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Full Characterization of Genuine 17-qubit Entanglement on the Superconducting Processor

Chang-Kang Hu1,2,3,*, Yong Wang4,5,6,*, YuXuan Zhou1,2,3,*, ChiLong Liu1,2,3, GuiXu Xie1,2,3, RuiYang Zhou1,2,3, HaoLan Yuan1,2,3, Lijun Liu7,†, Li Li6,8 et al.

Song Liu1,2,3,9,‡, Dian Tan1,2,3,9,§, Shuming Cheng6,8,10,∥, and Dapeng Yu1,2,3,9,11

  • *These authors contributed equally to this work.
  • †Contact author: lljcelia@126.com
  • ‡Contact author: lius3@sustech.edu.cn
  • §Contact author: tand@sustech.edu.cn
  • ∥Contact author: shuming_cheng@tongji.edu.cn
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Phys. Rev. Lett. 135, 070801 – Published 13 August, 2025

DOI: https://doi.org/10.1103/qy9y-7ywp

Abstract

We present a least square state estimator regularized by state purity to accomplish the task of quantum state tomography and entanglement verification, and report an experimental validation on a superconducting processor. First, a scalable full-state tomography is achieved with state fidelity 0.8217(1) for the 9-qubit W state and 0.6817(1) for the 17-qubit Greenberger-Horne-Zeilinger state, with the latter reaching 0.7587(1) under a pure-state assumption. Then, the presence of genuine multiqubit entanglement in the majority of generated states is conclusively certified by violating the corresponding entanglement witness. Finally, it is demonstrated that our method can achieve superior tomography accuracy with limited measurement settings and that mitigating error is essential for leveraging noisy quantum systems in quantum processing tasks. Our results pave the way for more accurate full characterization of larger-scale many-body quantum systems.

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References (78)

