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Multidimensional Quantum Estimation and Model Learning Framework Based on Variational Bayesian Inference

Federico Belliardo1,2,*, Erik M. Gauger1, Mohamed H. Abobeih3,4, Tim H. Taminiau3,4, Yoann Altmann5,†, and Cristian Bonato1,‡

  • *Contact author: federico.belliardo@gmail.com
  • †Contact author: y.altmann@hw.ac.uk
  • ‡Contact author: c.bonato@hw.ac.uk
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PRX Quantum 7, 020360 – Published 15 June, 2026

DOI: https://doi.org/10.1103/6yhz-9vr6

Abstract

The advancement and scaling of quantum technology has made the learning and identification of quantum systems and devices in highly multidimensional parameter spaces a pressing task for a variety of applications. In many cases, the integration of real-time feedback control and adaptive choice of measurement settings places strict demands on the speed of this task. Here we present a joint model selection and parameter estimation algorithm that is fast and operable on a large number of model parameters. The algorithm is based on variational Bayesian inference, which approximates the target posterior distribution by optimizing a tractable family of distributions, making it more scalable than exact inference methods relying on sampling and that generally suffer from high variance and computational cost in high-dimensional spaces. We show how a regularizing prior can be used to select between competing models, each comprising a different number of parameters, identifying the simplest model that explains the experimental data. The regularization can further separate the degrees of freedom, e.g., quantum systems in the environment or processes, which contribute to major features in the observed dynamics, with respect to others featuring small coupling, which only contribute to a background. As an application of the introduced framework, we consider the problem of the identification of multiple individual nuclear spins with a single electron spin quantum sensor, relevant for nanoscale nuclear magnetic resonance and for the implementation of multiqubit quantum networking nodes. With the number of environmental spins unknown a priori, our Bayesian approach is able to correctly identify the model, i.e., the number of spins and their couplings. We benchmark the algorithm on both simulated and experimental data, using standard figures of merit, and demonstrate that we can estimate dozens of parameters within minutes. Our methodology is compatible with the implementation of real-time adaptive choice of experimental settings in multiparameter estimation, which has the potential to greatly decrease the estimation time in a variety of metrology, tomography, and characterization tasks by prioritizing the measurements with the highest content of information.

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Physics Subject Headings (PhySH)

Popular Summary

Every quantum processor or sensor depends on many hidden parameters, for example, related to the internal architecture or to coupling to the environment. However, traditional methods for identifying such parameters quickly become too slow as the systems grow in size and complexity.

Here, we introduce a characterization technique based on variational Bayesian inference, which approximates the complex, multidimensional probability distribution of the model parameters with a simpler, tractable family of distributions. This technique scales favorably with the number of parameters and simultaneously enables real-time “smart” experiments that choose the most informative measurements on the fly. Our method further includes an automatic “Occam’s razor,” helping decide how many components actually affect the operation of the quantum device.

We tested our method on a quantum sensing experiment in which a single electron spin is used to identify nearby nuclear spins (nanoscale magnetic resonance). In the long run, this approach could help future quantum technologies tune and calibrate themselves with minimal human intervention.

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References (182)

  1. S. Bravyi, A. W. Cross, J. M. Gambetta, D. Maslov, P. Rall, and T. J. Yoder, High-threshold and low-overhead fault-tolerant quantum memory, Nature 627, 778 (2024).
  2. Google Quantum AI and Collaborators et al., Quantum error correction below the surface code threshold, Nature 638, 920 (2025).
  3. N. Maskara, S. Ostermann, J. Shee, M. Kalinowski, A. McClain Gomez, R. Araiza Bravo, D. S. Wang, A. I. Krylov, N. Y. Yao, M. Head-Gordon, M. D. Lukin, and S. F. Yelin, Programmable simulations of molecules and materials with reconfigurable quantum processors, Nat. Phys. 21, 289 (2025).
  4. D. Bluvstein et al., A fault-tolerant neutral-atom architecture for universal quantum computation, Nature 649, 39 (2026).
  5. N.-C. Chiu, E. C. Trapp, J. Guo, M. H. Abobeih, L. M. Stewart, S. Hollerith, P. L. Stroganov, M. Kalinowski, A. A. Geim, S. J. Evered, S. H. Li, X. Lyu, L. M. Peters, D. Bluvstein, T. T. Wang, M. Greiner, V. Vuletić, and M. D. Lukin, Continuous operation of a coherent 3,000-qubit system, Nature 646, 1075 (2025).
  6. P.-J. Stas, Y. Q. Huan, B. Machielse, E. N. Knall, A. Suleymanzade, B. Pingault, M. Sutula, S. W. Ding, C. M. Knaut, D. R. Assumpcao, Y.-C. Wei, M. K. Bhaskar, R. Riedinger, D. D. Sukachev, H. Park, M. Lončar, D. S. Levonian, and M. D. Lukin, Robust multi-qubit quantum network node with integrated error detection, Science 378, 557 (2022).
  7. S. L. N. Hermans, M. Pompili, H. K. C. Beukers, S. Baier, J. Borregaard, and R. Hanson, Qubit teleportation between non-neighbouring nodes in a quantum network, Nature 605, 663 (2022).
  8. C. Delle Donne et al., An operating system for executing applications on quantum network nodes, Nature 639, 321 (2025).
