Artificial intelligence for representing and characterizing quantum systems - Nature

Abstract

Efficient characterization of large-scale quantum systems, especially those produced by quantum analog simulators and megaquop quantum computers, poses a central challenge in quantum science owing to the exponential scaling of the Hilbert space with respect to system size. Recent advances in artificial intelligence (AI), with its aptitude for high-dimensional pattern recognition and function approximation, have emerged as a powerful tool to address this challenge. A growing body of research has leveraged AI to represent and characterize scalable quantum systems, spanning from theoretical foundations to experimental realizations. Depending on how previous knowledge and learning architectures are incorporated, the integration of AI into quantum system characterization can be categorized into three synergistic paradigms: machine learning, deep learning and language models. This Technical Review discusses how each of these AI paradigms contributes to two core tasks in representing and characterizing quantum systems: quantum property prediction and quantum system reconstruction. These tasks underlie a range of applications, from quantum certification and benchmarking to enhancing quantum algorithms and identifying critical quantum phenomena. We also discuss key challenges and open questions, together with future prospects at the interface of AI and quantum science.

Key points

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Artificial intelligence models can be leveraged to represent and characterize scalable quantum systems in a data-driven manner, enabling quantum property prediction and approximate quantum system reconstruction.

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Provably efficient machine learning models have been designed to characterize linear properties of scalable quantum systems and to classify quantum phases.

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Deep learning models offer powerful tools for predicting a wide range of quantum properties through representation learning, as well as for implicitly reconstructing quantum systems using generative modelling approaches.

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Language models, building on the generative pre-trained transformer architecture, provide a flexible framework for auto-regressively representing large families of quantum states, paving the way towards foundation models for quantum systems and enabling new directions for research and application.

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References

- Van Damme, J. et al. Advanced CMOS manufacturing of superconducting qubits on 300 mm wafers. Nature 634, 74–79 (2024).

- Shaw, A. L. et al. Benchmarking highly entangled states on a 60-atom analogue quantum simulator. Nature 628, 71–77 (2024).

- Manetsch, H. J. et al. tweezer array with 6100 highly coherent atomic qubits. Nature 647, 60 (2025).

- King, A. D. et al. Beyond-classical computation in quantum simulation. Science 388, 199–204 (2025).

- Bourgund, D. et al. Formation of individual stripes in a mixed-dimensional cold-atom Fermi–Hubbard system. Nature 637, 57–62 (2025).

- Xu, S. et al. Non-Abelian braiding of Fibonacci anyons with a superconducting processor. Nat. Phys. 20, 1469–1475 (2024).

- Manovitz, T. et al. Quantum coarsening and collective dynamics on a programmable simulator. Nature 638, 86–92 (2025).

- Andersen, T. I. et al. Thermalization and criticality on an analogue–digital quantum simulator. Nature 638, 79–85 (2025).

- DeCross, M. et al. Computational power of random quantum circuits in arbitrary geometries. Phys. Rev. X 15, 021052 (2025).

- Bluvstein, D. et al. Logical quantum processor based on reconfigurable atom arrays. Nature 626, 58–65 (2024).

- Google Quantum AI and Collaborators. Quantum error correction below the surface code threshold. Nature 638, 920–926 (2025).

- Gao, D. et al. Establishing a new benchmark in quantum computational advantage with 105-qubit Zuchongzhi 3.0 processor. Phys. Rev. Lett. 134, 090601 (2025).

- Preskill, J. Beyond NISQ: the megaquop machine. ACM Trans. Quantum Comput. 6, 18 (2025).

- Eisert, J. & Preskill, J. Mind the gaps: the fraught road to quantum advantage. Preprint at https://doi.org/10.48550/arXiv.2510.19928 (2025).

- Orús, R. Tensor networks for complex quantum systems. Nat. Rev. Phys. 1, 538–550 (2019).