  1. F. Arute, K. Arya, R. Babbush, D. Bacon, J. C. Bardin, R. Barends, R. Biswas, S. Boixo, F. G. Brandao, D. A. Buell et al., Quantum supremacy using a programmable superconducting processor, Nature (London) 574, 505 (2019).
  2. Y. Wu, W.-S. Bao, S. Cao, F. Chen, M.-C. Chen, X. Chen, T.-H. Chung, H. Deng, Y. Du, D. Fan et al., Strong quantum computational advantage using a superconducting quantum processor, Phys. Rev. Lett. 127, 180501 (2021).
  3. G. Q. AI, Suppressing quantum errors by scaling a surface code logical qubit, Nature (London) 614, 676 (2023).
  4. S. Cao, B. Wu, F. Chen, M. Gong, Y. Wu, Y. Ye, C. Zha, H. Qian, C. Ying, S. Guo et al., Generation of genuine entanglement up to 51 superconducting qubits, Nature (London) 619, 738 (2023).
  5. L. V. Abdurakhimov, I. Mahboob, H. Toida, K. Kakuyanagi, Y. Matsuzaki, and S. Saito, Identification of different types of high-frequency defects in superconducting qubits, PRX Quantum 3, 040332 (2022).
  6. L. Postler, S. Heußen, I. Pogorelov, M. Rispler, T. Feldker, M. Meth, C. D. Marciniak, R. Stricker, M. Ringbauer, R. Blatt et al., Demonstration of fault-tolerant universal quantum gate operations, Nature (London) 605, 675 (2022).
  7. S. Moses, C. Baldwin, M. Allman, R. Ancona, L. Ascarrunz, C. Barnes, J. Bartolotta, B. Bjork, P. Blanchard, M. Bohn et al., A race-track trapped-ion quantum processor, Phys. Rev. X 13, 041052 (2023).
  8. H.-S. Zhong, H. Wang, Y.-H. Deng, M.-C. Chen, L.-C. Peng, Y.-H. Luo, J. Qin, D. Wu, X. Ding, Y. Hu et al., Quantum computational advantage using photons, Science 370, 1460 (2020).
  9. L. S. Madsen, F. Laudenbach, M. F. Askarani, F. Rortais, T. Vincent, J. F. Bulmer, F. M. Miatto, L. Neuhaus, L. G. Helt, M. J. Collins et al., Quantum computational advantage with a programmable photonic processor, Nature (London) 606, 75 (2022).
  10. D. Bluvstein, S. J. Evered, A. A. Geim, S. H. Li, H. Zhou, T. Manovitz, S. Ebadi, M. Cain, M. Kalinowski, D. Hangleiter et al., Logical quantum processor based on reconfigurable atom arrays, Nature (London) 626, 58 (2023).
  11. A. Anshu and S. Arunachalam, A survey on the complexity of learning quantum states, Nat. Rev. Phys. 6, 59 (2024).
  12. K. Vogel and H. Risken, Determination of quasiprobability distributions in terms of probability distributions for the rotated quadrature phase, Phys. Rev. A 40, 2847 (1989).
  13. H. Häffner, W. Hänsel, C. Roos, J. Benhelm, D. Chek-al Kar, M. Chwalla, T. Körber, U. Rapol, M. Riebe, P. Schmidt et al., Scalable multiparticle entanglement of trapped ions, Nature (London) 438, 643 (2005).
  14. 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).
  15. M. Baur, A. Fedorov, L. Steffen, S. Filipp, M. P. da Silva, and A. Wallraff, Benchmarking a quantum teleportation protocol in superconducting circuits using tomography and an entanglement witness, Phys. Rev. Lett. 108, 040502 (2012).
  16. W.-T. Liu, T. Zhang, J.-Y. Liu, P.-X. Chen, and J.-M. Yuan, Experimental quantum state tomography via compressed sampling, Phys. Rev. Lett. 108, 170403 (2012).
  17. Z. Hou, H.-S. Zhong, Y. Tian, D. Dong, B. Qi, L. Li, Y. Wang, F. Nori, G.-Y. Xiang, C.-F. Li et al., Full reconstruction of a 14-qubit state within four hours, New J. Phys. 18, 083036 (2016).
  18. C. A. Riofrio, D. Gross, S. T. Flammia, T. Monz, D. Nigg, R. Blatt, and J. Eisert, Experimental quantum compressed sensing for a seven-qubit system, Nat. Commun. 8, 15305 (2017).