  9. A. J. Stolk et al., Metropolitan-scale heralded entanglement of solid-state qubits, Sci. Adv. 10, eadp6442 (2024).
  10. C. M. Knaut, A. Suleymanzade, Y.-C. Wei, D. R. Assumpcao, P.-J. Stas, Y. Q. Huan, B. Machielse, E. N. Knall, M. Sutula, G. Baranes, N. Sinclair, C. De-Eknamkul, D. S. Levonian, M. K. Bhaskar, H. Park, M. Lončar, and M. D. Lukin, Entanglement of nanophotonic quantum memory nodes in a telecom network, Nature 629, 573 (2024).
  11. E. Boto, N. Holmes, J. Leggett, G. Roberts, V. Shah, S. S. Meyer, L. D. Muñoz, K. J. Mullinger, T. M. Tierney, S. Bestmann, G. R. Barnes, R. Bowtell, and M. J. Brookes, Moving magnetoencephalography towards real-world applications with a wearable system, Nature 555, 657 (2018).
  12. B. Stray et al., Quantum sensing for gravity cartography, Nature 602, 590 (2022).
  13. N. Aslam, H. Zhou, E. K. Urbach, M. J. Turner, R. L. Walsworth, M. D. Lukin, and H. Park, Quantum sensors for biomedical applications, Nat. Rev. Phys. 5, 157 (2023).
  14. R. Budakian et al., Roadmap on nanoscale magnetic resonance imaging, Nanotechnology 35, 412001 (2024).
  15. K. D. Briegel, N. R. Von Grafenstein, J. C. Draeger, P. Blümler, R. D. Allert, and D. B. Bucher, Optical widefield nuclear magnetic resonance microscopy, Nat. Commun. 16, 1281 (2025).
  16. N. Mohseni, T. Fösel, L. Guo, C. Navarrete-Benlloch, and F. Marquardt, Deep learning of quantum many-body dynamics via random driving, Quantum 6, 714 (2022).
  17. V. Gebhart, R. Santagati, A. A. Gentile, E. M. Gauger, D. Craig, N. Ares, L. Banchi, F. Marquardt, L. Pezzè, and C. Bonato, Learning quantum systems, Nat. Rev. Phys. 5, 141 (2023).
  18. M. Krenn, J. Landgraf, T. Foesel, and F. Marquardt, Artificial intelligence and machine learning for quantum technologies, Phys. Rev. A 107, 010101 (2023).
  19. A. Dawid, Machine Learning in Quantum Sciences (Cambridge University, Cambridge, 2025), 1st ed.
  20. G. Acampora et al., Quantum computing and artificial intelligence: Status and perspectives, arXiv:2505.23860 [quant-ph].
  21. Y. Alexeev et al., Artificial intelligence for quantum computing, Nat. Commun. 16, 10829 (2025).
  22. Y. Du, Y. Zhu, Y.-H. Zhang, M.-H. Hsieh, P. Rebentrost, W. Gao, Y.-D. Wu, J. Eisert, G. Chiribella, D. Tao, and B. C. Sanders, Artificial intelligence for representing and characterizing quantum systems, arXiv:2509.04923.
  23. R. Blume-Kohout, Optimal, reliable estimation of quantum states, New J. Phys. 12, 043034 (2010).
  24. M. Paris and J. Řeháček, (Eds.), Quantum State Estimation, Lecture Notes in Physics (Springer, Berlin, Heidelberg, 2004), Vol. 649.
  25. N. R. von Grafenstein, K. D. Briegel, J. Casanova, and D. B. Bucher, Coherent signal detection in the statistical polarization regime enables high-resolution nanoscale NMR spectroscopy, arXiv:2501.02093 [physics].
  26. N. Wiebe, C. Granade, C. Ferrie, and D. Cory, Quantum Hamiltonian learning using imperfect quantum resources, Phys. Rev. A 89, 042314 (2014).
  27. J. Wang, S. Paesani, R. Santagati, S. Knauer, A. A. Gentile, N. Wiebe, M. Petruzzella, J. L. O’Brien, J. G. Rarity, A. Laing, and M. G. Thompson, Experimental quantum Hamiltonian learning, Nat. Phys. 13, 551 (2017).
  28. A. Anshu, S. Arunachalam, T. Kuwahara, and M. Soleimanifar, Sample-efficient learning of interacting quantum systems, Nat. Phys. 17, 931 (2021).
  29. M. Hayashi, Asymptotic Theory of Quantum Statistical Inference (World Scientific, Singapore, 2005).
  30. A. Holevo, Probabilistic and Statistical Aspects of Quantum Theory (Edizioni Della Normale, Pisa, 2011).
  31. A. A. Gentile, B. Flynn, S. Knauer, N. Wiebe, S. Paesani, C. E. Granade, J. G. Rarity, R. Santagati, and A. Laing, Learning models of quantum systems from experiments, Nat. Phys. 17, 837 (2021).
  32. S. Wallace, Y. Altmann, B. D. Gerardot, E. M. Gauger, and C. Bonato, Learning the dynamics of Markovian open quantum systems from experimental data, Phys. Rev. Res. 7, 033217 (2025).
  33. J. M. Lukens, K. J. H. Law, A. Jasra, and P. Lougovski, A practical and efficient approach for Bayesian quantum state estimation, New J. Phys. 22, 063038 (2020).
  34. K. Jones, Principles of quantum inference, Ann. Phys. 207, 140 (1991).
  35. V. Bužek, R. Derka, G. Adam, and P. Knight, Reconstruction of quantum states of spin systems: From quantum Bayesian inference to quantum tomography, Ann. Phys. 266, 454 (1998).
  36. R. Schack, T. A. Brun, and C. M. Caves, Quantum Bayes rule, Phys. Rev. A 64, 014305 (2001).
  37. K. S. Kravtsov, S. S. Straupe, I. V. Radchenko, N. M. T. Houlsby, F. Huszár, and S. P. Kulik, Experimental adaptive Bayesian tomography, Phys. Rev. A 87, 062122 (2013).
