Interpreting machine learning of topological quantum phase transitions
Abstract
There has been growing excitement over the possibility of employing artificial neural networks (ANNs) to gain new theoretical insight into the physics of quantum manybody problems. “Interpretability” remains a concern: can we understand the basis for the ANN’s decisionmaking criteria in order to inform our theoretical understanding? “Interpretable” machine learning in quantum matter has to date been restricted to linear models, such as support vector machines, due to the greater difficulty of interpreting nonlinear ANNs. Here we consider topological quantum phase transitions in models of Chern insulator, Z_{2} topological insulator, and Z_{2} quantum spin liquid, each using a shallow fully connected feedforward ANN. The use of quantum loop topography, a “domain knowledge”–guided approach to feature selection, facilitates the construction of faithful phase diagrams and semiquantitative interpretation of the criteria in certain cases. To identify the topological phases, the ANNs learn physically meaningful features, such as topological invariants and deconfinement of loops. The interpretability in these cases suggests hope for theoretical progress based on future uses of ANNbased machine learning on quantum manybody problems.
 Authors:

 Cornell Univ., Ithaca, NY (United States); Peking Univ., Beijing (China)
 Cornell Univ., Ithaca, NY (United States)
 Publication Date:
 Research Org.:
 Cornell Univ., Ithaca, NY (United States)
 Sponsoring Org.:
 USDOE Office of Science (SC), Basic Energy Sciences (BES). Materials Sciences & Engineering Division (MSE); USDOE
 OSTI Identifier:
 1632033
 Alternate Identifier(s):
 OSTI ID: 1733302; OSTI ID: 2322523
 Grant/Contract Number:
 SC0018946
 Resource Type:
 Published Article
 Journal Name:
 Physical Review Research
 Additional Journal Information:
 Journal Volume: 2; Journal Issue: 2; Journal ID: ISSN 26431564
 Publisher:
 American Physical Society (APS)
 Country of Publication:
 United States
 Language:
 English
 Subject:
 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; Machine learning; Topological insulators; Topological phases of matter; Artificial neural networks
Citation Formats
Zhang, Yi, Ginsparg, Paul, and Kim, EunAh. Interpreting machine learning of topological quantum phase transitions. United States: N. p., 2020.
Web. doi:10.1103/physrevresearch.2.023283.
Zhang, Yi, Ginsparg, Paul, & Kim, EunAh. Interpreting machine learning of topological quantum phase transitions. United States. https://doi.org/10.1103/physrevresearch.2.023283
Zhang, Yi, Ginsparg, Paul, and Kim, EunAh. Thu .
"Interpreting machine learning of topological quantum phase transitions". United States. https://doi.org/10.1103/physrevresearch.2.023283.
@article{osti_1632033,
title = {Interpreting machine learning of topological quantum phase transitions},
author = {Zhang, Yi and Ginsparg, Paul and Kim, EunAh},
abstractNote = {There has been growing excitement over the possibility of employing artificial neural networks (ANNs) to gain new theoretical insight into the physics of quantum manybody problems. “Interpretability” remains a concern: can we understand the basis for the ANN’s decisionmaking criteria in order to inform our theoretical understanding? “Interpretable” machine learning in quantum matter has to date been restricted to linear models, such as support vector machines, due to the greater difficulty of interpreting nonlinear ANNs. Here we consider topological quantum phase transitions in models of Chern insulator, Z2 topological insulator, and Z2 quantum spin liquid, each using a shallow fully connected feedforward ANN. The use of quantum loop topography, a “domain knowledge”–guided approach to feature selection, facilitates the construction of faithful phase diagrams and semiquantitative interpretation of the criteria in certain cases. To identify the topological phases, the ANNs learn physically meaningful features, such as topological invariants and deconfinement of loops. The interpretability in these cases suggests hope for theoretical progress based on future uses of ANNbased machine learning on quantum manybody problems.},
doi = {10.1103/physrevresearch.2.023283},
journal = {Physical Review Research},
number = 2,
volume = 2,
place = {United States},
year = {2020},
month = {6}
}
https://doi.org/10.1103/physrevresearch.2.023283
Works referenced in this record:
Quantum Entanglement in Neural Network States
journal, May 2017
 Deng, DongLing; Li, Xiaopeng; Das Sarma, S.
 Physical Review X, Vol. 7, Issue 2
Unsupervised identification of topological phase transitions using predictive models
journal, April 2020
 Greplova, Eliska; Valenti, Agnes; Boschung, Gregor
 New Journal of Physics, Vol. 22, Issue 4
Machine learning in electronicquantummatter imaging experiments
journal, June 2019
 Zhang, Yi; Mesaros, A.; Fujita, K.
 Nature, Vol. 570, Issue 7762
Kernel methods for interpretable machine learning of order parameters
journal, November 2017
 Ponte, Pedro; Melko, Roger G.
 Physical Review B, Vol. 96, Issue 20
Anyons in an exactly solved model and beyond
journal, January 2006
 Kitaev, Alexei
 Annals of Physics, Vol. 321, Issue 1
Deep learning topological invariants of band insulators
journal, August 2018
 Sun, Ning; Yi, Jinmin; Zhang, Pengfei
 Physical Review B, Vol. 98, Issue 8
Classifying snapshots of the doped Hubbard model with machine learning
journal, July 2019
 Bohrdt, Annabelle; Chiu, Christie S.; Ji, Geoffrey
 Nature Physics, Vol. 15, Issue 9
Machine learning quantum phases of matter beyond the fermion sign problem
journal, August 2017
 Broecker, Peter; Carrasquilla, Juan; Melko, Roger G.
 Scientific Reports, Vol. 7, Issue 1
Machine Learning Topological Invariants with Neural Networks
journal, February 2018
 Zhang, Pengfei; Shen, Huitao; Zhai, Hui
 Physical Review Letters, Vol. 120, Issue 6
Learning multiple order parameters with interpretable machines
journal, March 2019
 Liu, Ke; Greitemann, Jonas; Pollet, Lode
 Physical Review B, Vol. 99, Issue 10
Efficient representation of quantum manybody states with deep neural networks
journal, September 2017
 Gao, Xun; Duan, LuMing
 Nature Communications, Vol. 8, Issue 1
Interpretable machine learning study of the manybody localization transition in disordered quantum Ising spin chains
journal, February 2019
 Zhang, Wei; Wang, Lei; Wang, Ziqiang
 Physical Review B, Vol. 99, Issue 5
${Z}_{2}$ Topological Order and the Quantum Spin Hall Effect
journal, September 2005
 Kane, C. L.; Mele, E. J.
 Physical Review Letters, Vol. 95, Issue 14, Article No. 146802
Machine Learning ManyBody Localization: Search for the Elusive Nonergodic Metal
journal, December 2018
 Hsu, YiTing; Li, Xiao; Deng, DongLing
 Physical Review Letters, Vol. 121, Issue 24
Machine learning of explicit order parameters: From the Ising model to SU(2) lattice gauge theory
journal, November 2017
 Wetzel, Sebastian J.; Scherzer, Manuel
 Physical Review B, Vol. 96, Issue 18
Machine learning ${\mathbb{Z}}_{2}$ quantum spin liquids with quasiparticle statistics
journal, December 2017
 Zhang, Yi; Melko, Roger G.; Kim, EunAh
 Physical Review B, Vol. 96, Issue 24
Machine learning action parameters in lattice quantum chromodynamics
journal, May 2018
 Shanahan, Phiala E.; Trewartha, Daniel; Detmold, William
 Physical Review D, Vol. 97, Issue 9
Phase diagrams of lattice gauge theories with Higgs fields
journal, June 1979
 Fradkin, Eduardo; Shenker, Stephen H.
 Physical Review D, Vol. 19, Issue 12
Emergent Schrödinger equation in an introspective machine learning architecture
journal, September 2019
 Wang, Ce; Zhai, Hui; You, YiZhuang
 Science Bulletin, Vol. 64, Issue 17
Machine Learning Phases of Strongly Correlated Fermions
journal, August 2017
 Ch’ng, Kelvin; Carrasquilla, Juan; Melko, Roger G.
 Physical Review X, Vol. 7, Issue 3
Symmetries and ManyBody Excitations with NeuralNetwork Quantum States
journal, October 2018
 Choo, Kenny; Carleo, Giuseppe; Regnault, Nicolas
 Physical Review Letters, Vol. 121, Issue 16
NeuralNetwork Quantum States, StringBond States, and Chiral Topological States
journal, January 2018
 Glasser, Ivan; Pancotti, Nicola; August, Moritz
 Physical Review X, Vol. 8, Issue 1
Identifying quantum phase transitions using artificial neural networks on experimental data
journal, July 2019
 Rem, Benno S.; Käming, Niklas; Tarnowski, Matthias
 Nature Physics, Vol. 15, Issue 9
Approximation by superpositions of a sigmoidal function
journal, December 1989
 Cybenko, G.
 Mathematics of Control, Signals, and Systems, Vol. 2, Issue 4
Probing hidden spin order with interpretable machine learning
journal, February 2019
 Greitemann, Jonas; Liu, Ke; Pollet, Lode
 Physical Review B, Vol. 99, Issue 6
Identifying topological order through unsupervised machine learning
journal, May 2019
 RodriguezNieva, Joaquin F.; Scheurer, Mathias S.
 Nature Physics, Vol. 15, Issue 8
Model for a Quantum Hall Effect without Landau Levels: CondensedMatter Realization of the "Parity Anomaly"
journal, October 1988
 Haldane, F. D. M.
 Physical Review Letters, Vol. 61, Issue 18
Visualizing a neural network that develops quantum perturbation theory
journal, July 2018
 Wu, Yadong; Zhang, Pengfei; Shen, Huitao
 Physical Review A, Vol. 98, Issue 1
Deep Learning the Quantum Phase Transitions in Random TwoDimensional Electron Systems
journal, December 2016
 Ohtsuki, Tomoki; Ohtsuki, Tomi
 Journal of the Physical Society of Japan, Vol. 85, Issue 12
Selflearning Monte Carlo method and cumulative update in fermion systems
journal, June 2017
 Liu, Junwei; Shen, Huitao; Qi, Yang
 Physical Review B, Vol. 95, Issue 24
Quantum Loop Topography for Machine Learning
journal, May 2017
 Zhang, Yi; Kim, EunAh
 Physical Review Letters, Vol. 118, Issue 21
Solving the quantum manybody problem with artificial neural networks
journal, February 2017
 Carleo, Giuseppe; Troyer, Matthias
 Science, Vol. 355, Issue 6325
Machine learning phases of matter
journal, February 2017
 Carrasquilla, Juan; Melko, Roger G.
 Nature Physics, Vol. 13, Issue 5
Neuralnetwork quantum state tomography
journal, February 2018
 Torlai, Giacomo; Mazzola, Guglielmo; Carrasquilla, Juan
 Nature Physics, Vol. 14, Issue 5
Unsupervised learning using topological data augmentation
journal, March 2020
 Balabanov, Oleksandr; Granath, Mats
 Physical Review Research, Vol. 2, Issue 1
Mapping topological order in coordinate space
journal, December 2011
 Bianco, Raffaello; Resta, Raffaele
 Physical Review B, Vol. 84, Issue 24
Proper Definition of Spin Current in SpinOrbit Coupled Systems
journal, February 2006
 Shi, Junren; Zhang, Ping; Xiao, Di
 Physical Review Letters, Vol. 96, Issue 7
Regression Shrinkage and Selection Via the Lasso
journal, January 1996
 Tibshirani, Robert
 Journal of the Royal Statistical Society: Series B (Methodological), Vol. 58, Issue 1