- Physical Review Journals
All JournalsPhysics Magazine

Physical Review Letters

Reuse & Permissions

It is not necessary to obtain permission to reuse this article or its components as it is available under the terms of the Creative Commons Attribution 4.0 International license. This license permits unrestricted use, distribution, and reproduction in any medium, provided attribution to the author(s) and the published article's title, journal citation, and DOI are maintained. Please note that some figures may have been included with permission from other third parties. It is your responsibility to obtain the proper permission from the rights holder directly for these figures.

Export citation

Export citation

Choose format for download:

Download Citation
  • Open Access

Bell Sampling in Quantum Monte Carlo Simulations

Poetri Sonya Tarabunga1,2,*,† and Yi-Ming Ding3,4,5,†,‡

  • *Contact author: poetri.tarabunga@tum.de
  • †These authors contributed equally to this work.
  • ‡Contact author: dingyiming@westlake.edu.cn
PDF

Phys. Rev. Lett. 135, 200403 – Published 13 November, 2025

DOI: https://doi.org/10.1103/fq8z-y55j

Abstract

Quantum Monte Carlo (QMC) methods are essential for the numerical study of large-scale quantum many-body systems, yet their utility has been significantly hampered by the difficulty in computing key quantities such as off-diagonal operators and entanglement. This Letter introduces Bell-QMC, a novel QMC framework leveraging Bell sampling, a two-copy measurement protocol in the transversal Bell basis. We demonstrate that Bell-QMC enables an efficient and unbiased estimation of both challenging classes of observables, offering a significant advantage over previous QMC approaches. Notably, the entanglement across all system partitions can be computed in a single Bell-QMC simulation. We implement this method within the stochastic series expansion, where we design an efficient update scheme for sampling the configurations in the Bell basis. We demonstrate our algorithm in the one-dimensional transverse-field Ising model and the two-dimensional Z2 lattice gauge theory, extracting universal quantum features using only simple diagonal measurements. This Letter establishes Bell-QMC as a powerful framework that significantly expands the accessible quantum properties in QMC simulations, providing a substantial advantage over conventional QMC.

View figure in article

Physics Subject Headings (PhySH)

Article Text

Supplemental Material

References (92)