  38. O. Di Matteo, J. Gamble, C. Granade, K. Rudinger, and N. Wiebe, Operational, Gauge-free quantum tomography, Quantum 4, 364 (2020).
  39. M. Christandl and R. Renner, Reliable quantum state tomography, Phys. Rev. Lett. 109, 120403 (2012).
  40. C. Granade, J. Combes, and D. G. Cory, Practical Bayesian tomography, New J. Phys. 18, 033024 (2016).
  41. F. Huszár and N. M. T. Houlsby, Adaptive Bayesian quantum tomography, Phys. Rev. A 85, 052120 (2012).
  42. C. Granade, C. Ferrie, and S. T. Flammia, Practical adaptive quantum tomography, New J. Phys. 19, 113017 (2017).
  43. S. Mondal and A. K. Dutta, A Bayesian quantum state tomography along with adaptive frameworks based on linear minimum mean square error criterion, New J. Phys. 25, 123001 (2023).
  44. N. Wiebe, C. Granade, C. Ferrie, and D. G. Cory, Hamiltonian learning and certification using quantum resources, Phys. Rev. Lett. 112, 190501 (2014).
  45. N. Wiebe and C. Granade, Efficient Bayesian phase estimation, Phys. Rev. Lett. 117, 010503 (2016).
  46. M. D. Shulman, S. P. Harvey, J. M. Nichol, S. D. Bartlett, A. C. Doherty, V. Umansky, and A. Yacoby, Suppressing qubit dephasing using real-time Hamiltonian estimation, Nat. Commun. 5, 5156 (2014).
  47. V. Gebhart, A. Smerzi, and L. Pezzè, Bayesian quantum multiphase estimation algorithm, Phys. Rev. Appl. 16, 014035 (2021).
  48. F. Berritta, J. Benestad, L. Pahl, M. Mathews, J. A. Krzywda, R. Assouly, Y. Sung, D. K. Kim, B. M. Niedzielski, K. Serniak, M. E. Schwartz, J. L. Yoder, A. Chatterjee, J. A. Grover, J. Danon, W. D. Oliver, and F. Kuemmeth, Efficient qubit calibration by binary-search Hamiltonian tracking, PRX Quantum 6, 030335 (2025).
  49. L. Fioroni, I. Rojkov, and F. Reiter, A learning agent-based approach to the characterization of open quantum systems, Phys. Rev. Appl. 24, 034011 (2025).
  50. C. Bonato, M. S. Blok, H. T. Dinani, D. W. Berry, M. L. Markham, D. J. Twitchen, and R. Hanson, Optimized quantum sensing with a single electron spin using real-time adaptive measurements, Nat. Nanotechnol. 11, 247 (2016).
  51. D. T. Lennon, H. Moon, L. C. Camenzind, L. Yu, D. M. Zumbühl, G. A. D. Briggs, M. A. Osborne, E. A. Laird, and N. Ares, Efficiently measuring a quantum device using machine learning, npj Quantum Inf. 5, 79 (2019).
  52. R. Santagati, A. A. Gentile, S. Knauer, S. Schmitt, S. Paesani, C. Granade, N. Wiebe, C. Osterkamp, L. P. McGuinness, J. Wang, M. G. Thompson, J. G. Rarity, F. Jelezko, and A. Laing, Magnetic-field learning using a single electronic spin in diamond with one-photon readout at room temperature, Phys. Rev. X 9, 021019 (2019).
  53. M. Valeri, E. Polino, D. Poderini, I. Gianani, G. Corrielli, A. Crespi, R. Osellame, N. Spagnolo, and F. Sciarrino, Experimental adaptive Bayesian estimation of multiple phases with limited data, npj Quantum Inf. 6, 92 (2020).
  54. S. Dushenko, K. Ambal, and R. D. McMichael, Sequential Bayesian experiment design for optically detected magnetic resonance of nitrogen-vacancy centers, Phys. Rev. Appl. 14, 054036 (2020).
  55. R. Kaubruegger, D. V. Vasilyev, M. Schulte, K. Hammerer, and P. Zoller, Quantum variational optimization of ramsey interferometry and atomic clocks, Phys. Rev. X 11, 041045 (2021).
  56. T. Joas, S. Schmitt, R. Santagati, A. A. Gentile, C. Bonato, A. Laing, L. P. McGuinness, and F. Jelezko, Online adaptive quantum characterization of a nuclear spin, npj Quantum Inf. 7, 56 (2021).
  57. K. Craigie, E. M. Gauger, Y. Altmann, and C. Bonato, Resource-efficient adaptive Bayesian tracking of magnetic fields with a quantum sensor, J. Phys.: Condens. Matter 33, 195801 (2021).
  58. S. E. D’Aurelio, M. Valeri, E. Polino, V. Cimini, I. Gianani, M. Barbieri, G. Corrielli, A. Crespi, R. Osellame, F. Sciarrino, and N. Spagnolo, Experimental investigation of Bayesian bounds in multiparameter estimation, Quantum Sci. Technol. 7, 025011 (2022).
  59. M. Caouette-Mansour, A. Solyom, B. Ruffolo, R. D. McMichael, J. Sankey, and L. Childress, Robust spin relaxometry with fast adaptive Bayesian estimation, Phys. Rev. Appl. 17, 064031 (2022).
  60. I. Zohar, B. Haylock, Y. Romach, M. J. Arshad, N. Halay, N. Drucker, R. Stöhr, A. Denisenko, Y. Cohen, C. Bonato, and A. Finkler, Real-time frequency estimation of a qubit without single-shot-readout, Quantum Sci. Technol. 8, 035017 (2023).
