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  • Letter
  • Open Access

Learning algorithm reflecting universal scaling behavior near phase transitions

N. Maskara1,2,*, M. Buchhold1,3, M. Endres1, and E. van Nieuwenburg1,4

  • 1Division of Physics, Mathematics and Astronomy, California Institute of Technology, Pasadena, California 91125, USA
  • 2Department of Physics, Harvard University, Cambridge, Massachusetts 02138, USA
  • 3Institute for Theoretical Physics, University of Cologne, Zülpicher Strasse 77, 50937 Cologne, Germany
  • 4Niels Bohr International Academy, Niels Bohr Institute, University of Copenhagen, Blegdamsvej 17, 2100 Copenhagen, Denmark

  • *nmaskara@g.harvard.edu
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Phys. Rev. Research 4, L022032 – Published 9 May, 2022

DOI: https://doi.org/10.1103/PhysRevResearch.4.L022032

Abstract

Machine-learning-inspired techniques have emerged as a new paradigm for analysis of phase transitions in quantum matter. In this work, we introduce a supervised learning algorithm for studying critical phenomena from measurement data, “neural network scaling,” which is based on iteratively training convolutional networks of increasing (spatial) complexity and test it on the transverse field Ising chain and q =6 Potts model. At the continuous Ising transition, our scaling procedure directly reflects the hallmark of a continuous (second-order) phase transition, divergence of a characteristic length scale. Specifically, we extract a classification length scale by measuring the response of the classification accuracy while varying the largest convolution size (architecture of the network). We observe empirically the scaling exponent of the classification length is consistent with a power law with the correlation length exponent ν=1. Furthermore, we demonstrate the versatility of our algorithm by showing the universal scaling behaviors persist across a variety of measurement bases, including when the order parameter is nonlocal. Finally, we show that the classification length scale remains finite for the q = 6 Potts model, which has a first-order transition and lacks a divergent correlation length.

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References (68)