  1. U. Schollwöck, Ann. Phys. (Amsterdam) 326, 96 (2011), January 2011 Special Issue.
  2. R. Orús, Ann. Phys. (Amsterdam) 349, 117 (2014).
  3. A. W. Sandvik, A. Avella, and F. Mancini, in AIP Conference Proceedings (AIP, Vietri sul Mare, Italy, 2010).
  4. J. Gubernatis, N. Kawashima, and P. Werner, Quantum Monte Carlo Methods: Algorithms for Lattice Models (Cambridge University Press, Cambridge, England, 2016).
  5. F. Becca and S. Sorella, Quantum Monte Carlo Approaches for Correlated Systems (Cambridge University Press, Cambridge, England, 2017).
  6. M. M. Wolf, F. Verstraete, M. B. Hastings, and J. I. Cirac, Phys. Rev. Lett. 100, 070502 (2008).
  7. P. Calabrese and J. Cardy, J. Stat. Mech. (2004) P06002.
  8. M. A. Metlitski and T. Grover, arXiv:1112.5166.
  9. A. Kitaev and J. Preskill, Phys. Rev. Lett. 96, 110404 (2006).
  10. M. Levin and X.-G. Wen, Phys. Rev. Lett. 96, 110405 (2006).
  11. L. Amico, R. Fazio, A. Osterloh, and V. Vedral, Rev. Mod. Phys. 80, 517 (2008).
  12. N. Laflorencie, Phys. Rep. 646, 1 (2016).
  13. M. B. Hastings, I. González, A. B. Kallin, and R. G. Melko, Phys. Rev. Lett. 104, 157201 (2010).
  14. S. Humeniuk and T. Roscilde, Phys. Rev. B 86, 235116 (2012).
  15. X. Zhou, Z. Y. Meng, Y. Qi, and Y. Da Liao, Phys. Rev. B 109, 165106 (2024).
  16. R. G. Melko, A. B. Kallin, and M. B. Hastings, Phys. Rev. B 82, 100409 (2010).
  17. Y.-M. Ding, J.-S. Sun, N. Ma, G. Pan, C. Cheng, and Z. Yan, Phys. Rev. B 110, 165152 (2024).
  18. Z. Wang, Z. Wang, Y.-M. Ding, B.-B. Mao, and Z. Yan, Nat. Commun. 16, 5880 (2025).
  19. J. D’Emidio, Phys. Rev. Lett. 124, 110602 (2020).
  20. Z. Wang, Z. Liu, B.-B. Mao, Z. Wang, and Z. Yan, arXiv:2412.01384.
  21. D. Hangleiter and M. J. Gullans, Phys. Rev. Lett. 133, 020601 (2024).
  22. H.-Y. Huang, M. Broughton, J. Cotler, S. Chen, J. Li, M. Mohseni, H. Neven, R. Babbush, R. Kueng, J. Preskill, and J. R. McClean, Science 376, 1182 (2022).
  23. H.-Y. Huang, R. Kueng, and J. Preskill, Phys. Rev. Lett. 126, 190505 (2021).
  24. T. Haug and M. Kim, PRX Quantum 4, 010301 (2023).
  25. T. Haug, S. Lee, and M. Kim, Phys. Rev. Lett. 132, 240602 (2024).
  26. A. Montanaro, arXiv:1707.04012.
  27. D. Gross, S. Nezami, and M. Walter, Commun. Math. Phys. 385, 1325 (2021).
  28. S. Grewal, V. Iyer, W. Kretschmer, and D. Liang, arXiv:2305.13409.
  29. R. King, D. Gosset, R. Kothari, and R. Babbush, PRX Quantum 6, 010336 (2025).
  30. S. Chen, J. Cotler, H.-Y. Huang, and J. Li, in 2021 IEEE 62nd Annual Symposium on Foundations of Computer Science (FOCS) (IEEE Computer Society, Los Alamitos, CA, USA, 2022), pp. 574–585.
  31. 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., Nature (London) 626, 58 (2024).
  32. T. Haug and P. S. Tarabunga, arXiv:2504.18098.
  33. R. Islam, R. Ma, P. M. Preiss, M. Eric Tai, A. Lukin, M. Rispoli, and M. Greiner, Nature (London) 528, 77 (2015).
  34. A. M. Kaufman, M. E. Tai, A. Lukin, M. Rispoli, R. Schittko, P. M. Preiss, and M. Greiner, Science 353, 794 (2016).
  35. N. M. Linke, S. Johri, C. Figgatt, K. A. Landsman, A. Y. Matsuura, and C. Monroe, Phys. Rev. A 98, 052334 (2018).
  36. P. Horodecki and A. Ekert, Phys. Rev. Lett. 89, 127902 (2002).
  37. A. K. Ekert, C. M. Alves, D. K. L. Oi, M. Horodecki, P. Horodecki, and L. C. Kwek, Phys. Rev. Lett. 88, 217901 (2002).
  38. T. Haug, K. Bharti, and D. E. Koh, Quantum 9, 1759 (2025).
  39. A. W. Sandvik, Phys. Rev. E 68, 056701 (2003).
  40. See Supplemental Material at http://link.aps.org/supplemental/10.1103/fq8z-y55j for technical details of the algorithm and additional numerical data, which includes Refs. [41–46].
  41. A. W. Sandvik, Phys. Rev. B 59, R14157 (1999).
  42. O. F. Syljuåsen and A. W. Sandvik, Phys. Rev. E 66, 046701 (2002).
  43. O. F. Syljuåsen, Phys. Rev. E 67, 046701 (2003).
  44. J. Zhao, Z. Yan, M. Cheng, and Z. Y. Meng, Phys. Rev. Res. 3, 033024 (2021).
  45. A. W. Sandvik, Phys. Rev. B 56, 11678 (1997).
  46. R. G. Melko and A. W. Sandvik, Phys. Rev. E 72, 026702 (2005).
  47. A. W. Sandvik, arXiv:1909.10591.
  48. R. G. Melko, Stochastic series expansion quantum Monte Carlo, in Strongly Correlated Systems: Numerical Methods, edited by A. Avella and F. Mancini (Springer, Berlin, Heidelberg, 2013), pp. 185–206.