  61. M. J. Arshad, C. Bekker, B. Haylock, K. Skrzypczak, D. White, B. Griffiths, J. Gore, G. W. Morley, P. Salter, J. Smith, I. Zohar, A. Finkler, Y. Altmann, E. M. Gauger, and C. Bonato, Real-time adaptive estimation of decoherence timescales for a single qubit, Phys. Rev. Appl. 21, 024026 (2024).
  62. P. Cappellaro, Spin-bath narrowing with adaptive parameter estimation, Phys. Rev. A 85, 030301 (2012).
  63. E. Scerri, E. M. Gauger, and C. Bonato, Extending qubit coherence by adaptive quantum environment learning, New J. Phys. 22, 035002 (2020).
  64. F. Berritta, T. Rasmussen, J. A. Krzywda, J. Van Der Heijden, F. Fedele, S. Fallahi, G. C. Gardner, M. J. Manfra, E. Van Nieuwenburg, J. Danon, A. Chatterjee, and F. Kuemmeth, Real-time two-axis control of a spin qubit, Nat. Commun. 15, 1676 (2024).
  65. R. Bellman, Dynamic Programming (Princeton University, Princeton, NJ, 1984).
  66. T. Bengtsson, P. Bickel, and B. Li, in Institute of Mathematical Statistics Collections (Institute of Mathematical Statistics, Beachwood, Ohio, USA, 2008), pp. 316–334.
  67. H.-H. Lu, K. V. Myilswamy, R. S. Bennink, S. Seshadri, M. S. Alshaykh, J. Liu, T. J. Kippenberg, D. E. Leaird, A. M. Weiner, and J. M. Lukens, Bayesian tomography of high-dimensional on-chip biphoton frequency combs with randomized measurements, Nat. Commun. 13, 4338 (2022).
  68. J. C. Chapman, J. M. Lukens, B. Qi, R. C. Pooser, and N. A. Peters, Bayesian homodyne and heterodyne tomography, Opt. Express 30, 15184 (2022).
  69. M. Valeri, V. Cimini, S. Piacentini, F. Ceccarelli, E. Polino, F. Hoch, G. Bizzarri, G. Corrielli, N. Spagnolo, R. Osellame, and F. Sciarrino, Experimental multiparameter quantum metrology in adaptive regime, Phys. Rev. Res. 5, 013138 (2023).
  70. F. Belliardo, V. Cimini, E. Polino, F. Hoch, B. Piccirillo, N. Spagnolo, V. Giovannetti, and F. Sciarrino, Optimizing quantum-enhanced Bayesian multiparameter estimation of phase and noise in practical sensors, Phys. Rev. Res. 6, 023201 (2024).
  71. We stress that this exponential growth is not related to the exponential complexity in time and memory of simulating and certifying quantum systems, which we are not trying to address in this paper.
  72. D. M. Blei, A. Kucukelbir, and J. D. McAuliffe, Variational inference: A review for statisticians, J. Am. Stat. Assoc. 112, 859 (2017).
  73. P. P. Mazza, D. Zietlow, F. Carollo, S. Andergassen, G. Martius, and I. Lesanovsky, Machine learning time-local generators of open quantum dynamics, Phys. Rev. Res. 3, 023084 (2021).
  74. G. Cemin, F. Carnazza, S. Andergassen, G. Martius, F. Carollo, and I. Lesanovsky, Inferring interpretable dynamical generators of local quantum observables from projective measurements through machine learning, Phys. Rev. Appl. 21, L041001 (2024).
  75. L. Schorling, P. Vaidhyanathan, J. Schuff, M. J. Carballido, D. Zumbühl, G. Milburn, F. Marquardt, J. Foerster, M. A. Osborne, and N. Ares, Meta-learning characteristics and dynamics of quantum systems, arXiv:2503.10492 [quant-ph].
  76. G. L. Van De Stolpe, D. P. Kwiatkowski, C. E. Bradley, J. Randall, M. H. Abobeih, S. A. Breitweiser, L. C. Bassett, M. Markham, D. J. Twitchen, and T. H. Taminiau, Mapping a 50-spin-qubit network through correlated sensing, Nat. Commun. 15, 2006 (2024).
  77. C. E. Bradley, J. Randall, M. H. Abobeih, R. C. Berrevoets, M. J. Degen, M. A. Bakker, M. Markham, D. J. Twitchen, and T. H. Taminiau, A ten-qubit solid-state spin register with quantum memory up to one minute, Phys. Rev. X 9, 031045 (2019).
  78. J. Randall, C. E. Bradley, F. V. Van Der Gronden, A. Galicia, M. H. Abobeih, M. Markham, D. J. Twitchen, F. Machado, N. Y. Yao, and T. H. Taminiau, Many-body-localized discrete time crystal with a programmable spin-based quantum simulator, Science 374, 1474 (2021).
  79. K. Jung, M. H. Abobeih, J. Yun, G. Kim, H. Oh, A. Henry, T. H. Taminiau, and D. Kim, Deep learning enhanced individual nuclear-spin detection, npj Quantum Inf. 7, 41 (2021).
  80. B. Varona-Uriarte, C. Munuera-Javaloy, E. Terradillos, Y. Ban, A. Alvarez-Gila, E. Garrote, and J. Casanova, Automatic detection of nuclear spins at arbitrary magnetic fields via signal-to-image AI model, Phys. Rev. Lett. 132, 150801 (2024).