- Aaronson, S. & Gottesman, D. Improved simulation of stabilizer circuits. Phys. Rev. A 70, 052328 (2004).

- Bravyi, S. & Gosset, D. Improved classical simulation of quantum circuits dominated by Clifford gates. Phys. Rev. Lett. 116, 250501 (2016).

- Aharonov, D., Gao, X., Landau, Z., Liu, Y. & Vazirani, U. A polynomial-time classical algorithm for noisy random circuit sampling. In Proc. 55th Annual ACM Symposium on Theory of Computing 945–957 (ACM, 2023).

- Gebhart, V. et al. Learning quantum systems. Nat. Rev. Phys. 5, 141–156 (2023).

- Radford, A. et al. Improving language understanding by generative pre-training. Preprint at https://paperswithcode.com/paper/improving-language-understanding-by (2018).

- Anshu, A. & Arunachalam, S. A survey on the complexity of learning quantum states. Nat. Rev. Phys. 6, 59–69 (2024).

- Huang, H.-Y., Kueng, R., Torlai, G., Albert, V. V. & Preskill, J. Provably efficient machine learning for quantum many-body problems. Science 377, eabk3333 (2022). First to prove that a classical algorithm can predict ground-state properties of a gapped phase with polynomial sample and time complexity.

- Cho, G. & Kim, D. Machine learning on quantum experimental data toward solving quantum many-body problems. Nat. Commun. 15, 7552 (2024).

- Wu, Y.-D., Zhu, Y., Wang, Y. & Chiribella, G. Learning quantum properties from short-range correlations using multi-task networks. Nat. Commun. 15, 8796 (2024). A multi-task learning model for predicting quantum properties, which can be further used for transfer learning and out-of-distribution test.

- Qian, Y., Du, Y., He, Z., Hsieh, M.-H. & Tao, D. Multimodal deep representation learning for quantum cross-platform verification. Phys. Rev. Lett. 133, 130601 (2024).A multimodal deep network for cross-platform quantum verification that jointly processes heterogeneous measurement data from different quantum hardware.

- Du, Y., Hsieh, M.-H. & Tao, D. Efficient learning for linear properties of bounded-gate quantum circuits. Nat. Commun. 16, 3790 (2025). The first machine-learning algorithm with provable sample-complexity guarantees for predicting linear properties of bounded-gate quantum circuits.

- Eisert, J. et al. Quantum certification and benchmarking. Nat. Rev. Phys. 2, 382–390 (2020).

- Alexeev, Y. et al. Artificial intelligence for quantum computing. Nat. Commun. 16, 10829 (2025).

- Cerezo, M. et al. Variational quantum algorithms. Nat. Rev. Phys. 3, 625–644 (2021).

- Dawid, A. et al. Machine learning in quantum sciences (Cambridge Univ. Press, 2025).

- Arunachalam, S. & de Wolf, R. Guest Column: A survey of quantum learning theory. ACM SIGACT News 48, 41–67 (2017).

- Carleo, G. et al. Machine learning and the physical sciences. Rev. Mod. Phys. 91, 045002 (2019).

- Das Sarma, S., Deng, D.-L. & Duan, L.-M. Machine learning meets quantum physics. Phys. Today 72, 48–54 (2019).

- Wetzel, S. J., Ha, S., Iten, R., Klopotek, M. & Liu, Z. Interpretable machine learning in physics: a review. Preprint at https://doi.org/10.48550/arXiv.2503.23616 (2025).

- Acampora, G. et al. Quantum computing and artificial intelligence: status and perspectives. Preprint at https://doi.org/10.48550/arXiv.2505.23860 (2025).

- Krenn, M., Landgraf, J., Foesel, T. & Marquardt, F. Artificial intelligence and machine learning for quantum technologies. Phys. Rev. A 107, 010101 (2023).

- Varela, J. M., de Palhares Jr, A. B. & Duarte, D. H. Entanglement detection and quantification through machine learning: a comprehensive review. Braz. J. Phys. 56, 25 (2026).

