[PDF]  https://doi.org/10.3952/physics.v61i4.4639

Open access article / Atviros prieigos straipsnis
Lith. J. Phys. 61, 205–214 (2021)
 

AUTOENCODER-AIDED ANALYSIS OF LOW-DIMENSIONAL HILBERT SPACES
Giedrius Žlabys, Mantas Račiūnas, and Egidijus Anisimovas
  Institute of Theoretical Physics and Astronomy, Vilnius University, Saulėtekio 3, 10257 Vilnius, Lithuania
Emails: giedrius.zlabys@tfai.vu.lt, mantas.raciunas@tfai.vu.lt, egidijus.anisimovas@ff.vu.lt

Received 14 September 2021; accepted 23 September 2021

We study the applicability of feedforward autoencoders in determining the ground state of a quantum system from a noisy signal provided in a form of random superpositions sampled from a low-dimensional subspace of the system’s Hilbert space. The proposed scheme relies on a minimum set of assumptions: the presence of a finite number of orthogonal states in the samples and a weak statistical dominance of the targeted ground state. The provided data is compressed into a two-dimensional feature space and subsequently analyzed to determine the optimal approximation to the true ground state. The scheme is applicable to single- and many-particle quantum systems as well as in the presence of magnetic frustration.
Keywords: feedforward autoencoder, low-dimensional Hilbert spaces, numerical ground-state estimation
PACS: 03.65.Aa, 02.70.Rr, 02.90.+p

NEDIDELIO MATMENŲ SKAIČIAUS HILBERTO ERDVĖS ANALIZĖ PASITELKIANT AUTOENKODERĮ
Giedrius Žlabys, Mantas Račiūnas, Egidijus Anisimovas

Vilniaus universiteto Teorinės fizikos ir astronomijos institutas, Vilnius, Lietuva

Straipsnyje aprašomas tiesioginio sklidimo autoenkoderių pritaikomumo siekiant patikimai nustatyti kvantinės sistemos pagrindinę būseną iš triukšmingo signalo, kurį sudaro seka atsitiktinių superpozicijų, paimtų iš sistemos Hilberto erdvės nedidelio matmenų skaičiaus poerdvio, tyrimas. Siūloma schema remiasi tik minimaliomis prielaidomis: (i) laikoma, kad imties elementus sudaro baigtinio skaičiaus ortogonalių būsenų superpozicijos ir (ii) yra silpnas statistinis ieškomos pagrindinės būsenos dominavimas. Gaunami atsitiktiniai duomenys pasitelkiant autoenkoderį yra suspaudžiami į dvimatį požymių sluoksnį ir šis atvaizdavimas yra analizuojamas siekiant nustatyti optimalų tikrosios pagrindinės būsenos artinį. Siūlomas metodas yra universalus ir tinka tiek viendalelinėms, tiek daugiadalelinėms kvantinėms sistemoms tirti. Taip pat parodoma, kad jis yra pritaikomas ir daugiadalelinėms sistemoms stipriuose magnetiniuose laukuose, kai sistemos energijos spektras yra fraktalinio pobūdžio, o banginės funkcijos pasižymi nereguliaria struktūra.


References / Nuorodos

[1] A. Avella and F. Mancini, Strongly Correlated Systems: Numerical Methods, Springer Series in Solid-State Sciences (Springer Berlin Heidelberg, 2015),
https://doi.org/10.1007/978-3-662-44133-6
[2] P.A. Lee, From high temperature superconductivity to quantum spin liquid: progress in strong correlation physics, Rep. Prog. Phys. 71, 012501 (2007),
https://doi.org/10.1088/0034-4885/71/1/012501
[3] K. von Klitzing, Quantum Hall effect: Discovery and application, Annu. Rev. Condens. Matter Phys. 8, 13 (2017),
https://doi.org/10.1146/annurev-conmatphys-031016-025148
[4] R. Orús, A practical introduction to tensor networks: Matrix product states and projected entangled pair states, Ann. Phys. 349, 117 (2014),
https://doi.org/10.1016/j.aop.2014.06.013
[5] G. Carleo and M. Troyer, Solving the quantum many-body problem with artificial neural networks, Science 355, 602 (2017),
https://doi.org/10.1126/science.aag2302
[6] R.O. Jones, Density functional theory: Its origins, rise to prominence, and future, Rev. Mod. Phys. 87, 897 (2015),
https://doi.org/10.1103/RevModPhys.87.897
[7] W.M.C. Foulkes, L. Mitas, R.J. Needs, and G. Rajagopal, Quantum Monte Carlo simulations of solids, Rev. Mod. Phys. 73, 33 (2001),
https://doi.org/10.1103/RevModPhys.73.33
[8] Computational Many-Particle Physics, eds. H. Fehske, R. Schneider, and A. Weiße, Lecture Notes in Physics (Springer Berlin Heidelberg, 2008),
https://doi.org/10.1007/978-3-540-74686-7
[9] F. Becca and S. Sorella, Correlated models and wave functions, in: Quantum Monte Carlo Approaches for Correlated Systems (Cambridge University Press, 2017) pp. 3–36,
https://doi.org/10.1017/9781316417041.002
[10] L. Lehtovaara, J. Toivanen, and J. Eloranta, Solution of time-independent Schrödinger equation by the imaginary time propagation method, J. Comput. Phys. 221, 148 (2007),
https://doi.org/10.1016/j.jcp.2006.06.006
[11] R. Orús, Tensor networks for complex quantum systems, Nat. Rev. Phys. 1, 538 (2019),
https://doi.org/10.1038/s42254-019-0086-7
[12] U. Schollwöck, The density-matrix renormalization group in the age of matrix product states, Ann. Phys. 326, 96 (2011),
https://doi.org/10.1016/j.aop.2010.09.012
[13] H. Saito, Solving the Bose–Hubbard model with machine learning, J. Phys. Soc. Jpn. 86, 093001 (2017),
https://doi.org/10.7566/JPSJ.86.093001
[14] H. Saito and M. Kato, Solving the Bose–Hubbard model with machine learning, J. Phys. Soc. Jpn. 87, 014001 (2017),
https://doi.org/10.7566/JPSJ.87.014001
[15] G. Carleo, K. Choo, D. Hofmann, J.E.T. Smith, T. Westerhout, F. Alet, E.J. Davis, S. Efthymiou, I. Glasser, S.-H. Lin, et al., NetKet: A machine learning toolkit for many-body quantum systems, SoftwareX 10, 100311 (2019),
https://doi.org/10.1016/j.softx.2019.100311
[16] 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),
https://doi.org/10.1103/RevModPhys.91.045002
[17] F. Becca and S. Sorella, Quantum Monte Carlo Approaches for Correlated Systems (Cambridge University Press, 2017),
https://doi.org/10.1017/9781316417041
[18] G.E. Hinton and R.R. Salakhutdinov, Reducing the dimensionality of data with neural networks, Science 313, 504 (2006),
https://doi.org/10.1126/science.1127647
[19] I. Goodfellow, Y. Bengio, and A. Courville, Deep Learning (MIT Press, 2016),
https://www.deeplearningbook.org/
[20] M. Lewenstein, A. Sanpera, and V. Ahufinger, Ultracold Atoms in Optical Lattices: Simulating Quantum Many-body Systems (Oxford University Press, 2012),
https://doi.org/10.1093/acprof:oso/9780199573127.001.0001
[21] A. Eckardt, Colloquium: Atomic quantum gases in periodically driven optical lattices, Rev. Mod. Phys. 89, 011004 (2017),
https://arxiv.org/abs/1606.08041,
https://doi.org/10.1103/RevModPhys.89.011004
[22] E. J. Bergholtz and Z. Liu, Topological flat band models and fractional Chern insulators, Intl. J. Mod. Phys. B 27, 1330017 (2013),
https://arxiv.org/abs/1308.0343,
https://doi.org/10.1142/S021797921330017X
[23] M. Račiūnas, F.N. Ünal, E. Anisimovas, and A. Eckardt, Creating, probing, and manipulating fractionally charged excitations of fractional Chern insulators in optical lattices, Phys. Rev. A 98, 063621 (2018),
https://doi.org/10.1103/PhysRevA.98.063621
[24] D. Tong, Lectures on the quantum Hall effect (2016),
https://arxiv.org/abs/1606.06687
[25] R. Perline, Zipf’s law, the central limit theorem, and the random division of the unit interval, Phys. Rev. E 54, 220 (1996),
https://doi.org/10.1103/PhysRevE.54.220
[26] Keras: the Python deep learning library,
https://keras.io
[27] D. P. Kingma and J. Ba, Adam: A method for stochastic optimization (2014),
https://arxiv.org/abs/1412.6980