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Restricted Boltzmann machines in quantum physics

A type of stochastic neural network called a restricted Boltzmann machine has been widely used in artificial intelligence applications for decades. They are now finding new life in the simulation of complex wavefunctions in quantum many-body physics.

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References

1. 1.

White, S. R. Density matrix formulation for quantum renormalization groups. Phys. Rev. Lett. 69, 2863–2866 (1992).

2. 2.

Sandvik, A. W. Computational studies of quantum spin systems. AIP Conf. Proc. 1297, 135–338 (2010).

3. 3.

Becca, F. & Sorella, S. Quantum Monte Carlo Approaches for Correlated Systems (Cambridge Univ. Press, 2017).

4. 4.

Bernien, H. et al. Probing many-body dynamics on a 51-atom quantum simulator. Nature 551, 579–584 (2017).

5. 5.

Zhang, J. et al. Observation of a many-body dynamical phase transition with a 53-qubit quantum simulator. Nature 551, 601–604 (2017).

6. 6.

Torlai, G. et al. Integrating neural networks with a quantum simulator for state reconstruction. Preprint at https://arxiv.org/abs/1904.08441 (2019).

7. 7.

Carleo, G. & Troyer, M. Solving the quantum many-body problem with artificial neural networks. Science 355, 602–606 (2017).

8. 8.

Carrasquilla, J. & Melko, R. G. Machine learning phases of matter. Nat. Phys. 13, 431–434 (2017).

9. 9.

Torlai, G. et al. Neural-network quantum state tomography. Nat. Phys. 14, 447–450 (2018).

10. 10.

Torlai, G. & Melko, R. G. Learning thermodynamics with Boltzmann machines. Phys. Rev. B 94, 165134 (2016).

11. 11.

Terhal, B. M. Quantum supremacy, here we come. Nat. Phys. 14, 530–531 (2018).

12. 12.

Hopfield, J. J. Neural networks and physical systems with emergent collective computational abilities. Proc. Natl Acad. Sci. USA 79, 2554–2558 (1982).

13. 13.

Nguyen, H. C., Zecchina, R. & Berg, J. Inverse statistical problems: from the inverse ising problem to data science. Adv. Phys. 66, 197–261 (2017).

14. 14.

Ackley, D. H., Hinton, G. E. & Sejnowski, T. J. A learning algorithm for Boltzmann machines. Cogn. Sci. 9, 147–169 (1985).

15. 15.

Smolensky, P. in Parallel Distributed Processing: Explorations in the Microstructure of Cognition Vol. 1. (eds Rumelhart, D. E. & McClelland, J. L.) 194–281 (MIT Press, 1986).

16. 16.

Hinton, G. E. Training products of experts by minimizing contrastive divergence. Neural Comput. 14, 1771–1800 (2002).

17. 17.

Hinton, G. E. & Salakhutdinov, R. R. Reducing the dimensionality of data with neural networks. Science 313, 504–507 (2006).

18. 18.

Salakhutdinov, R., Mnih, A. & Hinton, G. Restricted Boltzmann machines for collaborative filtering. In Proc. 24th International Conference on Machine Learning (ed. Ghahramani, Z.) 791–798 (ACM, 2007).

19. 19.

Roux, N. L. & Bengio, Y. Representational power of restricted Boltzmann machines and deep belief networks. Neural Comput. 20, 1631–1649 (2008).

20. 20.

Orus, R. A practical introduction to tensor networks: matrix product states and projected entangled pair states. Ann. Phys. 349, 117–158 (2014).

21. 21.

Vidal, G. Efficient simulation of one-dimensional quantum many-body systems. Phys. Rev. Lett. 93, 040502 (2004).

22. 22.

Eisert, J., Cramer, M. & Plenio, M. B. Colloquium: area laws for the entanglement entropy. Rev. Mod. Phys. 82, 277–306 (2010).

23. 23.

Östlund, S. & Rommer, S. Thermodynamic limit of density matrix renormalization. Phys. Rev. Lett. 75, 3537–3540 (1995).

24. 24.

Verstraete, F. & Cirac, J. I. Renormalization algorithms for quantum-many body systems in two and higher dimensions. Preprint at https://arxiv.org/abs/cond-mat/0407066 (2004).