- Melko, R. G. & Carrasquilla, J. Language models for quantum simulation. Nat. Comput. Sci. 4, 11–18 (2024).

- Carrasquilla, J. & Torlai, G. How to use neural networks to investigate quantum many-body physics. PRX Quantum 2, 040201 (2021).

- Lange, H., Van de Walle, A., Abedinnia, A. & Bohrdt, A. From architectures to applications: a review of neural quantum states. Quantum Sci. Tech. 9, 040501 (2024).

- Bharti, K. et al. Noisy intermediate-scale quantum algorithms. Rev. Mod. Phys. 94, 015004 (2022).

- Du, Y. et al. A Gentle Introduction to Quantum Machine Learning (Springer, 2025).

- Dunjko, V. & Briegel, H. J. Machine learning & artificial intelligence in the quantum domain: a review of recent progress. Rep. Prog. Phys. 81, 074001 (2018).

- Krenn, M., Malik, M., Fickler, R., Lapkiewicz, R. & Zeilinger, A. Automated search for new quantum experiments. Phys. Rev. Lett. 116, 090405 (2016).

- Melnikov, A. A. et al. Active learning machine learns to create new quantum experiments. Proc. Natl Acad. Sci. USA 115, 1221–1226 (2018).

- Sachdev, S. Quantum phase transitions. Phys. World 12, 33 (1999).

- Nielsen, M. A. & Chuang, I. L. Quantum Computation and Quantum Information (Cambridge Univ. Press, 2010).

- Elben, A. et al. The randomized measurement toolbox. Nat. Rev. Phys. 5, 9–24 (2023).

- Zhu, Y. et al. Flexible learning of quantum states with generative query neural networks. Nat. Commun. 13, 6222 (2022). Uses generative query networks to build implicit, query-driven representations of quantum states for downstream property prediction.

- Kim, H. et al. Attention to quantum complexity. Sci. Adv. 11, eadu0059 (2025).

- Mohri, M., Rostamizadeh, A. & Talwalkar, A. Foundations of Machine Learning (MIT Press, 2018).

- Bishop, C. M. & Nasrabadi, N. M. Pattern Recognition and Machine Learning Vol. 4 (Springer, 2006).

- Zou, H. & Hastie, T. Regularization and variable selection via the elastic net. J. R. Stat. Soc. B 67, 301–320 (2005).

- Huang, H.-Y., Kueng, R. & Preskill, J. Predicting many properties of a quantum system from very few measurements. Nat. Phys. 16, 1050–1057 (2020). Enables prediction of quantum properties from a number of measurements logarithmic in the number of target properties.

- Grafakos, L. Classical Fourier Analysis (Springer, 2008).

- Lewis, L. et al. Improved machine learning algorithm for predicting ground state properties. Nat. Commun. 15, 895 (2024). Reduces the sample complexity for predicting ground state properties of gapped local Hamiltonians from polynomial in system size to logarithmic.

- Wanner, M., Lewis, L., Bhattacharyya, C., Dubhashi, D. & Gheorghiu, A. Predicting ground state properties: constant sample complexity and deep learning algorithms. In Proc. Advances in Neural Information Processing Systems 33962–34024 (Curran Associates, 2024).

- Che, Y., Gneiting, C. & Nori, F. Exponentially improved efficient machine learning for quantum many-body states with provable guarantees. Phys. Rev. Res. 6, 033035 (2024).

- Šmíd, Š & Bondesan, R. Efficient learning of long-range and equivariant quantum systems. Quantum 9, 1597 (2025).

- Šmíd, Š. & Bondesan, R. Accurate learning of equivariant quantum systems from a single ground state. Preprint at https://doi.org/10.48550/arXiv.2405.12309 (2024).

- Schuld, M., Sweke, R. & Meyer, J. J. Effect of data encoding on the expressive power of variational quantum-machine-learning models. Phys. Rev. A 103, 032430 (2021).

- Vidal, J. G. & Theis, D. O. Input redundancy for parameterized quantum circuits. Preprint at https://doi.org/10.48550/arXiv.1901.11434 (2020).

- Schreiber, F. J., Eisert, J. & Meyer, J. J. Classical surrogates for quantum learning models. Phys. Rev. Lett. 131, 100803 (2023). Shows that broad families of parameterized quantum learning models admit classical surrogates with provable approximation guarantees.

- Landman, J., Thabet, S., Dalyac, C., Mhiri, H. & Kashefi, E. Classically approximating variational quantum machine learning with random Fourier features. Preprint at https://doi.org/10.48550/arXiv.2210.13200 (2022).

- Sweke, R. et al. Potential and limitations of random Fourier features for dequantizing quantum machine learning. Quantum 9, 1640 (2025).

- Gan, B. Y., Huang, P.-W., Gil-Fuster, E. & Rebentrost, P. Concept learning of parameterized quantum models from limited measurements. Preprint at https://doi.org/10.48550/arXiv.2408.05116 (2024).

- Liao, W.-Y. et al. Demonstration of efficient predictive surrogates for large-scale quantum processors. Nat. Commun. 17, 4731 (2026).

- Servedio, R. A. & Gortler, S. J. Equivalences and separations between quantum and classical learnability. SIAM J. Comput. 33, 1067–1092 (2004).

- Gyurik, C. & Dunjko, V. On establishing learning separations between classical and quantum machine learning with classical data. Preprint at https://doi.org/10.48550/arXiv.2208.06339 (2022).

- Gyurik, C. & Dunjko, V. Exponential separations between classical and quantum learners. Preprint at https://doi.org/10.48550/arXiv.2306.16028 (2023).

- Molteni, R., Gyurik, C. & Dunjko, V. Exponential quantum advantages in learning quantum observables from classical data. npj Quantum Inf. 12, 19 (2026).

- Bouland, A. et al. Public-key pseudoentanglement and the hardness of learning ground state entanglement structure. Preprint at https://doi.org/10.48550/arXiv.2311.12017 (2023).

- Bouland, A., Zhang, C. & Zhou, Z. On the hardness of learning ground state entanglement of geometrically local Hamiltonians. Preprint at https://doi.org/10.48550/arXiv.2411.04353 (2024).

- Thabet, S., Monbroussou, L., Mamon, E. Z. & Landman, J. When quantum and classical models disagree: learning beyond minimum norm least square. npj Quantum Inf. 12, 81 (2026).

- Gil-Fuster, E., Gyurik, C., Perez-Salinas, A. & Dunjko, V. On the relation between trainability and dequantization of variational quantum learning models. In Proc. The Thirteenth International Conference on Learning Representations (ICLR ’25) (Curran Associates, 2025).

- Sadoune, N., Giudici, G., Liu, K. & Pollet, L. Unsupervised interpretable learning of phases from many-qubit systems. Phys. Rev. Res. 5, 013082 (2023).

- Che, Y., Gneiting, C., Wang, X. & Nori, F. Quantum circuit complexity and unsupervised machine learning of topological order. Nat. Commun. 17, 5179 (2026).

- Dawid, A. Machine Learning in Quantum Sciences (Cambridge Univ. Press, 2025).

- Rocchetto, A. Stabiliser states are efficiently PAC-learnable. Quantum Inf. Comput. 18, 541–552 (2018).

- Grewal, S., Iyer, V., Kretschmer, W. & Liang, D. Efficient learning of quantum states prepared with few non-Clifford gates. Quantum 9, 1907 (2025).

- Leone, L., Oliviero, S. F. & Hamma, A. Learning t-doped stabilizer states. Quantum 8, 1361 (2024).

- Landau, Z. & Liu, Y. Learning quantum states prepared by shallow circuits in polynomial time. In Proc. 57th Annual ACM Symposium on Theory of Computing (STOC ’25) 1828–1838 (ACM, 2025).

- Huang, H.-Y. et al. Learning shallow quantum circuits. In Proc. 56th Annual ACM Symposium on Theory of Computing (STOC ’24) 1343–1351 (ACM, 2024).

- Huang, H.-Y., Chen, S. & Preskill, J. Learning to predict arbitrary quantum processes. PRX Quantum 4, 040337 (2023).

- Chen, S. et al. Predicting quantum channels over general product distributions. In Proc. 38th Conference on Learning Theory (COLT '25) 986–1007 (PMLR, 2025).

- Pouyanfar, S. et al. A survey on deep learning: Algorithms, techniques, and applications. ACM Comput. Surv. 51, 1–36 (2018).

- Goodfellow, I., Bengio, Y. & Courville, A. Deep Learning (MIT Press, 2016).

- Wang, H. et al. TorchQuantum case study for robust quantum circuits. In Proc. 41st IEEE/ACM International Conference on Computer-Aided Design (ICCAD ’22) https://doi.org/10.1145/3508352.3561118 (ACM, 2022).

- Carrasquilla, J., Torlai, G., Melko, R. G. & Aolita, L. Reconstructing quantum states with generative models. Nat. Mach. Intell. 1, 155–161 (2019). Reduces quantum state tomography to the unsupervised learning of outcome statistics of an informationally complete measurement.

- Smith, A. W. R., Gray, J. & Kim, M. S. Efficient quantum state sample tomography with basis-dependent neural networks. PRX Quantum 2, 020348 (2021).

- Mohseni, N., Fösel, T., Guo, L., Navarrete-Benlloch, C. & Marquardt, F. Deep learning of quantum many-body dynamics via random driving. Quantum 6, 714 (2022).

- Mohseni, N., Shi, J., Byrnes, T. & Hartmann, M. J. Deep learning of many-body observables and quantum information scrambling. Quantum 8, 1417 (2024).

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

- Cha, P. et al. Attention-based quantum tomography. Mach. Learn. Sci. Technol. 3, 01LT01 (2021).

- Schmale, T., Reh, M. & Gärttner, M. Efficient quantum state tomography with convolutional neural networks. npj Quantum Inf. 8, 115 (2022).

- Ahmed, S., Muñoz, C. S., Nori, F. & Kockum, A. F. Quantum state tomography with conditional generative adversarial networks. Phys. Rev. Lett. 127, 140502 (2021).

- Ahmed, S., Sánchez Muñoz, C., Nori, F. & Kockum, A. F. Classification and reconstruction of optical quantum states with deep neural networks. Phys. Rev. Res. 3, 033278 (2021).

- Du, Y. et al. ShadowNet for data-centric quantum system learning. Preprint at https://doi.org/10.48550/arXiv.2308.11290 (2023).

- Vaswani, A. et al. Attention is all you need. In Proc. Advances in Neural Information Processing Systems 6000–6010 (Curran Associates, 2017).

- Zhang, X. et al. Direct fidelity estimation of quantum states using machine learning. Phys. Rev. Lett. 127, 130503 (2021).

- Qin, H. et al. Experimental direct quantum fidelity learning via a data-driven approach. Phys. Rev. Lett. 132, 190801 (2024).

- Elben, A. et al. Cross-platform verification of intermediate scale quantum devices. Phys. Rev. Lett. 124, 010504 (2020).

- Wu, Y.-D., Zhu, Y., Bai, G., Wang, Y. & Chiribella, G. Quantum similarity testing with convolutional neural networks. Phys. Rev. Lett. 130, 210601 (2023).

- Vadali, A., Kshirsagar, R., Shyamsundar, P. & Perdue, G. N. Quantum circuit fidelity estimation using machine learning. Quantum Mach. Intell. 6, 1 (2024).

- Ma, Y.-C. & Yung, M.-H. Transforming Bell’s inequalities into state classifiers with machine learning. npj Quantum Inf. 4, 34 (2018).

