circuit combining effective optical nonlinearities, Proc. Natl. Acad. Sci. USA 108, 10067 (2011).
  58. S. Slussarenko and G. J. Pryde, Photonic quantum information processing: A concise review, Appl. Phys. Rev. 6, 041303 (2019).
  59. G. Chiribella, G. M. D’Ariano, and P. Perinotti, Quantum circuit architecture, Phys. Rev. Lett. 101, 060401 (2008).
  60. G. Chiribella, G. M. D’Ariano, and P. Perinotti, Theoretical framework for quantum networks, Phys. Rev. A 80, 022339 (2009).
  61. A. Bisio and P. Perinotti, Theoretical framework for higher-order quantum theory, Proc. R. Soc. A 475, 20180706 (2019).
  62. L. Apadula, A. Bisio, and P. Perinotti, No-signalling constraints quantum computation with indefinite causal structure, Quantum 8, 1241 (2024).
  63. K. Goswami and J. Romero, Experiments on quantum causality, AVS Quantum Sci. 2, 037101 (2020).
  64. H. Cao, N.-N. Wang, Z. Jia, C. Zhang, Y. Guo, B.-H. Liu, Y.-F. Huang, C.-F. Li, and G.-C. Guo, Quantum simulation of indefinite causal order induced quantum refrigeration, Phys. Rev. Res. 4, L032029 (2022).
  65. X. Nie, X. Zhu, K. Huang, K. Tang, X. Long, Z. Lin, Y. Tian, C. Qiu, C. Xi, X. Yang, J. Li, Y. Dong, T. Xin, and D. Lu, Experimental realization of a quantum refrigerator driven by indefinite causal orders, Phys. Rev. Lett. 129, 100603 (2022).
  66. M. Antesberger, M. T. Quintino, P. Walther, and L. A. Rozema, Higher-order process matrix tomography of a passively-stable quantum switch, PRX Quantum 5, 010325 (2024).
  67. L. A. Rozema, T. Strömberg, H. Cao, Y. Guo, B.-H. Liu, and P. Walther, Experimental aspects of indefinite causal order in quantum mechanics, Nat. Rev. Phys. 6, 483 (2024).
  68. N. Gao, D. Li, A. Mishra, J. Yan, K. Simonov, and G. Chiribella, Measuring incompatibility and clustering quantum observables with a quantum switch, Phys. Rev. Lett. 130, 170201 (2023).
  69. M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information, 10th ed. (Cambridge University Press, Cambridge, 2011).
  70. The results obtained in what follows can be straightforwardly extended to arbitrary (mixed) state ρ by considering the action of S(A,B) on it as S(A,B)ρS†(A,B).
  71. T. van der Lugt, J. Barrett, and G. Chiribella, Device-independent certification of indefinite causal order in the quantum switch, Nat. Commun. 14, 5811 (2023).
  72. We note that the notion of equivalence given in Definition t5, also referred to in literature as LU equivalence [73, 74, 75], restricts the allowed unitary operators to tensor product of single-qubit unitary gates only. The rationale for this constraint lies in the aim of enabling universal quantum computing via superposed orders of single-qubits gates.
  73. J. Zhang, J. Vala, S. Sastry, and K. B. Whaley, Geometric theory of nonlocal two-qubit operations, Phys. Rev. A 67, 042313 (2003).
  74. M. Musz, M. Kuś, and K. Życzkowski, Unitary quantum gates, perfect entanglers, and unistochastic maps, Phys. Rev. A 87, 022111 (2013).
  75. Y. Shen, L. Chen, and L. Yu, Classification of Schmidt-rank-two multipartite unitary gates by singular number, J. Phys. A: Math. Theor. 55, 465302 (2022).
  76. M. Caleffi, M. Amoretti, D. Ferrari, J. Illiano, A. Manzalini, and A. S. Cacciapuoti, Distributed quantum computing: A survey, Comput. Netw. 254, 110672 (2024).
  77. A. Hashim, R. Rines, V. Omole, R. K. Naik, J. M. Kreikebaum, D. I. Santiago, F. T. Chong, I. Siddiqi, and P. Gokhale, Optimized swap networks with equivalent circuit averaging for qaoa, Phys. Rev. Res. 4, 033028 (2022).
  78. C. G. Almudéver, A quantum computer architecture perspective: Executing quantum applications on real quantum processors, in Proceedings of SPIE 11918, Photonics for Quantum (2020), p. 1191809.
  79. C. G. Almudever and E. Alarcon, Structured optimized architecting of full-stack quantum systems in the NISQ era, in 2021 Design, Automation & Test in Europe Conference & Exhibition (DATE) (Grenoble, France, 2021), pp. 762–767.
  80. C. G. Almudever, R. Wille, F. Sebastiano, N. Haider and E. Alarcon, From Designing Quantum Processors to Large-Scale Quantum Computing Systems, in 2024 Design, Automation & Test in Europe Conference & Exhibition (DATE) (Valencia, Spain, 2024), pp. 1–10.
  81. D. Gottesman, Fault-tolerant quantum computation with higher-dimensional systems, Chaos Solitons Frac. 10, 1749 (1999).
  82. J. Daboul, X. Wang, and B. C. Sanders, Quantum gates on hybrid qudits, J. Phys. A: Math. Gen. 36, 2525 (2003).
  83. Y. Wang, Z. Hu, B. C. Sanders, and S. Kais, Qudits and high-dimensional quantum computing, Front. Phys. 8, 589504 (2020).
  84. C. P. Williams, Explorations in Quantum Computing (Springer, London, 2011), p. 93.
  85. X.-F. Shi, Universal Barenco quantum gates via a tunable noncollinear interaction, Phys. Rev. A 97, 032310 (2018).

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Phys. Rev. Research 8, 013239– Published 5 March, 2026
  • Received 18 April 2025
  • Accepted 19 December 2025
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