  1. J. Carrasquilla, Machine learning for quantum matter, Adv. Phys.: X 5, 1797528 (2020).
  2. P. Mehta, M. Bukov, C.-H. Wang, A. G. R. Day, C. Richardson, C. K. Fisher, and D. J. Schwab, A high-bias, low-variance introduction to machine learning for physicists, Phys. Rep. 810, 1 (2019).
  3. J. Schmidt, M. R. G. Marques, S. Botti, and M. A. L. Marques, Recent advances and applications of machine learning in solid-state materials science, npj Comput. Mater. 5, 83 (2019).
  4. G. Carleo, I. Cirac, K. Cranmer, L. Daudet, M. Schuld, N. Tishby, L. Vogt-Maranto, and L. Zdeborová, Machine learning and the physical sciences, Rev. Mod. Phys. 91, 045002 (2019).
  5. E. van Nieuwenburg, Y.-H. Liu, and S. Huber, Learning phase transitions by confusion, Nat. Phys. 13, 435 (2017).
  6. J. Carrasquilla and R. G. Melko, Machine learning phases of matter, Nat. Phys. 13, 431 (2017).
  7. L. Wang, Discovering phase transitions with unsupervised learning, Phys. Rev. B 94, 195105 (2016).
  8. C. Giannetti, B. Lucini, and D. Vadacchino, Machine learning as a universal tool for quantitative investigations of phase transition, Nucl. Phys. B 944, 114639 (2019).
  9. P. Ponte and R. G. Melko, Kernel methods for interpretable machine learning of order parameters, Phys. Rev. B 96, 205146 (2017).
  10. M. J. S. Beach, A. Golubeva, and R. G. Melko, Machine learning vortices at the Kosterlitz-Thouless transition, Phys. Rev. B 97, 045207 (2018).
  11. K. Liu, J. Greitemann, and L. Pollet, Learning multiple order parameters with interpretable machines, Phys. Rev. B 99, 104410 (2019).
  12. S. J. Wetzel and M. Scherzer, Machine learning of explicit order parameters: From the Ising model to SU(2) lattice gauge theory, Phys. Rev. B 96, 184410 (2017).
  13. F. Schäfer and N. Lörch, Divergence of predictive model output as indication of phase transitions, Phys. Rev. E 99, 062107 (2019).
  14. T. Mendes-Santos, X. Turkeshi, M. Dalmonte, and A. Rodriguez, Unsupervised Learning Universal Critical Behavior via the Intrinsic Dimension, Phys. Rev. X 11, 011040 (2021).
  15. S. S. Lee and B. J. Kim, Confusion scheme in machine learning detects double phase transitions and quasi-long-range order, Phys. Rev. E 99, 043308 (2019).
  16. E. Bedolla, L. C. Padierna, and R. Castañeda-Priego, Machine learning for condensed matter physics, J. Phys.: Condens. Matter 33, 053001 (2021).
  17. M. S. Scheurer and R.-J. Slager, Unsupervised Machine Learning and Band Topology, Phys. Rev. Lett. 124, 226401 (2020).
  18. R. A. Vargas-Hernández, J. Sous, M. Berciu, and R. V. Krems, Extrapolating Quantum Observables with Machine Learning: Inferring Multiple Phase Transitions from Properties of a Single Phase, Phys. Rev. Lett. 121, 255702 (2018).
  19. W. Zhang, J. Liu, and T.-C. Wei, Machine learning of phase transitions in the percolation and X Y models, Phys. Rev. E 99, 032142 (2019).
  20. J. Greitemann, K. Liu, and L. Pollet, Probing hidden spin order with interpretable machine learning, Phys. Rev. B 99, 060404(R) (2019).
  21. S. Efthymiou, M. J. S. Beach, and R. G. Melko, Super-resolving the Ising model with convolutional neural networks, Phys. Rev. B 99, 075113 (2019).
  22. S. Ebadi, T. T. Wang, H. Levine, A. Keesling, G. Semeghini, A. Omran, D. Bluvstein, R. Samajdar, H. Pichler, W. W. Ho et al., Quantum phases of matter on a 256-atom programmable quantum simulator, Nature (London) 595, 227 (2021).
  23. J. Zhang, G. Pagano, P. W. Hess, A. Kyprianidis, P. Becker, H. Kaplan, A. V. Gorshkov, Z.-X. Gong, and C. Monroe, Observation of a many-body dynamical phase transition with a 53-qubit quantum simulator, Nature (London) 551, 601 (2017).
  24. F. Arute, K. Arya, R. Babbush, D. Bacon, J. C. Bardin, R. Barends, R. Biswas, S. Boixo, F. G. S. L. Brandao, D. A. Buell, B. Burkett, Y. Chen, Z. Chen, B. Chiaro, R. Collins, W. Courtney, A. Dunsworth, E. Farhi, B. Foxen, A. Fowler et al., Quantum supremacy using a programmable superconducting processor, Nature (London) 574, 505 (2019).