  19. A. Steffens, C. Riofrío, W. McCutcheon, I. Roth, B. A. Bell, A. McMillan, M. Tame, J. Rarity, and J. Eisert, Experimentally exploring compressed sensing quantum tomography, Quantum Sci. Technol. 2, 025005 (2017).
  20. C. Song, K. Xu, W. Liu, C.-p. Yang, S.-B. Zheng, H. Deng, Q. Xie, K. Huang, Q. Guo, L. Zhang, P. Zhang, D. Xu, D. Zheng, X. Zhu, H. Wang, Y. A. Chen, C. Y. Lu, S. Han, and J. W. Pan, 10-qubit entanglement and parallel logic operations with a superconducting circuit, Phys. Rev. Lett. 119, 180511 (2017).
  21. B. Lanyon, C. Maier, M. Holzäpfel, T. Baumgratz, C. Hempel, P. Jurcevic, I. Dhand, A. Buyskikh, A. Daley, M. Cramer et al., Efficient tomography of a quantum many-body system, Nat. Phys. 13, 1158 (2017).
  22. J. G. Titchener, M. Gräfe, R. Heilmann, A. S. Solntsev, A. Szameit, and A. A. Sukhorukov, Scalable on-chip quantum state tomography, npj Quantum Inf. 4, 19 (2018).
  23. M. Naghiloo, M. Abbasi, Y. N. Joglekar, and K. Murch, Quantum state tomography across the exceptional point in a single dissipative qubit, Nat. Phys. 15, 1232 (2019).
  24. B. Fang, M. Menotti, M. Liscidini, J. E. Sipe, and V. O. Lorenz, Three-photon discrete-energy-entangled W state in an optical fiber, Phys. Rev. Lett. 123, 070508 (2019).
  25. J.-C. Besse, K. Reuer, M. C. Collodo, A. Wulff, L. Wernli, A. Copetudo, D. Malz, P. Magnard, A. Akin, M. Gabureac et al., Realizing a deterministic source of multipartite-entangled photonic qubits, Nat. Commun. 11, 4877 (2020).
  26. C.-J. Huang, G.-Y. Xiang, Y. Guo, K.-D. Wu, B.-H. Liu, C.-F. Li, G.-C. Guo, and A. Tavakoli, Nonlocality, steering, and quantum state tomography in a single experiment, Phys. Rev. Lett. 127, 020401 (2021).
  27. 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).
  28. R. Stricker, M. Meth, L. Postler, C. Edmunds, C. Ferrie, R. Blatt, P. Schindler, T. Monz, R. Kueng, and M. Ringbauer, Experimental single-setting quantum state tomography, PRX Quantum 3, 040310 (2022).
  29. D. Gross, Y.-K. Liu, S. T. Flammia, S. Becker, and J. Eisert, Quantum state tomography via compressed sensing, Phys. Rev. Lett. 105, 150401 (2010).
  30. A. Kalev, R. L. Kosut, and I. H. Deutsch, Quantum tomography protocols with positivity are compressed sensing protocols, npj Quantum Inf. 1, 15018 (2015).
  31. F. Verstraete, J. J. García-Ripoll, and J. I. Cirac, Matrix product density operators: Simulation of finite-temperature and dissipative systems, Phys. Rev. Lett. 93, 207204 (2004).
  32. M. Cramer, M. B. Plenio, S. T. Flammia, R. Somma, D. Gross, S. D. Bartlett, O. Landon-Cardinal, D. Poulin, and Y.-K. Liu, Efficient quantum state tomography, Nat. Commun. 1, 149 (2010).
  33. M. K. Kurmapu, V. V. Tiunova, E. S. Tiunov, M. Ringbauer, C. Maier, R. Blatt, T. Monz, A. K. Fedorov, and A. I. Lvovsky, Reconstructing complex states of a 20-qubit quantum simulator, PRX Quantum 4, 040345 (2023).
  34. G. Tóth, W. Wieczorek, D. Gross, R. Krischek, C. Schwemmer, and H. Weinfurter, Permutationally invariant quantum tomography, Phys. Rev. Lett. 105, 250403 (2010).
  35. M. Christandl and R. Renner, Reliable quantum state tomography, Phys. Rev. Lett. 109, 120403 (2012).
  36. S. De Nicola, A. A. Michailidis, and M. Serbyn, Entanglement view of dynamical quantum phase transitions, Phys. Rev. Lett. 126, 040602 (2021).
  37. J. Chen, E. M. Stoudenmire, and S. R. White, Quantum Fourier transform has small entanglement, PRX Quantum 4, 040318 (2023).