  81. C.-W. Huang, D. Krueger, A. Lacoste, and A. Courville, Neural autoregressive flows, in Proceedings of the 35th International Conference on Machine Learning, edited by J. Dy and A. Krause (PMLR, 2018), Vol. 80, pp. 2078–2087.
  82. M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information (Cambridge University, 2012).
  83. A. S. Holevo, Quantum Systems, Channels, Information: A Mathematical Introduction (De Gruyter, 2019).
  84. F. Belliardo, F. Zoratti, F. Marquardt, and V. Giovannetti, Model-aware reinforcement learning for high-performance Bayesian experimental design in quantum metrology, Quantum 8, 1555 (2024).
  85. F. Belliardo, F. Zoratti, and V. Giovannetti, Applications of model-aware reinforcement learning in Bayesian quantum metrology, Phys. Rev. A 109, 062609 (2024).
  86. N. Gordon, D. Salmond, and A. Smith, Novel approach to nonlinear/non-gaussian Bayesian state estimation, IEE Proc. F (Radar Signal Process.) 140, 107 (1993).
  87. A. Doucet, N. Freitas, and N. Gordon, (Eds.), Sequential Monte Carlo Methods in Practice (Springer New York, New York, NY, 2001).
  88. P. Del Moral, A. Doucet, and A. Jasra, Sequential Monte Carlo samplers, J. R. Stat. Soc. Ser. B: Stat. Methodol. 68, 411 (2006).
  89. C. E. Granade, C. Ferrie, N. Wiebe, and D. G. Cory, Robust online Hamiltonian learning, New J. Phys. 14, 103013 (2012).
  90. G. Kitagawa, Monte Carlo filter and smoother for Non-Gaussian nonlinear state space models, J. Comput. Graph. Stat. 5, 1 (1996).
  91. J. S. Liu and R. Chen, Sequential Monte Carlo methods for dynamic systems, J. Am. Stat. Assoc. 93, 1032 (1998).
  92. J. Liu and M. West, in Sequential Monte Carlo Methods in Practice (Springer New York, New York, NY, 2001), pp. 197–223.
  93. O. Cappe, S. J. Godsill, and E. Moulines, An overview of existing methods and recent advances in sequential Monte Carlo, Proc. IEEE 95, 899 (2007).
  94. T. Li, M. Bolic, and P. M. Djuric, Resampling methods for particle filtering: Classification, implementation, and strategies, IEEE Signal Process. Mag. 32, 70 (2015).
  95. L. M. Murray, A. Lee, and P. E. Jacob, Parallel resampling in the particle filter, J. Comput. Graph. Stat. 25, 789 (2016).
  96. A. E. Gelfand and A. F. M. Smith, Sampling-based approaches to calculating marginal densities, J. Am. Stat. Assoc. 85, 398 (1990).
  97. S. Brooks, A. Gelman, G. Jones, and X.-L. Meng, Handbook of Markov Chain Monte Carlo (Chapman and Hall/CRC, New York, 2011), 1st ed.
  98. G. L. Jones and Q. Qin, Markov chain Monte Carlo in practice, Annu. Rev. Stat. Appl. 9, 557 (2022).
  99. A. N. Poteshman, M. Onizhuk, C. Egerstrom, D. P. Mark, D. D. Awschalom, F. J. Heremans, and G. Galli, High-throughput spin-bath characterization of spin-defects in semiconductors, Phys. Rev. Appl. 24, 054048 (2025).
  100. J. Skilling, in AIP Conference Proceedings (AIP, Garching (Germany), 2004), Vol. 735, pp. 395–405.
  101. J. Skilling, Nested sampling for general Bayesian computation, Bayesian Anal. 1, 833 (2006).
  102. G. Ashton et al., Nested sampling for physical scientists, Nat. Rev. Methods Primers 2, 39 (2022).
  103. C. Peterson and J. Anderson, A mean field theory learning algorithm for neural networks, Complex Syst. 1, 995 (1987).
  104. G. E. Hinton and D. Van Camp, in Proceedings of the Sixth Annual Conference on Computational Learning Theory—COLT ’93 (ACM, Santa Cruz, California, USA, 1993), pp. 5–13.
  105. R. M. Neal and G. E. Hinton, in Learning in Graphical Models (Springer Netherlands, Dordrecht, 1998), pp. 355–368.
  106. L. K. Saul, T. Jaakkola, and M. I. Jordan, Mean field theory for sigmoid belief networks, J. Artif. Intell. Res. 4, 61 (1996).
  107. T. S. Jaakkola and M. I. Jordan, A variational approach to Bayesian logistic regression models and their extensions, in Proceedings of the Sixth International Workshop on Artificial Intelligence and Statistics, edited by D. Madigan and P. Smyth (PMLR, 1997), Vol. R1, pp. 283–294..
  108. D. P. Kingma and M. Welling, Auto-encoding variational Bayes, in 2nd International Conference on Learning Representations, ICLR 2014, Banff, AB, Canada, April 14–16, 2014, Conference Track Proceedings (2014).
  109. D. Rezende and S. Mohamed, Variational inference with normalizing flows, in Proceedings of the 32nd International Conference on Machine Learning, edited by F. Bach and D. Blei (PMLR, Lille, France, 2015).
  110. F. Belliardo and V. Giovannetti, Achieving Heisenberg scaling with maximally entangled states: An analytic upper bound for the attainable root-mean-square error, Phys. Rev. A 102, 042613 (2020).
  111. R. M. Neal, Slice sampling, Ann. Stat. 31, 705 (2003).
  112. E. Higson, W. Handley, M. Hobson, and A. Lasenby, Dynamic nested sampling: An improved algorithm for parameter estimation and evidence calculation, Stat. Comput. 29, 891 (2019).
  113. H. Haario, E. Saksman, and J. Tamminen, An adaptive metropolis algorithm, Bernoulli 7, 223 (2001).
  114. J. Burnett, L. Faoro, I. Wisby, V. L. Gurtovoi, A. V. Chernykh, G. M. Mikhailov, V. A. Tulin, R. Shaikhaidarov, V. Antonov, P. J. Meeson, A. Ya. Tzalenchuk, and T. Lindström, Evidence for interacting two-level systems from the 1/f noise of a superconducting resonator, Nat. Commun. 5, 4119 (2014).
  115. S. E. De Graaf, L. Faoro, L. B. Ioffe, S. Mahashabde, J. J. Burnett, T. Lindström, S. E. Kubatkin, A. V. Danilov, and A. Ya. Tzalenchuk, Two-level systems in superconducting quantum devices due to trapped quasiparticles, Sci. Adv. 6, eabc5055 (2020).
  116. M. Lucas, A. V. Danilov, L. V. Levitin, A. Jayaraman, A. J. Casey, L. Faoro, A. Ya. Tzalenchuk, S. E. Kubatkin, J. Saunders, and S. E. De Graaf, Quantum bath suppression in a superconducting circuit by immersion cooling, Nat. Commun. 14, 3522 (2023).
  117. B.-J. Liu, Y.-Y. Wang, T. Sheffer, and C. Wang, Observation of discrete charge states of a coherent two-level system in a superconducting qubit, Phys. Rev. Lett. 133, 160602 (2024).
  118. A. Gelman, A. Jakulin, M. G. Pittau, and Y.-S. Su, A weakly informative default prior distribution for logistic and other regression models, Ann. Appl. Stat. 2, 1360 (2008).
  119. R. Tibshirani, Regression shrinkage and selection via the Lasso, J. R. Stat. Soc.: Ser. B (Methodol.) 58, 267 (1996).
  120. 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).
  121. N. A. Nemkov, E. O. Kiktenko, I. A. Luchnikov, and A. K. Fedorov, Efficient variational synthesis of quantum circuits with coherent multi-start optimization, Quantum 7, 993 (2023).
  122. R. Kress, Numerical Analysis (Springer Science & Business Media, 2012).
  123. J. C. Marcks, B. Pingault, J. Zhang, C. Zeledon, F. J. Heremans, and D. D. Awschalom, Nuclear spin engineering for quantum information science, J. Mater. Res. 40, 1433 (2025).
  124. T. Staudacher, F. Shi, S. Pezzagna, J. Meijer, J. Du, C. A. Meriles, F. Reinhard, and J. Wrachtrup, Nuclear magnetic resonance spectroscopy on a (5-nanometer)3 sample volume, Science 339, 561 (2013).
  125. F. Shi, X. Kong, P. Wang, F. Kong, N. Zhao, R.-B. Liu, and J. Du, Sensing and atomic-scale structure analysis of single nuclear-spin clusters in diamond, Nat. Phys. 10, 21 (2014).
  126. J. Zopes, K. S. Cujia, K. Sasaki, J. M. Boss, K. M. Itoh, and C. L. Degen, Three-dimensional localization spectroscopy of individual nuclear spins with sub-Angstrom resolution, Nat. Commun. 9, 4678 (2018).
  127. J. Zopes, K. Herb, K. S. Cujia, and C. L. Degen, Three-dimensional nuclear spin positioning using coherent radio-frequency control, Phys. Rev. Lett. 121, 170801 (2018).
  128. K. S. Cujia, J. M. Boss, K. Herb, J. Zopes, and C. L. Degen, Tracking the precession of single nuclear spins by weak measurements, Nature 571, 230 (2019).
  129. K. S. Cujia, K. Herb, J. Zopes, J. M. Abendroth, and C. L. Degen, Parallel detection and spatial mapping of large nuclear spin clusters, Nat. Commun. 13, 1260 (2022).
  130. K. Herb, T. F. Segawa, L. A. Völker, J. M. Abendroth, E. Janitz, T. Zhu, and C. L. Degen, Multidimensional spectroscopy of nuclear spin clusters in diamond, Phys. Rev. Lett. 132, 133002 (2024).
  131. J. Du, F. Shi, X. Kong, F. Jelezko, and J. Wrachtrup, Single-molecule scale magnetic resonance spectroscopy using quantum diamond sensors, Rev. Mod. Phys. 96, 025001 (2024).
  132. N. Zhao, J. Honert, B. Schmid, M. Klas, J. Isoya, M. Markham, D. Twitchen, F. Jelezko, R.-B. Liu, H. Fedder, and J. Wrachtrup, Sensing single remote nuclear spins, Nat. Nanotechnol. 7, 657 (2012).
  133. G. Wolfowicz, P.-A. Mortemousque, R. Guichard, S. Simmons, M. L. W. Thewalt, K. M. Itoh, and J. J. L. Morton, 29Si nuclear spins as a resource for donor spin qubits in silicon, New J. Phys. 18, 023021 (2016).
  134. N. Kalb, A. A. Reiserer, P. C. Humphreys, J. J. W. Bakermans, S. J. Kamerling, N. H. Nickerson, S. C. Benjamin, D. J. Twitchen, M. Markham, and R. Hanson, Entanglement distillation between solid-state quantum network nodes, Science 356, 928 (2017).
  135. S. Asaad, V. Mourik, B. Joecker, M. A. I. Johnson, A. D. Baczewski, H. R. Firgau, M. T. Mądzik, V. Schmitt, J. J. Pla, F. E. Hudson, K. M. Itoh, J. C. McCallum, A. S. Dzurak, A. Laucht, and A. Morello, Coherent electrical control of a single high-spin nucleus in silicon, Nature 579, 205 (2020).
  136. M. H. Abobeih, Y. Wang, J. Randall, S. J. H. Loenen, C. E. Bradley, M. Markham, D. J. Twitchen, B. M. Terhal, and T. H. Taminiau, Fault-tolerant operation of a logical qubit in a diamond quantum processor, Nature 606, 884 (2022).
  137. C. E. Bradley, S. W. De Bone, P. F. W. Möller, S. Baier, M. J. Degen, S. J. H. Loenen, H. P. Bartling, M. Markham, D. J. Twitchen, R. Hanson, D. Elkouss, and T. H. Taminiau, Robust quantum-network memory based on spin qubits in isotopically engineered diamond, npj Quantum Inf. 8, 122 (2022).
  138. J. Reiner, Y. Chung, S. H. Misha, C. Lehner, C. Moehle, D. Poulos, S. Monir, K. J. Charde, P. Macha, L. Kranz, I. Thorvaldson, B. Thorgrimsson, D. Keith, Y. L. Hsueh, R. Rahman, S. K. Gorman, J. G. Keizer, and M. Y. Simmons, High-fidelity initialization and control of electron and nuclear spins in a four-qubit register, Nat. Nanotechnol. 19, 605 (2024).
  139. A. N. Poteshman, J. Yun, T. H. Taminiau, and G. Galli, Trans-dimensional Hamiltonian model selection and parameter estimation from sparse, noisy data, Quantum 10, 2055 (2026).
  140. M. H. Abobeih, J. Cramer, M. A. Bakker, N. Kalb, M. Markham, D. J. Twitchen, and T. H. Taminiau, One-second coherence for a single electron spin coupled to a multi-qubit nuclear-spin environment, Nat. Commun. 9, 2552 (2018).
  141. M. W. Doherty, N. B. Manson, P. Delaney, F. Jelezko, J. Wrachtrup, and L. C. Hollenberg, The nitrogen-vacancy colour centre in diamond, Phys. Rep. 528, 1 (2013).
  142. R. Schirhagl, K. Chang, M. Loretz, and C. L. Degen, Nitrogen-vacancy centers in diamond: Nanoscale sensors for physics and biology, Annu. Rev. Phys. Chem. 65, 83 (2014).
  143. J. F. Barry, J. M. Schloss, E. Bauch, M. J. Turner, C. A. Hart, L. M. Pham, and R. L. Walsworth, Sensitivity optimization for NV-diamond magnetometry, Rev. Mod. Phys. 92, 015004 (2020).
  144. T. H. Taminiau, J. J. T. Wagenaar, T. Van Der Sar, F. Jelezko, V. V. Dobrovitski, and R. Hanson, Detection and control of individual nuclear spins using a weakly coupled electron spin, Phys. Rev. Lett. 109, 137602 (2012).
  145. M. J. Biercuk, A. C. Doherty, and H. Uys, Dynamical decoupling sequence construction as a filter-design problem, J. Phys. B: At. Mol. Opt. Phys. 44, 154002 (2011).
  146. C. L. Degen, F. Reinhard, and P. Cappellaro, Quantum sensing, Rev. Mod. Phys. 89, 035002 (2017).
  147. A. Laraoui, F. Dolde, C. Burk, F. Reinhard, J. Wrachtrup, and C. A. Meriles, High-resolution correlation spectroscopy of 13C spins near a nitrogen-vacancy centre in diamond, Nat. Commun. 4, 1651 (2013).
  148. J. Casanova, Z.-Y. Wang, J. F. Haase, and M. B. Plenio, Robust dynamical decoupling sequences for individual-nuclear-spin addressing, Phys. Rev. A 92, 042304 (2015).
  149. M. Zahedian, V. Vorobyov, and J. Wrachtrup, Blueprint for efficient nuclear spin characterization with color centers, Phys. Rev. B 109, 214111 (2024).
  150. L. Robledo, L. Childress, H. Bernien, B. Hensen, P. F. A. Alkemade, and R. Hanson, High-fidelity projective read-out of a solid-state spin quantum register, Nature 477, 574 (2011).
  151. H. T. Dinani, D. W. Berry, R. Gonzalez, J. R. Maze, and C. Bonato, Bayesian estimation for quantum sensing in the absence of single-shot detection, Phys. Rev. B 99, 125413 (2019).
  152. P. C. Mahalanobis, On the generalized distance in statistics, Proc. Natl. Inst. Sci. India 2, 49 (1936).
  153. E. Bingham, J. P. Chen, M. Jankowiak, F. Obermeyer, N. Pradhan, T. Karaletsos, R. Singh, P. Szerlip, P. Horsfall, and N. D. Goodman, Pyro: Deep universal probabilistic programming, JMLR 20, (2019).
  154. D. Phan, N. Pradhan, and M. Jankowiak, Composable effects for flexible and accelerated probabilistic programming in NumPyro, arXiv:1912.11554 [stat].
  155. G. Baudart, M. Hirzel, and L. Mandel, Deep probabilistic programming languages: A qualitative study, CoRR, arXiv:1804.06458 [cs].
  156. A. Valenti, E. Van Nieuwenburg, S. Huber, and E. Greplova, Hamiltonian learning for quantum error correction, Phys. Rev. Res. 1, 033092 (2019).
  157. A. Valenti, G. Jin, J. Léonard, S. D. Huber, and E. Greplova, Scalable Hamiltonian learning for large-scale out-of-equilibrium quantum dynamics, Phys. Rev. A 105, 023302 (2022).