  25. C. S. Wang, J. C. Curtis, B. J. Lester, Y. Zhang, Y. Y. Gao, J. Freeze, V. S. Batista, P. H. Vaccaro, I. L. Chuang, L. Frunzio, L. Jiang, S. Girvin, and R. J. Schoelkopf, Efficient Multiphoton Sampling of Molecular Vibronic Spectra on a Superconducting Bosonic Processor, Phys. Rev. X 10, 021060 (2020).
  26. C. Gross and I. Bloch, Quantum simulations with ultracold atoms in optical lattices, Science 357, 995 (2017).
  27. J.-y. Choi, S. Hild, J. Zeiher, P. Schauß, A. Rubio-Abadal, T. Yefsah, V. Khemani, D. A. Huse, I. Bloch, and C. Gross, Exploring the many-body localization transition in two dimensions, Science 352, 1547 (2016).
  28. A. Mazurenko, C. S. Chiu, G. Ji, M. F. Parsons, M. Kanász-Nagy, R. Schmidt, F. Grusdt, E. Demler, D. Greif, and M. Greiner, Experimental realization of a long-range antiferromagnet in the Hubbard model with ultracold atoms, Nature (London) 545, 462 (2017).
  29. H. Bernien, S. Schwartz, A. Keesling, H. Levine, A. Omran, H. Pichler, S. Choi, A. S. Zibrov, M. Endres, M. Greiner, V. Vuletić, and M. D. Lukin, Probing many-body dynamics on a 51-atom quantum simulator, Nature (London) 551, 579 (2017).
  30. A. Browaeys and T. Lahaye, Many-body physics with individually controlled Rydberg atoms, Nat. Phys. 16, 132 (2020).
  31. H. Labuhn, D. Barredo, S. Ravets, S. de Léséleuc, T. Macr, T. Lahaye, and A. Browaeys, Tunable two-dimensional arrays of single Rydberg atoms for realizing quantum Ising models, Nature (London) 534, 667 (2016).
  32. N. Friis, O. Marty, C. Maier, C. Hempel, M. Holzäpfel, P. Jurcevic, M. B. Plenio, M. Huber, C. Roos, R. Blatt, and B. Lanyon, Observation of Entangled States of a Fully Controlled 20-Qubit System, Phys. Rev. X 8, 021012 (2018).
  33. G. Pagano, A. Bapat, P. Becker, K. S. Collins, A. De, P. W. Hess, H. B. Kaplan, A. Kyprianidis, W. L. Tan, C. Baldwin, L. T. Brady, A. Deshpande, F. Liu, S. Jordan, A. V. Gorshkov, and C. Monroe, Quantum approximate optimization of the long-range Ising model with a trapped-ion quantum simulator, Proc. Natl. Acad. Sci. 117, 25396 (2020).
  34. J. Venderley, V. Khemani, and E.-A. Kim, Machine Learning Out-of-Equilibrium Phases of Matter, Phys. Rev. Lett. 120, 257204 (2018).
  35. H. Théveniaut and F. Alet, Neural network setups for a precise detection of the many-body localization transition: Finite-size scaling and limitations, Phys. Rev. B 100, 224202 (2019).
  36. E. van Nieuwenburg, E. Bairey, and G. Refael, Learning phase transitions from dynamics, Phys. Rev. B 98, 060301 (2018).
  37. M. Matty, Y. Zhang, Z. Papić, and E.-A. Kim, Multifaceted machine learning of competing orders in disordered interacting systems, Phys. Rev. B 100, 155141 (2019).
  38. W. Zhang, L. Wang, and Z. Wang, Interpretable machine learning study of the many-body localization transition in disordered quantum Ising spin chains, Phys. Rev. B 99, 054208 (2019).
  39. D.-L. Deng, X. Li, and S. Das Sarma, Machine learning topological states, Phys. Rev. B 96, 195145 (2017).
  40. J. F. Rodriguez-Nieva and M. S. Scheurer, Identifying topological order through unsupervised machine learning, Nat. Phys. 15, 790 (2019).
  41. O. Balabanov and M. Granath, Unsupervised interpretable learning of topological indices invariant under permutations of atomic bands, arXiv:2008.07268.
  42. E. Greplova, A. Valenti, G. Boschung, F. Schäfer, N. Lörch, and S. Huber, Unsupervised identification of topological order using predictive models, arXiv:1910.10124.
  43. T. Vieijra, C. Casert, J. Nys, W. De Neve, J. Haegeman, J. Ryckebusch, and F. Verstraete, Restricted Boltzmann Machines for Quantum States with Non-Abelian or Anyonic Symmetries, Phys. Rev. Lett. 124, 097201 (2020).
  44. H.-Y. Hu, S.-H. Li, L. Wang, and Y.-Z. You, Machine learning holographic mapping by neural network renormalization group, Phys. Rev. Res. 2, 023369 (2020).
  45. B. S. Rem, N. Käming, M. Tarnowski, L. Asteria, N. Fläschner, C. Becker, K. Sengstock, and C. Weitenberg, Identifying quantum phase transitions using artificial neural networks on experimental data, Nat. Phys. 15, 917 (2019).
  46. A. Bohrdt, C. S. Chiu, G. Ji, M. Xu, D. Greif, M. Greiner, E. Demler, F. Grusdt, and M. Knap, Classifying snapshots of the doped Hubbard model with machine learning, Nat. Phys. 15, 921 (2019).
  47. Y. Zhang, A. Mesaros, K. Fujita, S. D. Edkins, M. H. Hamidian, K. Ch'ng, H. Eisaki, S. Uchida, J. C. S. Davis, E. Khatami et al., Machine learning in electronic-quantum-matter imaging experiments, Nature (London) 570, 484 (2019).
  48. C. Szegedy, W. Liu, Y. Jia, P. Sermanet, S. Reed, D. Anguelov, D. Erhan, V. Vanhoucke, and A. Rabinovich, Going deeper with convolutions, arXiv:1409.4842.
  49. M. Lin, Q. Chen, and S. Yan, Network in network, arXiv:1312.4400.
  50. C. Miles, A. Bohrdt, R. Wu, C. Chiu, M. Xu, G. Ji, M. Greiner, K. Q. Weinberger, E. Demler, and E.-A. Kim, Correlator convolutional neural networks: An interpretable architecture for image-like quantum matter data, arXiv:2011.03474.
  51. K. Mills, K. Ryczko, I. Luchak, A. Domurad, C. Beeler, and I. Tamblyn, Extensive deep neural networks for transferring small scale learning to large scale systems, Chem. Sci. 10, 4129 (2019).
  52. D. A. Martin, T. L. Ribeiro, S. A. Cannas, T. S. Grigera, D. Plenz, and D. R. Chialvo, Box-scaling as a proxy of finite-size correlations, arXiv:2007.08236.
  53. G. Vidal, Classical Simulation of Infinite-Size Quantum Lattice Systems in One Spatial Dimension, Phys. Rev. Lett. 98, 070201 (2007).
  54. F. Chollet et al., Keras, https://keras.io (2015).
  55. See Supplemental Material at http://link.aps.org/supplemental/10.1103/PhysRevResearch.4.L022032 for additional information about network architecture, observables in different bases, and different partitions.
  56. U. Wolff, Collective Monte Carlo Updating for Spin Systems, Phys. Rev. Lett. 62, 361 (1989).
  57. F. Y. Wu, The Potts model, Rev. Mod. Phys. 54, 235 (1982).
  58. M. Kardar, Statistical Physics of Fields (Cambridge University Press, Cambridge, England, 2007).
  59. S. Sachdev, Quantum Phase Transitions, 2nd ed. (Cambridge University Press, Cambridge, England, 2011).
  60. V. R. Vieira, Quantum information and phase transitions: Fidelity and state distinguishability, J. Phys.: Conf. Ser. 213, 012005 (2010).
  61. S.-J. Gu, Fidelity approach to quantum phase transitions, Int. J. Mod. Phys. B 24, 4371 (2010).
  62. P. Peczak and D. P. Landau, Monte Carlo study of finite-size effects at a weakly first-order phase transition, Phys. Rev. B 39, 11932 (1989).
  63. K. Liu, N. Sadoune, N. Rao, J. Greitemann, and L. Pollet, Revealing the phase diagram of Kitaev materials by machine learning: Cooperation and competition between spin liquids, arXiv:2004.14415.
  64. I. Cong, S. Choi, and M. D. Lukin, Quantum convolutional neural networks, Nat. Phys. 15, 1273 (2019).
  65. H.-Y. Huang, R. Kueng, G. Torlai, V. V. Albert, and J. Preskill, Provably efficient machine learning for quantum many-body problems, arXiv:2106.12627.
  66. A. Seif, M. Hafezi, and C. Jarzynski, Machine learning the thermodynamic arrow of time, Nat. Phys. 17, 105 (2021).
  67. Y. Zhang and E.-A. Kim, Quantum Loop Topography for Machine Learning, Phys. Rev. Lett. 118, 216401 (2017).
  68. M. Iqbal and N. Schuch, Entanglement Order Parameters and Critical Behavior for Topological Phase transitions and Beyond, Phys. Rev. X 11, 041014 (2021).

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Phys. Rev. Research 4, L022032– Published 9 May, 2022
  • Received 12 April 2021
  • Revised 14 December 2021
  • Accepted 15 February 2022
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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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