  49. P. Di Francesco, P. Mathieu, and D. Sénéchal, Conformal Field Theory, Graduate Texts in Contemporary Physics (Springer, New York, 1997).
  50. B. Zeng and X.-G. Wen, Phys. Rev. B 91, 125121 (2015).
  51. B. Zeng, X. Chen, D.-L. Zhou, and X.-G. Wen, Quantum Information Meets Quantum Matter: From Quantum Entanglement to Topological Phases of Many-Body Systems (Springer, New York, 2019).
  52. A. Lavasani, Y. Alavirad, and M. Barkeshli, Nat. Phys. 17, 342 (2021).
  53. K. Klocke and M. Buchhold, Phys. Rev. B 106, 104307 (2022).
  54. H. W. J. Blöte and Y. Deng, Phys. Rev. E 66, 066110 (2002).
  55. J. Zhao, B.-B. Chen, Y.-C. Wang, Z. Yan, M. Cheng, and Z. Y. Meng, npj Quantum Mater. 7, 69 (2022).
  56. S. J. Garratt, Z. Weinstein, and E. Altman, Phys. Rev. X 13, 021026 (2023).
  57. J. Y. Lee, C.-M. Jian, and C. Xu, PRX Quantum 4, 030317 (2023).
  58. S. Murciano, P. Sala, Y. Liu, R. S. Mong, and J. Alicea, Phys. Rev. X 13, 041042 (2023).
  59. Z. Weinstein, R. Sajith, E. Altman, and S. J. Garratt, Phys. Rev. B 107, 245132 (2023).
  60. Z. Yang, D. Mao, and C.-M. Jian, Phys. Rev. B 108, 165120 (2023).
  61. M. Hoshino, M. Oshikawa, and Y. Ashida, Phys. Rev. B 111, 155143 (2025).
  62. K. Baweja, D. J. Luitz, and S. J. Garratt, arXiv:2410.13844.
  63. L. A. Lessa, R. Ma, J.-H. Zhang, Z. Bi, M. Cheng, and C. Wang, PRX Quantum 6, 010344 (2025).
  64. P. Sala, S. Gopalakrishnan, M. Oshikawa, and Y. You, Phys. Rev. B 110, 155150 (2024).
  65. E. Chitambar and G. Gour, Rev. Mod. Phys. 91, 025001 (2019).
  66. L. Leone, S. F. E. Oliviero, and A. Hamma, Phys. Rev. Lett. 128, 050402 (2022).
  67. T. Haug and L. Piroli, Phys. Rev. B 107, 035148 (2023).
  68. G. Lami and M. Collura, Phys. Rev. Lett. 131, 180401 (2023).
  69. T. Haug and L. Piroli, Quantum 7, 1092 (2023).
  70. P. S. Tarabunga, E. Tirrito, T. Chanda, and M. Dalmonte, PRX Quantum 4, 040317 (2023).
  71. P. S. Tarabunga, E. Tirrito, M. C. Bañuls, and M. Dalmonte, Phys. Rev. Lett. 133, 010601 (2024).
  72. P. S. Tarabunga, Quantum 8, 1413 (2024).
  73. P. S. Tarabunga and C. Castelnovo, Quantum 8, 1347 (2024).
  74. P. R. N. Falcão, P. S. Tarabunga, M. Frau, E. Tirrito, J. Zakrzewski, and M. Dalmonte, Phys. Rev. B 111, L081102 (2025).
  75. P. S. Tarabunga and T. Haug, SciPost Phys. 19, 085 (2025).
  76. M. Collura, J. D. Nardis, V. Alba, and G. Lami, arXiv:2412.05367.
  77. Z. Liu and B. K. Clark, Phys. Rev. B 111, 085144 (2025).
  78. Y.-M. Ding, Z. Wang, and Z. Yan, PRX Quantum 6, 030328 (2025).
  79. A. W. Sandvik, Phys. Rev. B 57, 10287 (1998).
  80. O. F. Syljuåsen, Phys. Rev. B 78, 174429 (2008).
  81. A. W. Sandvik, Phys. Rev. E 94, 063308 (2016).
  82. H. Shao and A. W. Sandvik, Phys. Rep. 1003, 1 (2023), progress on stochastic analytic continuation of quantum Monte Carlo data.
  83. P. Calabrese, J. Cardy, and E. Tonni, Phys. Rev. Lett. 109, 130502 (2012).
  84. Y.-M. Ding, Y. Tang, Z. Wang, Z. Wang, B.-B. Mao, and Z. Yan, Phys. Rev. B 111, L241108 (2025).
  85. K.-H. Wu, T.-C. Lu, C.-M. Chung, Y.-J. Kao, and T. Grover, Phys. Rev. Lett. 125, 140603 (2020).
  86. F.-H. Wang and X. Y. Xu, Nat. Commun. 16, 2637 (2025).
  87. C.-M. Chung, V. Alba, L. Bonnes, P. Chen, and A. M. Läuchli, Phys. Rev. B 90, 064401 (2014).
  88. A. Elben, R. Kueng, H.-Y. R. Huang, R. van Bijnen, C. Kokail, M. Dalmonte, P. Calabrese, B. Kraus, J. Preskill, P. Zoller, and B. Vermersch, Phys. Rev. Lett. 125, 200501 (2020).
  89. A. Neven, J. Carrasco, V. Vitale, C. Kokail, A. Elben, M. Dalmonte, P. Calabrese, P. Zoller, B. Vermersch, R. Kueng, and B. Kraus, npj Quantum Inf. 7, 152 (2021).
  90. X.-D. Yu, S. Imai, and O. Gühne, Phys. Rev. Lett. 127, 060504 (2021).
  91. P. S. Tarabunga and T. Haug, arXiv:2507.13840.
  92. P. S. Tarabunga and Y.-M. Ding, Repository: Zenodo, 10.5281/zenodo.15470953 (2025).

Outline

Information

Phys. Rev. Lett. 135, 200403– Published 13 November, 2025
  • Received 2 June 2025
  • Revised 1 September 2025
  • Accepted 24 October 2025
Crossmark - Check for updates button
Creative Commons Logo

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.

Published by the American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation