Solving the quantum many-body problem with artificial neural networks
Open accessAbstract
Machine learning and quantum physics: Elucidating the behavior of quantum interacting systems of many particles remains one of the biggest challenges in physics. Traditional numerical methods often work well, but some of the most interesting problems leave them stumped. Carleo and Troyer harnessed the power of machine learning to develop a variational approach to the quantum many-body problem (see the Perspective by Hush). The method performed at least as well as state-of-the-art approaches, setting a benchmark for a prototypical two-dimensional problem. With further development, it may well prove a valuable piece in the quantum toolbox. Science , this issue p. 602 ; see also p. 580
The claims and statistics of this paper haven’t been checked yet.
Run the check — free, no account needed. It reads the open-access text and checks the sources it cites.
Integrity signals
Registry
Cites
Statistics
Claims the paper makes
The claims and statistics of this paper haven’t been checked yet.
Run the check — free, no account needed. It reads the open-access text and checks the sources it cites.
Claims other papers make about this one
Each is checked against this paper’s own text: does it say what the citing paper says it does?
The matrix S is typically ill-conditioned and needs to be regularized.
“In practice, the matrix S is typically ill-conditioned and needs to be regularized, for which there exist several approaches 47,56,121 .”
From Beyond-classical computation in quantum simulation (2025) · cites this paper as [47]
Appendix A (p. 8) says the covariance matrix S can be non-invertible and describes the authors’ preferred regularization. It does not say S is typically ill-conditioned or that regularization is necessary: it also gives a pseudo-inverse as an alternative. (LLM, conf=0.86)
Experimental — accuracy not yet measured
Self-check
Percentages, sums and reported differences re-computed from the paper’s own numbers.
The self-check needs the paper’s full text.
Cites
LeCun, Y., Bengio, Y., & Hinton, G. (2015) · Nature · doi:10.1038/nature14539 · Cited by 84,126 · see citing works
Equation of State Calculations by Fast Computing Machines
Metropolis, N., Rosenbluth, A. W., Rosenbluth, M. N., Teller, A. H., & Teller, E. (1953) · The Journal of Chemical Physics · doi:10.1063/1.1699114 · Cited by 37,192 · see citing works
Reducing the dimensionality of data with neural networks
Hinton, G. E., & Salakhutdinov, R. R. (2006) · Science · doi:10.1126/science.1127647 · Cited by 20,928 · see citing works
Mastering the game of Go with deep neural networks and tree search
Silver, D., Huang, A., Maddison, C. J., Guez, A., Sifre, L., van den Driessche, G., Schrittwieser, J., Antonoglou, I., Panneershelvam, V., Lanctot, M., et al. (2016) · Nature · doi:10.1038/nature16961 · Cited by 15,612 · see citing works
Messiah, A. (1961) · Cited by 15,553 · see citing works
Density matrix formulation for quantum renormalization groups
White, S. R. (1992) · Physical Review Letters · doi:10.1103/physrevlett.69.2863 · Cited by 7,702 · see citing works
Approximation capabilities of multilayer feedforward networks
Hornik, K. (1991) · Neural Networks · doi:10.1016/0893-6080(91)90009-t · Cited by 6,091 · see citing works
The density-matrix renormalization group in the age of matrix product states
Schollwöck, U. (2011) · Annals of Physics · doi:10.1016/j.aop.2010.09.012 · Cited by 4,818 · see citing works
Note on Exchange Phenomena in the Thomas Atom
Dirac, P. A. M. (1930) · Mathematical Proceedings of the Cambridge Philosophical Society · doi:10.1017/s0305004100016108 · Cited by 3,302 · see citing works
Colloquium: Nonequilibrium dynamics of closed interacting quantum systems
Polkovnikov, A., Sengupta, K., Silva, A., & Vengalattore, M. (2011) · Reviews of Modern Physics · doi:10.1103/revmodphys.83.863 · Cited by 2,785 · see citing works
Quantum Monte Carlo simulations of solids
Foulkes, W. M. C., Mitáš, L., Needs, R. J., & Rajagopal, G. (2001) · Reviews of Modern Physics · doi:10.1103/revmodphys.73.33 · Cited by 2,318 · see citing works
Orús, R. (2014) · Annals of Physics · doi:10.1016/j.aop.2014.06.013 · Cited by 2,255 · see citing works
Efficient Simulation of One-Dimensional Quantum Many-Body Systems
Vidal, G. (2004) · Physical Review Letters · doi:10.1103/physrevlett.93.040502 · Cited by 1,779 · see citing works
Verstraete, F., Murg, V., & Cirac, J. I. (2008) · Advances In Physics · doi:10.1080/14789940801912366 · Cited by 1,710 · see citing works
Equation of state calculations by fast computing machines
Metropolis, N., Rosenbluth, A. W., Rosenbluth, M. N., Teller, A. H., & Teller, E. (1953) · doi:10.2172/4390578 · Cited by 1,701 · see citing works
Machine learning phases of matter
Carrasquilla, J., & Melko, R. G. (2017) · Nature Physics · doi:10.1038/nphys4035 · Cited by 1,448 · see citing works
Real-Time Evolution Using the Density Matrix Renormalization Group
White, S. R., & Feiguin, A. E. (2004) · Physical Review Letters · doi:10.1103/physrevlett.93.076401 · Cited by 1,365 · see citing works
Quantum many-body systems out of equilibrium
Eisert, J., Friesdorf, M., & Gogolin, C. (2015) · Nature Physics · doi:10.1038/nphys3215 · Cited by 1,255 · see citing works
Computational Complexity and Fundamental Limitations to Fermionic Quantum Monte Carlo Simulations
Troyer, M., & Wiese, U. ‐. (2005) · Physical Review Letters · doi:10.1103/physrevlett.94.170201 · Cited by 1,251 · see citing works
Self-citationTime-dependent density-matrix renormalization-group using adaptive effective Hilbert spaces
Daley, A. J., Kollath, C., Schollwöck, U., & Vidal, G. (2004) · Journal of Statistical Mechanics Theory and Experiment · doi:10.1088/1742-5468/2004/04/p04005 · Cited by 1,133 · see citing works
Representational Power of Restricted Boltzmann Machines and Deep Belief Networks
Roux, N. L., & Bengio, Y. (2008) · Neural Computation · doi:10.1162/neco.2008.04-07-510 · Cited by 821 · see citing works
Quantum Monte Carlo methods for nuclear physics
Carlson, J., Gandolfi, S., Pederiva, F., Pieper, S. C., Schiavilla, R., Schmidt, K. E., & Wiringa, R. B. (2015) · Reviews of Modern Physics · doi:10.1103/revmodphys.87.1067 · Cited by 804 · see citing works
McMillan, W. L. (1965) · Physical Review · doi:10.1103/physrev.138.a442 · Cited by 788 · see citing works
The ALPS project release 1.3: Open-source software for strongly correlated systems
Albuquerque, A. F., Alet, F., Corboz, P., Dayal, P., Feiguin, A. E., Fuchs, S., Gamper, L., Gull, E., Gürtler, S., Honecker, A., et al. (2006) · Journal of Magnetism and Magnetic Materials · doi:10.1016/j.jmmm.2006.10.304 · Cited by 729 · see citing works
Self-citationThe ALPS project release 2.0: open source software for strongly correlated systems
Bauer, B., Carr, L. D., Evertz, H. G., Feiguin, A., Freire, J., Fuchs, S., Gamper, L., Gukelberger, J., Gull, E., Guertler, S., et al. (2011) · Journal of Statistical Mechanics Theory and Experiment · doi:10.1088/1742-5468/2011/05/p05001 · Cited by 677 · see citing works
Self-citationDiscovering phase transitions with unsupervised learning
Wang, L. (2016) · Physical Review B · doi:10.1103/physrevb.94.195105 · Cited by 618 · see citing works
Finite-size scaling of the ground-state parameters of the two-dimensional Heisenberg model
Sandvik, A. W. (1997) · Physical review. B, Condensed matter · doi:10.1103/physrevb.56.11678 · Cited by 491 · see citing works
A structural approach to relaxation in glassy liquids
Schoenholz, S. S., Cubuk, E. D., Sussman, D. M., Kaxiras, E., & Liu, A. J. (2016) · Nature Physics · doi:10.1038/nphys3644 · Cited by 464 · see citing works
Rommer, S., & Ostlund, S. S. (1997) · Physical review. B, Condensed matter · doi:10.1103/physrevb.55.2164 · Cited by 422 · see citing works
Ceperley, D. M., & Alder, B. J. (1986) · Science · doi:10.1126/science.231.4738.555 · Cited by 362 · see citing works
The ALPS project release 2.0: open source software for strongly correlated systems
Bauer, B., Carr, L. D., Hehn, A., Igarashi, R., Isakov, S. V., Koop, D., Ma, P. N., Mates, P., Matsuo, H., Parcollet, O., et al. (2011) · RWTH Publications (RWTH Aachen) · Cited by 335 · see citing works
Self-citationLearning thermodynamics with Boltzmann machines
Torlai, G., & Melko, R. G. (2016) · Physical review. B./Physical review. B · doi:10.1103/physrevb.94.165134 · Cited by 252 · see citing works
Sorella, S., Casula, M., & Rocca, D. (2007) · The Journal of Chemical Physics · doi:10.1063/1.2746035 · Cited by 241 · see citing works
Kolmogorov, A. N. (1961) · Translations - American Mathematical Society/Translations · doi:10.1090/trans2/017/12 · Cited by 241 · see citing works
Localization and Glassy Dynamics Of Many-Body Quantum Systems
Carleo, G., Becca, F., Schiró, M., & Fabrizio, M. (2012) · Scientific Reports · doi:10.1038/srep00243 · Cited by 211 · see citing works
Self-citationAlgorithms for finite projected entangled pair states
Lubasch, M., Cirac, J. I., & Bañuls, M.-C. (2014) · Physical Review B · doi:10.1103/physrevb.90.064425 · Cited by 205 · see citing works
Stacks of convolutional Restricted Boltzmann Machines for shift-invariant feature learning
Norouzi, M., Ranjbar, M., & Mori, G. (2009) · IEEE Computer Society Conference on Computer Vision and Pattern Recognition/Proceedings - IEEE Computer Society Conference on Computer Vision and Pattern Recognition/Proceedings · doi:10.1109/cvpr.2009.5206577 · Cited by 186 · see citing works
Light-cone effect and supersonic correlations in one- and two-dimensional bosonic superfluids
Carleo, G., Becca, F., Sanchez-Palencia, L., Sorella, S., & Fabrizio, M. (2014) · Physical Review A · doi:10.1103/physreva.89.031602 · Cited by 144 · see citing works
Self-citationLearning Invariant Representations with Local TransformationsPreprint
Sohn, K., & Lee, H. (2012) · arXiv (Cornell University) · doi:10.48550/arxiv.1206.6418 · Cited by 109 · see citing works
Mezzacapo, F., Schuch, N., Boninsegni, M., & Cirac, J. I. (2009) · New Journal of Physics · doi:10.1088/1367-2630/11/8/083026 · Cited by 105 · see citing works
Matrix product state applications for the ALPS project
Dolfi, M., Bauer, B., Keller, S. F., Kosenkov, A., Ewart, T., Kantian, A., Giamarchi, T., & Troyer, M. (2014) · Computer Physics Communications · doi:10.1016/j.cpc.2014.08.019 · Cited by 98 · see citing works
Self-citationComplete-graph tensor network states: A new fermionic wave function ansatz for moleculesPreprint
Marti, K. H., Bauer, B., Reiher, M., Troyer, M., & Verstraete, F. (2010) · doi:10.3929/ethz-b-000159263 · Cited by 89 · see citing works
Self-citationComplete-graph tensor network states: a new fermionic wave function ansatz for molecules
Marti, K. H., Bauer, B., Reiher, M., Troyer, M., & Verstraete, F. (2010) · New Journal of Physics · doi:10.1088/1367-2630/12/10/103008 · Cited by 87 · see citing works
Self-citationStochastic Gradient Approximation: An Efficient Method to Optimize Many-Body Wave Functions
Harju, A., Barbiellini, B., Siljamaki, S., Nieminen, R. M., & Ortíz, G. (1997) · Physical Review Letters · doi:10.1103/physrevlett.79.1173 · Cited by 75 · see citing works
Montorsi, A. (1992) · WORLD SCIENTIFIC eBooks · doi:10.1142/1346 · Cited by 59 · see citing works
Mezzacapo, F., Schuch, N., Boninsegni, M., & Cirac, J. (2009) · MPG.PuRe (Max Planck Society) · Cited by 54 · see citing works
Karp, I. L. (1959) · Physical Review · doi:10.1103/physrev.115.223 · Cited by 51 · see citing works
Choi, S.-C. T., & Saunders, M. A. (2014) · ACM Transactions on Mathematical Software · doi:10.1145/2527267 · Cited by 51 · see citing works
Stacks of convolutional Restricted Boltzmann Machines for shift-invariant feature learning
Norouzi, M., Ranjbar, M. A., & Mori, G. (2009) · IEEE Computer Society Conference on Computer Vision and Pattern Recognition/Proceedings - IEEE Computer Society Conference on Computer Vision and Pattern Recognition/Proceedings · doi:10.1109/cvprw.2009.5206577 · Cited by 45 · see citing works
Hubbard Model: The A Collection Of Reprints
(1992) · WORLD SCIENTIFIC eBooks · doi:10.1142/9789814360371 · Cited by 11 · see citing works
The nonequilibrium quantum many-body problem as a paradigm for extreme data science
Freericks, J. K., Nikolić, B. K., & Frieder, O. (2014) · International Journal of Modern Physics B · doi:10.1142/s0217979214300217 · Cited by 8 · see citing works
Flagged first, then by citations
2 of the 53 references couldn’t be matched to a known paper.
Research works printed but not linked: 2
- (2016). ArXiv: 1606.00318. · arXiv
- [35]Frenkel, I. Wave Mechanics: Advanced General Theory . No. v. 2 in The International series of monographs on nuclear energy: Reactor design physics (The Clarendon Press, 1934).
Cited by
Sorted by citations
Biamonte, J., Wittek, P., Pancotti, N., Rebentrost, P., Wiebe, N., & Lloyd, S. (2017) · Nature · doi:10.1038/nature23474 · Cited by 4,779 · see citing works
Recent advances and applications of machine learning in solid-state materials science
Schmidt, J., Marques, M. R. G., Botti, S., & Marques, M. A. L. (2019) · npj Computational Materials · doi:10.1038/s41524-019-0221-0 · Cited by 2,655 · see citing works
Machine learning and the physical sciences
Carleo, G., Cirac, J. I., Cranmer, K. S., Daudet, L., Schuld, M., Tishby, N., Vogt-Maranto, L., & Zdeborová, L. (2019) · Reviews of Modern Physics · doi:10.1103/revmodphys.91.045002 · Cited by 2,453 · see citing works
Self-citationScientific discovery in the age of artificial intelligence.
Wang, H., Fu, T., Du, Y., Gao, W., Huang, K., Liu, Z., Chandak, P., Liu, S., Van Katwyk, P., Deac, A., et al. (2023) · Nature · doi:10.1038/s41586-023-06221-2 · Cited by 1,823 · see citing works
Noisy intermediate-scale quantum algorithms
Bharti, K., Cervera-Lierta, A., Kyaw, T. H., Haug, T., Alperin-Lea, S., Anand, A., Degroote, M., Heimonen, H., Kottmann, J. S., Menke, T., et al. (2022) · Reviews of Modern Physics · doi:10.1103/revmodphys.94.015004 · Cited by 1,726 · see citing works
Unke, O. T., Chmiela, S., Sauceda, H. E., Gastegger, M., Poltavsky, I., Schütt, K. T., Tkatchenko, A., & Müller, K.-R. (2021) · Chemical Reviews · doi:10.1021/acs.chemrev.0c01111 · Cited by 1,647 · see citing works
Mitarai, K., Negoro, M., Kitagawa, M., & Fujii, K. (2018) · Physical Review A · doi:10.1103/physreva.98.032309 · Cited by 1,639 · see citing works
Quantum convolutional neural networks
Cong, I., Choi, S., & Lukin, M. D. (2019) · Nature Physics · doi:10.1038/s41567-019-0648-8 · Cited by 1,517 · see citing works
Predicting many properties of a quantum system from very few measurements
Huang, H.-Y., Kueng, R., & Preskill, J. (2020) · Nature Physics · doi:10.1038/s41567-020-0932-7 · Cited by 1,240 · see citing works
A high-bias, low-variance introduction to Machine Learning for physicists
Mehta, P., Bukov, M., Wang, C.-H., Day, A. G. R., Richardson, C., Fisher, C. K., & Schwab, D. J. (2019) · Physics Reports · doi:10.1016/j.physrep.2019.03.001 · Cited by 1,104 · see citing works
Machine learning & artificial intelligence in the quantum domain: a review of recent progress
Dunjko, V., & Briegel, H. J. (2018) · Reports on Progress in Physics · doi:10.1088/1361-6633/aab406 · Cited by 1,132 · see citing works
Sun, L., Gao, H., Pan, S., & Wang, J. (2019) · Computer Methods in Applied Mechanics and Engineering · doi:10.1016/j.cma.2019.112732 · Cited by 1,064 · see citing works
A Unifying Review of Deep and Shallow Anomaly Detection
Ruff, L., Kauffmann, J., Vandermeulen, R. A., Montavon, G., Samek, W., Kloft, M., Dietterich, T. G., & Muller, K.-R. (2021) · Proceedings of the IEEE · doi:10.1109/jproc.2021.3052449 · Cited by 1,080 · see citing works
The Variational Quantum Eigensolver: A review of methods and best practices
Tilly, J., Chen, H., Cao, S., Picozzi, D., Setia, K., Li, Y., Grant, E., Wossnig, L., Rungger, I., Booth, G. H., et al. (2022) · Physics Reports · doi:10.1016/j.physrep.2022.08.003 · Cited by 1,031 · see citing works
Quantum information processing with superconducting circuits: a review
Wendin, G. P. (2017) · Reports on Progress in Physics · doi:10.1088/1361-6633/aa7e1a · Cited by 970 · see citing works
Nanophotonic particle simulation and inverse design using artificial neural networks
Peurifoy, J., Shen, Y., Jing, L., Yang, Y., Cano-Renteria, F., DeLacy, B. G., Joannopoulos, J. D., Tegmark, M., & Marin Soljacic, A. (2018) · Science Advances · doi:10.1126/sciadv.aar4206 · Cited by 977 · see citing works
Integrating Scientific Knowledge with Machine Learning for Engineering and Environmental Systems
Willard, J., Jia, X., Xu, S., Steinbach, M. S., & Kumar, V. (2022) · ACM Computing Surveys · doi:10.1145/3514228 · Cited by 945 · see citing works
Deep-Learning-Enabled On-Demand Design of Chiral Metamaterials
Ma, W., Cheng, F., & Liu, Y. (2018) · ACS Nano · doi:10.1021/acsnano.8b03569 · Cited by 920 · see citing works
Neural-network quantum state tomography
Torlai, G., Mazzola, G., Carrasquilla, J., Troyer, M., Melko, R. G., & Carleo, G. (2018) · Nature Physics · doi:10.1038/s41567-018-0048-5 · Cited by 855 · see citing works
Self-citationFrom DFT to machine learning: recent approaches to materials science–a review
Schleder, G. R., Padilha, A. C. M., Acosta, C. M., Costa, M., & Fazzio, A. (2019) · Journal of Physics: Materials · doi:10.1088/2515-7639/ab084b · Cited by 852 · see citing works
Combining Machine Learning and Computational Chemistry for Predictive Insights Into Chemical Systems
Keith, J. A., Vassilev-Galindo, V., Cheng, B., Chmiela, S., Gastegger, M., Müller, K.-R., & Tkatchenko, A. (2021) · Chemical Reviews · doi:10.1021/acs.chemrev.1c00107 · Cited by 849 · see citing works
Protein–Ligand Scoring with Convolutional Neural Networks
Ragoza, M., Hochuli, J. E., Idrobo, E., Sunseri, J., & Koes, D. R. (2017) · Journal of Chemical Information and Modeling · doi:10.1021/acs.jcim.6b00740 · Cited by 835 · see citing works
Training deep quantum neural networks
Beer, K., Bondarenko, D., Farrelly, T., Osborne, T. J., Salzmann, R., Scheiermann, D., & Wolf, R. (2020) · Nature Communications · doi:10.1038/s41467-020-14454-2 · Cited by 726 · see citing works
A Survey on quantum computing technology
Gyongyosi, L., & Imre, S. (2019) · Computer Science Review · doi:10.1016/j.cosrev.2018.11.002 · Cited by 646 · see citing works
Ma, W., Cheng, F., Xu, Y., Wen, Q., & Liu, Y. (2019) · Advanced Materials · doi:10.1002/adma.201901111 · Cited by 617 · see citing works
Self-verifying variational quantum simulation of lattice models
Kokail, C., Maier, C., van Bijnen, R., Brydges, T., Joshi, M. K., Jurcevic, P., Muschik, C. A., Silvi, P., Blatt, R., Roos, C. F., et al. (2019) · Nature · doi:10.1038/s41586-019-1177-4 · Cited by 594 · see citing works
Deep-learning-enabled self-adaptive microwave cloak without human intervention
Qian, C., Zheng, B., Shen, Y., Jing, L., Li, E., Shen, L., & Chen, H. (2020) · Nature Photonics · doi:10.1038/s41566-020-0604-2 · Cited by 585 · see citing works
Deep-neural-network solution of the electronic Schrödinger equation
Hermann, J., Schätzle, Z., & Noé, F. (2020) · Nature Chemistry · doi:10.1038/s41557-020-0544-y · Cited by 559 · see citing works
Ab initio solution of the many-electron Schrödinger equation with deep neural networks
Pfau, D., Spencer, J. S., Matthews, A. G. D. G., & Foulkes, W. M. C. (2020) · Physical Review Research · doi:10.1103/physrevresearch.2.033429 · Cited by 555 · see citing works
Machine learning in medical applications: a review of state-of-the-art methods
Shehab, M., Abualigah, L., Shambour, Q., Abu-Hashem, M. A., Shambour, M. K. Y., Alsalibi, A. I., & Gandomi, A. H. (2022) · Computers in Biology and Medicine · doi:10.1016/j.compbiomed.2022.105458 · Cited by 542 · see citing works
Why Does Deep and Cheap Learning Work So Well?Preprint
Lin, H. W., Tegmark, M., & Rolnick, D. (2017) · DSpace@MIT (Massachusetts Institute of Technology) · Cited by 538 · see citing works
Colloquium: Nonthermal pathways to ultrafast control in quantum materials
de la Torre, A., Kennes, D. M., Claassen, M., Gerber, S., McIver, J., & Sentef, M. A. (2021) · Reviews of Modern Physics · doi:10.1103/revmodphys.93.041002 · Cited by 492 · see citing works
Wetzel, S. J. (2017) · Physical review. E · doi:10.1103/physreve.96.022140 · Cited by 491 · see citing works
Schütt, K. T., Gastegger, M., Tkatchenko, A., Müller, K. R., & Maurer, R. J. (2019) · Nature Communications · doi:10.1038/s41467-019-12875-2 · Cited by 476 · see citing works
All-optical neural network with nonlinear activation functions
Zuo, Y., Li, B., Zhao, Y., Jiang, Y., Chen, Y.-C., Chen, P., Jo, G.-B., Liu, J., & Du, S. (2019) · Optica · doi:10.1364/optica.6.001132 · Cited by 469 · see citing works
Discovering Physical Concepts with Neural Networks
Iten, R., Metger, T., Wilming, H., del Rio, L., & Renner, R. (2020) · Physical Review Letters · doi:10.1103/physrevlett.124.010508 · Cited by 464 · see citing works
Quantum Entanglement in Neural Network States
Deng, D.-L., Li, X., & Sarma, S. D. (2017) · Physical Review X · doi:10.1103/physrevx.7.021021 · Cited by 433 · see citing works
Machine Learning for Electronically Excited States of Molecules
Westermayr, J., & Marquetand, P. (2020) · Chemical Reviews · doi:10.1021/acs.chemrev.0c00749 · Cited by 430 · see citing works
Quantum Chemistry in the Age of Machine Learning
Dral, P. O. (2020) · The Journal of Physical Chemistry Letters · doi:10.1021/acs.jpclett.9b03664 · Cited by 422 · see citing works
Polino, E., Valeri, M., Spagnolo, N., & Sciarrino, F. (2020) · AVS Quantum Science · doi:10.1116/5.0007577 · Cited by 413 · see citing works
Koch, C. P., Boscain, U., Calarco, T., Dirr, G., Filipp, S., Glaser, S. J., Kosloff, R., Montangero, S., Schulte-Herbrüggen, T., Sugny, D., et al. (2022) · EPJ Quantum Technology · doi:10.1140/epjqt/s40507-022-00138-x · Cited by 404 · see citing works
Why Does Deep and Cheap Learning Work So Well?
Lin, H. W., Tegmark, M., & Rolnick, D. (2017) · Journal of Statistical Physics · doi:10.1007/s10955-017-1836-5 · Cited by 393 · see citing works
Machine learning quantum phases of matter beyond the fermion sign problem
Broecker, P., Carrasquilla, J., Melko, R. G., & Trebst, S. (2017) · Scientific Reports · doi:10.1038/s41598-017-09098-0 · Cited by 391 · see citing works
Exploring chemical compound space with quantum-based machine learning
von Lilienfeld, O. A., Müller, K., & Tkatchenko, A. (2020) · Nature Reviews Chemistry · doi:10.1038/s41570-020-0189-9 · Cited by 376 · see citing works
Deep Fluids: A Generative Network for Parameterized Fluid Simulations
Kim, B., Azevedo, V. C., Thuerey, N., Kim, T., Gross, M., & Solenthaler, B. (2019) · Computer Graphics Forum · doi:10.1111/cgf.13619 · Cited by 372 · see citing works
Predicting disruptive instabilities in controlled fusion plasmas through deep learning
Kates-Harbeck, J., Svyatkovskiy, A., & Tang, W. (2019) · Nature · doi:10.1038/s41586-019-1116-4 · Cited by 367 · see citing works
Deep learning for determining a near-optimal topological design without any iteration
Yu, Y., Hur, T., Jung, J., & Jang, I. G. (2018) · Structural and Multidisciplinary Optimization · doi:10.1007/s00158-018-2101-5 · Cited by 355 · see citing works
Dimiduk, D. M., Holm, E. A., & Niezgoda, S. R. (2018) · Integrating Materials and Manufacturing Innovation · doi:10.1007/s40192-018-0117-8 · Cited by 347 · see citing works
Efficient representation of quantum many-body states with deep neural networks
Gao, X., & Duan, L.-M. (2017) · Nature Communications · doi:10.1038/s41467-017-00705-2 · Cited by 372 · see citing works
Hu, W., Singh, R. R. P., & Scalettar, R. T. (2017) · Physical review. E · doi:10.1103/physreve.95.062122 · Cited by 330 · see citing works
Machine Learning Phases of Strongly Correlated Fermions
Ch’ng, K., Carrasquilla, J., Melko, R. G., & Khatami, E. (2017) · Physical Review X · doi:10.1103/physrevx.7.031038 · Cited by 333 · see citing works
Reinforcement Learning in Different Phases of Quantum Control
Bukov, M., Day, A. G. R., Sels, D., Weinberg, P., Polkovnikov, A., & Mehta, P. (2018) · Physical Review X · doi:10.1103/physrevx.8.031086 · Cited by 348 · see citing works
Solving parametric PDE problems with artificial neural networks
Khoo, Y., LU, J., & Lexing Ying, B. (2020) · European Journal of Applied Mathematics · doi:10.1017/s0956792520000182 · Cited by 330 · see citing works
Reconstructing quantum states with generative models
Carrasquilla, J., Torlai, G., Melko, R. G., & Aolita, L. (2019) · Nature Machine Intelligence · doi:10.1038/s42256-019-0028-1 · Cited by 321 · see citing works
Neural networks for topology optimization
Sosnovik, I., & Oseledets, I. V. (2019) · Russian Journal of Numerical Analysis and Mathematical Modelling · doi:10.1515/rnam-2019-0018 · Cited by 324 · see citing works
Restricted Boltzmann machine learning for solving strongly correlated quantum systems
Nomura, Y., Darmawan, A. S., Yamaji, Y., & Imada, M. (2017) · Physical review. B./Physical review. B · doi:10.1103/physrevb.96.205152 · Cited by 313 · see citing works
Expressive power of parametrized quantum circuits
Du, Y., Hsieh, M.-H., Liu, T., & Tao, D. (2020) · Physical Review Research · doi:10.1103/physrevresearch.2.033125 · Cited by 318 · see citing works
Beck, C., Weinan, E., & Jentzen, A. (2019) · Journal of Nonlinear Science · doi:10.1007/s00332-018-9525-3 · Cited by 318 · see citing works
Integrating Physics-Based Modeling with Machine Learning: A Survey
Willard, J., Jia, X., Xu, S., Steinbach, M. S., & Kumar, V. (2020) · Cited by 305 · see citing works
Deep learning electromagnetic inversion with convolutional neural networks
Puzyrev, V. (2019) · Geophysical Journal International · doi:10.1093/gji/ggz204 · Cited by 301 · see citing works
Active learning machine learns to create new quantum experiments
Melnikov, A. A., Nautrup, H. P., Krenn, M., Dunjko, V., Tiersch, M., Zeilinger, A., & Briegel, H. J. (2018) · Proceedings of the National Academy of Sciences · doi:10.1073/pnas.1714936115 · Cited by 290 · see citing works
Identifying topological order through unsupervised machine learning
Rodriguez-Nieva, J. F., & Scheurer, M. S. (2019) · Nature Physics · doi:10.1038/s41567-019-0512-x · Cited by 285 · see citing works
Artificial Intelligence in Chemistry: Current Trends and Future Directions
Baum, Z. J., Yu, X., Ayala, P. Y., Zhao, Y., Watkins, S. P., & Zhou, Q. A. (2021) · Journal of Chemical Information and Modeling · doi:10.1021/acs.jcim.1c00619 · Cited by 300 · see citing works
Machine learning topological states
Deng, D.-L., Li, X., & Sarma, S. D. (2017) · Physical review. B./Physical review. B · doi:10.1103/physrevb.96.195145 · Cited by 294 · see citing works
Zhu, M., He, T., & Lee, C. (2020) · Applied Physics Reviews · doi:10.1063/5.0016485 · Cited by 289 · see citing works
Quantum Loop Topography for Machine Learning
Zhang, Y., & Kim, E.-A. (2017) · Physical Review Letters · doi:10.1103/physrevlett.118.216401 · Cited by 263 · see citing works
Recurrent neural network wave functions
Hibat-Allah, M., Ganahl, M., Hayward, L. E., Melko, R. G., & Carrasquilla, J. (2020) · Physical Review Research · doi:10.1103/physrevresearch.2.023358 · Cited by 274 · see citing works
Unsupervised Generative Modeling Using Matrix Product States
Han, Z., Wang, J., Fan, H., Wang, L., & Zhang, P. (2018) · Physical Review X · doi:10.1103/physrevx.8.031012 · Cited by 273 · see citing works
Barren plateaus in variational quantum computing
Larocca, M., Thanasilp, S., Wang, S., Sharma, K., Biamonte, J., Coles, P. J., Cincio, Ł., McClean, J. R., Holmes, Z., & Cerezo, M. (2025) · Nature Reviews Physics · doi:10.1038/s42254-025-00813-9 · Cited by 318 · see citing works
Neural-Network Approach to Dissipative Quantum Many-Body Dynamics
Hartmann, M. J., & Carleo, G. (2019) · Physical Review Letters · doi:10.1103/physrevlett.122.250502 · Cited by 263 · see citing works
Self-citationA guidance to intelligent metamaterials and metamaterials intelligence
Qian, C., Kaminer, I. E., & Chen, H. (2025) · Nature Communications · doi:10.1038/s41467-025-56122-3 · Cited by 267 · see citing works
Deep Autoregressive Models for the Efficient Variational Simulation of Many-Body Quantum Systems
Sharir, O., Levine, Y., Wies, N., Carleo, G., & Shashua, A. (2020) · Physical Review Letters · doi:10.1103/physrevlett.124.020503 · Cited by 259 · see citing works
Self-citationInformation-Theoretic Bounds on Quantum Advantage in Machine Learning
Huang, H.-Y., Kueng, R., & Preskill, J. P. (2021) · Physical Review Letters · doi:10.1103/physrevlett.126.190505 · Cited by 258 · see citing works
Fermionic neural-network states for ab-initio electronic structure
Choo, K., Mezzacapo, A., & Carleo, G. (2020) · Nature Communications · doi:10.1038/s41467-020-15724-9 · Cited by 252 · see citing works
Self-citationLi, C. (2019) · Chapman & Hall/CRC eBooks · doi:10.1201/9781351006620-6 · Cited by 251 · see citing works
Exploring Entanglement and Optimization within the Hamiltonian Variational Ansatz
Wiersema, R., Zhou, C., de Sereville, Y., Carrasquilla, J., Kim, Y. B., & Yuen, H. (2020) · PRX Quantum · doi:10.1103/prxquantum.1.020319 · Cited by 248 · see citing works
Neural-Network Quantum States, String-Bond States, and Chiral Topological States
Glasser, I., Pancotti, N., August, M., Rodríguez, I., & Cirac, J. I. (2018) · Max Planck Digital Library · Cited by 250 · see citing works
Symmetries and Many-Body Excitations with Neural-Network Quantum States
Choo, K., Carleo, G., Regnault, N., & Neupert, T. (2018) · Physical Review Letters · doi:10.1103/physrevlett.121.167204 · Cited by 247 · see citing works
Self-citationQuantum Machine Learning in Chemical Compound Space
von Lilienfeld, O. A. (2017) · Angewandte Chemie International Edition · doi:10.1002/anie.201709686 · Cited by 240 · see citing works
Equivalence of restricted Boltzmann machines and tensor network states
Chen, J., Cheng, S., Xie, H., Wang, L., & Xiang, T. (2018) · Physical review. B./Physical review. B · doi:10.1103/physrevb.97.085104 · Cited by 245 · see citing works
Indira, D. N. V. S. L. S., Ganiya, R. K., Babu, P. A., Xavier, A. J., Kavisankar, L., Hemalatha, S., Senthilkumar, V., Kavitha, T., Rajaram, A., Karthik, A., et al. (2022) · BioMed Research International · doi:10.1155/2022/7799812 · Cited by 245 · see citing works
RetractedTwo-dimensional frustrated J1−J2 model studied with neural network quantum states
Choo, K., Neupert, T., & Carleo, G. (2019) · Physical review. B./Physical review. B · doi:10.1103/physrevb.100.125124 · Cited by 238 · see citing works
Self-citationLearning-Based Quantum Error Mitigation
Strikis, A., Qin, D., Chen, Y., Benjamin, S. C., & Li, Y. (2021) · PRX Quantum · doi:10.1103/prxquantum.2.040330 · Cited by 237 · see citing works
Identifying quantum phase transitions using artificial neural networks on experimental data
Rem, B. S., Käming, N., Tarnowski, M., Asteria, L., Fläschner, N., Becker, C., Sengstock, K., & Weitenberg, C. (2019) · Nature Physics · doi:10.1038/s41567-019-0554-0 · Cited by 237 · see citing works
Simulation methods for open quantum many-body systems
Weimer, H., Kshetrimayum, A., & Orús, R. (2021) · Reviews of Modern Physics · doi:10.1103/revmodphys.93.015008 · Cited by 237 · see citing works
Reinforcement Learning with Neural Networks for Quantum Feedback
Fösel, T., Tighineanu, P., Weiss, T., & Marquardt, F. (2018) · Max Planck Digital Library · Cited by 234 · see citing works
Provably efficient machine learning for quantum many-body problems
Huang, H.-Y., Kueng, R., Torlai, G., Albert, V. V., & Preskill, J. P. (2022) · Science · doi:10.1126/science.abk3333 · Cited by 233 · see citing works
Reinforcement Learning with Neural Networks for Quantum Feedback
Fösel, T., Tighineanu, P., Weiss, T., & Marquardt, F. (2018) · Physical Review X · doi:10.1103/physrevx.8.031084 · Cited by 240 · see citing works
Machine learning for quantum matter
Carrasquilla, J. (2020) · Advances in Physics X · doi:10.1080/23746149.2020.1797528 · Cited by 232 · see citing works
Variational Quantum Monte Carlo Method with a Neural-Network Ansatz for Open Quantum Systems
Nagy, A., & Savona, V. (2019) · Physical Review Letters · doi:10.1103/physrevlett.122.250501 · Cited by 227 · see citing works
Machine-Learning Methods for Computational Science and Engineering
Frank, M., Drikakis, D., & Charissis, V. (2020) · Computation · doi:10.3390/computation8010015 · Cited by 225 · see citing works
Quantum Many-Body Dynamics in Two Dimensions with Artificial Neural Networks
Schmitt, M., & Heyl, M. (2020) · Physical Review Letters · doi:10.1103/physrevlett.125.100503 · Cited by 221 · see citing works
Solving Statistical Mechanics Using Variational Autoregressive Networks
Wu, D., Wang, L., & Zhang, P. (2019) · Physical Review Letters · doi:10.1103/physrevlett.122.080602 · Cited by 217 · see citing works
Variational Neural-Network Ansatz for Steady States in Open Quantum Systems
Vicentini, F., Biella, A., Regnault, N., & Ciuti, C. (2019) · Physical Review Letters · doi:10.1103/physrevlett.122.250503 · Cited by 213 · see citing works
Solving many-electron Schrödinger equation using deep neural networks
Han, J., Zhang, L., & Weinan, E. (2019) · Journal of Computational Physics · doi:10.1016/j.jcp.2019.108929 · Cited by 216 · see citing works
Learning to predict the cosmological structure formation
He, S., Li, Y., Feng, Y., Ho, S., Ravanbakhsh, S., Chen, W., & Póczos, B. (2019) · Proceedings of the National Academy of Sciences · doi:10.1073/pnas.1821458116 · Cited by 216 · see citing works
Tackling Photonic Inverse Design with Machine Learning
Liu, Z., Zhu, D., Raju, L., & Cai, W. (2021) · Advanced Science · doi:10.1002/advs.202002923 · Cited by 214 · see citing works
Approximating quantum many-body wave functions using artificial neural networks
Cai, Z., & Liu, J.-G. (2018) · Physical review. B./Physical review. B · doi:10.1103/physrevb.97.035116 · Cited by 215 · see citing works
Neural-Network Quantum States, String-Bond States, and Chiral Topological States
Glasser, I., Pancotti, N., August, M., Rodriguez, I. D., & Cirac, J. I. (2018) · Physical Review X · doi:10.1103/physrevx.8.011006 · Cited by 215 · see citing works
Is Quantum Advantage the Right Goal for Quantum Machine Learning?
Schuld, M., & Killoran, N. (2022) · PRX Quantum · doi:10.1103/prxquantum.3.030101 · Cited by 227 · see citing works
- Authors
- Carleo, G., & Troyer, M.
- Affiliations
- ETH Zurich (CH)
- Microsoft (United States) (US)
- Corresponding author
- Carleo, Giuseppe
- Venue
- Science
- Published
- 2017-02-10
- Volume
- 355
- Issue
- 6325
- Pages
- 602-606
- ISSN
- 0036-8075 (ISSN-L)
- 1095-9203
- OpenAlex
- W2419175238
- Cited by
- 2,215
- References
- 53
- Citations by year
- 2026: 178 · 2025: 297 · 2024: 257 · 2023: 250 · 2022: 289 · 2021: 303 · 2020: 255 · 2019: 208 · 2018: 132 · 2017: 46
- FWCI
- 86.50
- Type
- journal article
- Language
- en
- Open access
- green
- arxiv.org
- Locations
- Science (publishedVersion)
- PubMed (publishedVersion)
- arXiv (Cornell University) (submittedVersion)
- Infoscience (Ecole Polytechnique Fédérale de Lausanne) (submittedVersion, cc-by-nc-nd)
- arXiv (Cornell University)
- Funders
- European Commission Directorate-General for Research and Innovation (award287220)
- Microsoft Research (award287221)
- Lincoln Laboratory (award287222)
- Swiss National Science Foundation (award287223)
- Topics
- Physical Sciences › Physics and Astronomy › Atomic and Molecular Physics, and Optics › Quantum many-body systems
- Physical Sciences › Computer Science › Artificial Intelligence › Quantum Computing Algorithms and Architecture
- Physical Sciences › Physics and Astronomy › Atomic and Molecular Physics, and Optics › Cold Atom Physics and Bose-Einstein Condensates
- Keywords
- quantum many-body problem, artificial neural networks, variational quantum states, reinforcement learning, ground state, unitary evolution
About these checks
Registry, retraction, Cites and statistics results are computed, not judged, with one exception: a printed reference that no index lists is read once by a model before it counts as a citation error. Claims are extracted by a model and kept only when their quote appears verbatim in the text; each cited source is then checked, with a model’s help, for whether it says what the claim relies on. Those verdicts can be wrong — read the claim’s quote and the cited source before relying on one.
Statistics and arithmetic are recomputed, not judged.
Confidence bands are shown only for fields calibrated on our annotated test set. None is calibrated yet, so no band is shown.
What counts as a claim ›How we measure confidence ›How we check this ›
Are you an author, or do you see a problem with this page? Request a review