25. 25.

Clark, S. R. Unifying neural-network quantum states and correlator product states via tensor networks. J. Phys. A 51, 135301 (2018).

26. 26.

Glasser, I., Pancotti, N., August, M., Rodriguez, I. D. & Cirac, J. I. Neural-network quantum states, string-bond states, and chiral topological states. Phys. Rev. X 8, 011006 (2018).

27. 27.

Chen, J., Cheng, S., Xie, H., Wang, L. & Xiang, T. Equivalence of restricted Boltzmann machines and tensor network states. Phys. Rev. B 97, 085104 (2018).

28. 28.

Deng, D.-L., Li, X. & Das Sarma, S. Quantum entanglement in neural network states. Phys. Rev. X 7, 021021 (2017).

29. 29.

Sfondrini, A., Cerrillo, J., Schuch, N. & Cirac, J. I. Simulating two- and three-dimensional frustrated quantum systems with string-bond states. Phys. Rev. B 81, 214426 (2010).

30. 30.

Pastori, L., Kaubruegger, R. & Budich, J. C. Generalized transfer matrix states from artificial neural networks. Phys. Rev. B 99, 165123 (2019).

31. 31.

Salakhutdinov, R. Learning deep generative models. Ann. Rev. Stat. Appl. 2, 361–385 (2015).

32. 32.

Gao, X. & Duan, L.-M. Efficient representation of quantum many-body states with deep neural networks. Nat. Commun. 8, 662 (2017).

33. 33.

Carleo, G., Nomura, Y. & Imada, M. Constructing exact representations of quantum many-body systems with deep neural networks. Nat. Commun. 9, 5322 (2018).

34. 34.

Sorella, S. Green function Monte Carlo with stochastic reconfiguration. Phys. Rev. Lett. 80, 4558–4561 (1998).

35. 35.

Deng, D.-L., Li, X. & Das Sarma, S. Machine learning topological states. Phys. Rev. B 96, 195145 (2017).

36. 36.

Nomura, Y., Darmawan, A. S., Yamaji, Y. & Imada, M. Restricted Boltzmann machine learning for solving strongly correlated quantum systems. Phys. Rev. B 96, 205152 (2017).

37. 37.

Saito, H. Solving the Bose–Hubbard model with machine learning. J. Phys. Soc. Jpn 86, 093001 (2017).

38. 38.

Luo, D. & Clark, B. K. Backflow transformations via neural networks for quantum many-body wave-functions. Preprint at https://arxiv.org/abs/1807.10770 (2018).

39. 39.

Han, J., Zhang, L. & E, W. Solving many-electron Schrodinger equation using deep neural networks. Preprint at https://arxiv.org/abs/1807.07014 (2018).

40. 40.

Teng, P. Machine-learning quantum mechanics: solving quantum mechanics problems using radial basis function networks. Phys. Rev. E 98, 033305 (2018).

41. 41.

Saito, H. Method to solve quantum few-body problems with artificial neural networks. J. Phys. Soc. Jpn 87, 074002 (2018).

42. 42.

Choo, K., Carleo, G., Regnault, N. & Neupert, T. Symmetries and many-body excitations with neural-network quantum states. Phys. Rev. Lett. 121, 167204 (2018).

43. 43.

Carleo, G., Becca, F., Schiro, M. & Fabrizio, M. Localization and glassy dynamics of many-body quantum systems. Sci. Rep. 2, 243 (2012).

44. 44.

Jónsson, B., Bauer, B. & Carleo, G. Neural-network states for the classical simulation of quantum computing. Preprint at https://arxiv.org/abs/1808.05232 (2018).

45. 45.

Kaubruegger, R., Pastori, L. & Budich, J. C. Chiral topological phases from artificial neural networks. Phys. Rev. B 97, 195136 (2018).

46. 46.

Schmitt, M. & Heyl, M. Quantum dynamics in transverse-field Ising models from classical networks. SciPost Phys. 4, 013 (2018).

47. 47.

Czischek, S., Gärttner, M. & Gasenzer, T. Quenches near Ising quantum criticality as a challenge for artificial neural networks. Phys. Rev. B 98, 024311 (2018).

48. 48.

Vieijra, T. et al. Restricted Boltzmann machines for quantum states with nonabelian or anyonic symmetries. Preprint at https://arxiv.org/abs/1905.06034 (2019).

49. 49.

Vogel, K. & Risken, H. Determination of quasiprobability distributions in terms of probability distributions for the rotated quadrature phase. Phys. Rev. A 40, 2847–2849 (1989).

50. 50.

James, D. F. V., Kwiat, P. G., Munro, W. J. & White, A. G. Measurement of qubits. Phys. Rev. A 64, 052312 (2001).

51. 51.

Roos, C. F. et al. Bell states of atoms with ultralong lifetimes and their tomographic state analysis. Phys. Rev. Lett. 92, 220402 (2004).

52. 52.

Cramer, M. et al. Efficient quantum state tomography. Nat. Commun. 1, 149 (2009).

53. 53.

Kim, K. et al. Quantum simulation of frustrated ising spins with trapped ions. Nature 465, 590–593 (2010).

54. 54.

Harris, R. et al. Phase transitions in a programmable quantum spin glass simulator. Science 361, 162–165 (2018).

55. 55.

King, A. D. et al. Observation of topological phenomena in a programmable lattice of 1,800 qubits. Nature 560, 456–460 (2018).

56. 56.

Hastings, M. B., González, I., Kallin, A. B. & Melko, R. G. Measuring Renyi entanglement entropy in quantum Monte Carlo simulations. Phys. Rev. Lett. 104, 157201 (2010).

57. 57.

Torlai, G. & Melko, R. G. Latent space purification via neural density operators. Phys. Rev. Lett. 120, 240503 (2018).

58. 58.

Rocchetto, A., Grant, E., Strelchuk, S., Carleo, G. & Severini, S. Learning hard quantum distributions with variational autoencoders. npj Quantum Inf. 4, 28 (2018).

59. 59.

Goodfellow, I. NIPS 2016 tutorial: generative adversarial networks. Preprint at https://arxiv.org/abs/1701.00160 (2016).

60. 60.

Kingma, D. P. & Welling, M. Auto-encoding variational Bayes. Preprint at https://arxiv.org/abs/1312.6114 (2013).

61. 61.

van den Oord, A., Kalchbrenner, N. & Kavukcuoglu, K. Pixel recurrent neural networks. Preprint at https://arxiv.org/abs/1601.06759 (2016).

62. 62.

Carrasquilla, J., Torlai, G., Melko, R. G. & Aolita, L. Reconstructing quantum states with generative models. Nat. Mach. Intell. 1, 155–161 (2019).

63. 63.

Sharir, O., Levine, Y., Wies, N., Carleo, G. & Shashua, A. Deep autoregressive models for the efficient variational simulation of many-body quantum systems. Preprint at https://arxiv.org/abs/1902.04057 (2019).

64. 64.

Goodfellow, I. et al. in Advances in Neural Information Processing Systems Vol. 27 (eds Ghahramani, Z., Welling, M., Cortes, C., Lawrence, N. D. & Weinberger, K. Q.) 2672–2680 (Curran Associates, 2014).

Acknowledgements

We thank G. Torlai, A. Rocchetto and M. Albergo for fruitful discussions. This research was supported by NSERC of Canada, the Perimeter Institute for Theoretical Physics, the Shared Hierarchical Academic Research Computing Network (SHARCNET) and the National Science Foundation under grant no. PHY-1748958. Research at Perimeter Institute is supported in part by the Government of Canada through the Department of Innovation, Science and Economic Development Canada and by the Province of Ontario through the Ministry of Economic Development, Job Creation and Trade. R.G.M. acknowledges support from a Canada Research Chair. J.C. acknowledges financial and computational support from the AI grant. J.I.C. was funded from the European Research Council (grant agreement no. 742102).

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Correspondence to Roger G. Melko.

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Melko, R.G., Carleo, G., Carrasquilla, J. et al. Restricted Boltzmann machines in quantum physics. Nat. Phys. 15, 887–892 (2019). https://doi.org/10.1038/s41567-019-0545-1

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