  38. G. Carleo and M. Troyer, Solving the quantum many-body problem with artificial neural networks, Science 355, 602 (2017).
  39. G. Torlai, G. Mazzola, J. Carrasquilla, M. Troyer, R. Melko, and G. Carleo, Neural-network quantum state tomography, Nat. Phys. 14, 447 (2018).
  40. G. Torlai and R. G. Melko, Latent space purification via neural density operators, Phys. Rev. Lett. 120, 240503 (2018).
  41. Y. Wang, S. Cheng, L. Li, and J. Chen, Learning quantum distributions with variational diffusion models, IFAC-PapersOnLine 56, 5888 (2023).
  42. A. Elben, S. T. Flammia, H.-Y. Huang, R. Kueng, J. Preskill, B. Vermersch, and P. Zoller, The randomized measurement toolbox, Nat. Rev. Phys. 5, 9 (2023).
  43. S. T. Flammia and Y.-K. Liu, Direct fidelity estimation from few Pauli measurements, Phys. Rev. Lett. 106, 230501 (2011).
  44. M. P. da Silva, O. Landon-Cardinal, and D. Poulin, Practical characterization of quantum devices without tomography, Phys. Rev. Lett. 107, 210404 (2011).
  45. S. Aaronson, Shadow Tomography of Quantum States (Association for Computing Machinery, New York, 2018), pp. 325–338.
  46. H.-Y. Huang, R. Kueng, and J. Preskill, Predicting many properties of a quantum system from very few measurements, Nat. Phys. 16, 1050 (2020).
  47. Y. Zhou and Q. Liu, Performance analysis of multi-shot shadow estimation, Quantum 7, 1044 (2023).
  48. S. Endo, S. C. Benjamin, and Y. Li, Practical quantum error mitigation for near-future applications, Phys. Rev. X 8, 031027 (2018).
  49. G. J. Mooney, G. A. L. White, C. D. Hill, and L. C. L. Hollenberg, Generation and verification of 27-qubit Greenberger-Horne-Zeilinger states in a superconducting quantum computer, J. Phys. Commun. 5, 095004 (2021).
  50. B. Yang, R. Raymond, and S. Uno, Efficient quantum readout-error mitigation for sparse measurement outcomes of near-term quantum devices, Phys. Rev. A 106, 012423 (2022).
  51. Z. Cai, R. Babbush, S. C. Benjamin, S. Endo, W. J. Huggins, Y. Li, J. R. McClean, and T. E. O’Brien, Quantum error mitigation, Rev. Mod. Phys. 95, 045005 (2023).
  52. O. Gühne and G. Tóth, Entanglement detection, Phys. Rep. 474, 1 (2009).
  53. Y. Chen, M. Farahzad, S. Yoo, and T.-C. Wei, Detector tomography on IBM quantum computers and mitigation of an imperfect measurement, Phys. Rev. A 100, 052315 (2019).
  54. A. Kandala, K. Temme, A. D. Córcoles, A. Mezzacapo, J. M. Chow, and J. M. Gambetta, Error mitigation extends the computational reach of a noisy quantum processor, Nature (London) 567, 491 (2019).
  55. A. Strikis, D. Qin, Y. Chen, S. C. Benjamin, and Y. Li, Learning-based quantum error mitigation, PRX Quantum 2, 040330 (2021).
  56. W. P. Livingston, M. S. Blok, E. Flurin, J. Dressel, A. N. Jordan, and I. Siddiqi, Experimental demonstration of continuous quantum error correction, Nat. Commun. 13, 2307 (2022).
  57. D. Qin, X. Xu, and Y. Li, An overview of quantum error mitigation formulas, Chin. Phys. B 31, 090306 (2022).
  58. J. A. Smolin, J. M. Gambetta, and G. Smith, Efficient method for computing the maximum-likelihood quantum state from measurements with additive Gaussian noise, Phys. Rev. Lett. 108, 070502 (2012).
  59. B. Qi, Z. Hou, L. Li, D. Dong, G. Xiang, and G. Guo, Quantum state tomography via linear regression estimation, Sci. Rep. 3, 3496 (2013).
  60. B. Qi, Z. Hou, Y. Wang, D. Dong, H.-S. Zhong, L. Li, G.-Y. Xiang, H. M. Wiseman, C.-F. Li, and G.-C. Guo, Adaptive quantum state tomography via linear regression estimation: Theory and two-qubit experiment, npj Quantum Inf. 3, 19 (2017).