  158. S. Duffield, M. Benedetti, and M. Rosenkranz, Bayesian learning of parameterised quantum circuits, Mach. Learn.: Sci. Technol. 4, 025007 (2023).
  159. L. Pezzè and A. Smerzi, Advances in multiparameter quantum sensing and metrology, arXiv:2502.17396.
  160. N. Zhao, J.-L. Hu, S.-W. Ho, J. T. K. Wan, and R. B. Liu, Atomic-scale magnetometry of distant nuclear spin clusters via nitrogen-vacancy spin in diamond, Nat. Nanotechnol. 6, 242 (2011).
  161. J. R. Maze, J. M. Taylor, and M. D. Lukin, Electron spin decoherence of single nitrogen-vacancy defects in diamond, Phys. Rev. B 78, 094303 (2008).
  162. W. Yang and R.-B. Liu, Quantum many-body theory of qubit decoherence in a finite-size spin bath, Phys. Rev. B 78, 085315 (2008).
  163. W. Yang and R.-B. Liu, Quantum many-body theory of qubit decoherence in a finite-size spin bath. II. Ensemble dynamics, Phys. Rev. B 79, 115320 (2009).
  164. W. Yang, W.-L. Ma, and R.-B. Liu, Quantum many-body theory for electron spin decoherence in nanoscale nuclear spin baths, Rep. Prog. Phys. 80, 016001 (2017).
  165. M. Onizhuk and G. Galli, PyCCE: A python package for cluster correlation expansion simulations of spin qubit dynamics, Adv. Theor. Simul. 4, 2100254 (2021).
  166. T. Rainforth, A. Foster, D. R. Ivanova, and F. B. Smith, Modern Bayesian experimental design, Stat. Sci. 39, 100 (2024).
  167. K. Reuer, J. Landgraf, T. Fösel, J. O’Sullivan, L. Beltrán, A. Akin, G. J. Norris, A. Remm, M. Kerschbaum, J.-C. Besse, F. Marquardt, A. Wallraff, and C. Eichler, Realizing a deep reinforcement learning agent for real-time quantum feedback, Nat. Commun. 14, 7138 (2023).
  168. L. Sarra and F. Marquardt, Deep Bayesian experimental design for quantum many-body systems, Mach. Learn.: Sci. Technol. 4, 045022 (2023).
  169. B. Varona-Uriarte, F. Belliardo, T. H. Taminiau, C. Bonato, E. Garrote, and J. Casanova, Computationally tractable offline quantum experimental design for nuclear spin detection, arXiv:2508.21450 [quant-ph].
  170. T. Bohlin, Practical Grey-Box Process Identification, Advances in Industrial Control (Springer, London, 2006).
  171. É. Genois, J. A. Gross, A. Di Paolo, N. J. Stevenson, G. Koolstra, A. Hashim, I. Siddiqi, and A. Blais, Quantum-tailored machine-learning characterization of a superconducting qubit, PRX Quantum 2, 040355 (2021).
  172. A. Youssry, Y. Yang, R. J. Chapman, B. Haylock, F. Lenzini, M. Lobino, and A. Peruzzo, Experimental graybox quantum system identification and control, npj Quantum Inf. 10, 9 (2024).
  173. M. Biloš and S. Günnemann, Scalable normalizing flows for permutation invariant densities, in Proceedings of the 38th International Conference on Machine Learning, edited by M. Meila and T. Zhang (PMLR, 2021), pp. 957–967.
  174. K. Rasul, I. Schuster, R. Vollgraf, and U. Bergmann, Set flow: A permutation invariant normalizing flow, arXiv:1909.02775 [cs].
  175. B. Zwartsenberg, A. Ścibior, M. Niedoba, V. Lioutas, Y. Liu, J. Sefas, S. Dabiri, J. W. Lavington, T. Campbell, and F. Wood, Conditional permutation invariant flows, Trans. Machine Learn. Res. (2023).
  176. H. Maron, O. Litany, G. Chechik, and E. Fetaya, On learning sets of symmetric elements, edited by Hal Daumé III and A. Singh (PMLR, 2020), Vol. 119, pp. 6734–6744.
  177. E. H. Thiede, T. S. Hy, and R. Kondor, The general theory of permutation equivariant neural networks and higher order graph variational encoders, arXiv:2004.03990 [cs].
  178. A. Zhou, K. Yang, K. Burns, A. Cardace, Y. Jiang, S. Sokota, J. Z. Kolter, and C. Finn, Permutation equivariant neural functionals, Advances in Neural Information Processing Systems, edited by A. Oh, T. Naumann, A. Globerson, K. Saenko, M. Hardt, and S. Levine (Curran Associates, Inc., 2023), pp. 24966–24992.
  179. B. Tolooshams, A. Chandrashekar, R. Zirvi, A. Mammadov, J. Yao, C. Wang, and A. Anandkumar, EquiReg: Equivariance regularized diffusion for inverse problems, arXiv:2505.22973 [cs].
  180. J. Köhler, L. Klein, and F. Noe, Equivariant Flows: Exact Likelihood Generative Learning for Symmetric Densities, in Proceedings of the 37th International Conference on Machine Learning (PMLR, 2020), pp. 5361–5370.
  181. https://doi.org/10.5281/zenodo.20189740.
  182. R. R. Allen, F. Machado, I. L. Chuang, H.-Y. Huang, and S. Choi, Quantum computing enhanced sensing, arXiv:2501.07625 [quant-ph].

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PRX Quantum 7, 020360– Published 15 June, 2026
  • Received 9 August 2025
  • Accepted 16 April 2026
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