  61. R. Jozsa, Fidelity for mixed quantum states, J. Mod. Opt. 41, 2315 (1994).
  62. A. J. Baldwin and J. A. Jones, Efficiently computing the Uhlmann fidelity for density matrices, Phys. Rev. A 107, 012427 (2023).
  63. Y. Wang, L. Liu, T. Dou, L. Li, and S. Cheng, Quantum state tomography based on infidelity estimation, Quantum Sci. Technol. 10, 015035 (2024).
  64. K. Banaszek, G. M. D’Ariano, M. G. A. Paris, and M. F. Sacchi, Maximum-likelihood estimation of the density matrix, Phys. Rev. A 61, 010304(R) (1999).
  65. J. Shang, Z. Zhang, and H. K. Ng, Superfast maximum-likelihood reconstruction for quantum tomography, Phys. Rev. A 95, 062336 (2017).
  66. See Supplemental Material at http://link.aps.org/supplemental/10.1103/qy9y-7ywp for a detailed description of the device performance, state preparation circuits, proposed state estimator, readout error mitigation, as well as reconstructed density matrices.
  67. P. C. Hansen and D. P. O’Leary, The use of the L-curve in the regularization of discrete ill-posed problems, SIAM J. Sci. Comput. 14, 1487 (1993).
  68. P. Johnston and R. Gulrajani, Selecting the corner in the L-curve approach to Tikhonov regularization, IEEE Trans. Biomed. Eng. 47, 1293 (2000).
  69. T. Brydges, A. Elben, P. Jurcevic, B. Vermersch, C. Maier, B. P. Lanyon, P. Zoller, R. Blatt, and C. F. Roos, Probing Rényi entanglement entropy via randomized measurements, Science 364, 260 (2019).
  70. S. Vidal Romero and J. Santos-Suárez, PauliComposer: Compute tensor products of Pauli matrices efficiently, Quantum Inf Process. 22, 449 (2023).
  71. C. Lanczos, An iteration method for the solution of the eigenvalue problem of linear differential and integral operators, J. Res. Natl. Bur. Stand. 45, 2133 (1950).
  72. F. Yan, P. Krantz, Y. Sung, M. Kjaergaard, D. L. Campbell, T. P. Orlando, S. Gustavsson, and W. D. Oliver, Tunable coupling scheme for implementing high-fidelity two-qubit gates, Phys. Rev. Appl. 10, 054062 (2018).
  73. Y. Xu, J. Chu, J. Yuan, J. Qiu, Y. Zhou, L. Zhang, X. Tan, Y. Yu, S. Liu, J. Li, F. Yan, and D. Yu, High-fidelity, high-scalability two-qubit gate scheme for superconducting qubits, Phys. Rev. Lett. 125, 240503 (2020).
  74. J. Chu, X. He, Y. Zhou, J. Yuan, L. Zhang, Q. Guo, Y. Hai, Z. Han, C.-K. Hu, W. Huang et al., Scalable algorithm simplification using quantum AND logic, Nat. Phys. 19, 126 (2023).
  75. D. Cruz, R. Fournier, F. Gremion, A. Jeannerot, K. Komagata, T. Tosic, J. Thiesbrummel, C. L. Chan, N. Macris, M.-A. Dupertuis et al., Efficient quantum algorithms for GHZ and W states, and implementation on the IBM quantum computer, Adv. Quantum Technol. 2, 1900015 (2019).
  76. S. Kumar, Wishart and random density matrices: Analytical results for the mean-square Hilbert-Schmidt distance, Phys. Rev. A 102, 012405 (2020).
  77. E. Bolduc, G. C. Knee, E. M. Gauger, and J. Leach, Projected gradient descent algorithms for quantum state tomography, npj Quantum Inf. 3, 44 (2017).
  78. Y. Wang, L. Liu, S. Cheng, L. Li, and J. Chen, Efficient factored gradient descent algorithm for quantum state tomography, Phys. Rev. Res. 6, 033034 (2024).

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Phys. Rev. Lett. 135, 070801– Published 13 August, 2025
  • Received 2 February 2024
  • Revised 6 June 2024
  • Accepted 7 July 2025
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