- Gao, J. et al. Experimental machine learning of quantum states. Phys. Rev. Lett. 120, 240501 (2018).

- Horodecki, R., Horodecki, P., Horodecki, M. & Horodecki, K. Quantum entanglement. Rev. Mod. Phys. 81, 865–942 (2009).

- Harney, C., Pirandola, S., Ferraro, A. & Paternostro, M. Entanglement classification via neural network quantum states. New J. Phys. 22, 045001 (2020).

- Chen, Y., Pan, Y., Zhang, G. & Cheng, S. Detecting quantum entanglement with unsupervised learning. Quantum Sci. Tech. 7, 015005 (2021).

- Chen, C., Ren, C., Lin, H. & Lu, H. Entanglement structure detection via machine learning. Quantum Sci. Tech. 6, 035017 (2021).

- Chen, Z., Lin, X. & Wei, Z. Certifying unknown genuine multipartite entanglement by neural networks. Quantum Sci. Tech. 8, 035029 (2023).

- Huang, Y. et al. Direct entanglement detection of quantum systems using machine learning. npj Quantum Inf. 11, 29 (2025).

- Gao, X. et al. Correlation-pattern-based continuous variable entanglement detection through neural networks. Phys. Rev. Lett. 132, 220202 (2024).

- Koutny`, D. et al. Deep learning of quantum entanglement from incomplete measurements. Sci. Adv. 9, eadd7131 (2023).

- Lin, X., Chen, Z. & Wei, Z. Quantifying quantum entanglement via a hybrid quantum-classical machine learning framework. Phys. Rev. A 107, 062409 (2023).

- Rieger, M., Reh, M. & Gärttner, M. Sample-efficient estimation of entanglement entropy through supervised learning. Phys. Rev. A 109, 012403 (2024).

- Taghadomi, N., Mani, A., Fahim, A., Bakoui, A. & Salami, M. S. Quantum Mach. Intell. 7, 40 (2025).

- Krawczyk, M., Pawłowski, J., Maśka, M. M. & Roszak, K. Data-driven criteria for quantum correlations. Phys. Rev. A 109, 022405 (2024).

- Cimini, V., Barbieri, M., Treps, N., Walschaers, M. & Parigi, V. Neural networks for detecting multimode Wigner negativity. Phys. Rev. Lett. 125, 160504 (2020).

- Mello, A. F., Lami, G. & Collura, M. Retrieving nonstabilizerness with neural networks. Phys. Rev. A 111, 012440 (2025).

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

- Van Nieuwenburg, E. P., Liu, Y.-H. & Huber, S. D. Learning phase transitions by confusion. Nat. Phys. 13, 435 (2017).

- Bohrdt, A. et al. Analyzing nonequilibrium quantum states through snapshots with artificial neural networks. Phys. Rev. Lett. 127, 150504 (2021).

- Miles, C. et al. Machine learning discovery of new phases in programmable quantum simulator snapshots. Phys. Rev. Res. 5, 013026 (2023).

- Zhang, H. et al. Experimental demonstration of adversarial examples in learning topological phases. Nat. Commun. 13, 4993 (2022).

- Jiang, S., Lu, S. & Deng, D.-L. Adversarial machine learning phases of matter. Quantum Front. 2, 15 (2023).

- Skinner, B., Ruhman, J. & Nahum, A. Measurement-induced phase transitions in the dynamics of entanglement. Phys. Rev. X 9, 031009 (2019).

- Li, Y., Chen, X. & Fisher, M. P. A. Measurement-driven entanglement transition in hybrid quantum circuits. Phys. Rev. B 100, 134306 (2019).

- Barratt, F., Agrawal, U., Potter, A. C., Gopalakrishnan, S. & Vasseur, R. Transitions in the learnability of global charges from local measurements. Phys. Rev. Lett. 129, 200602 (2022).

- Ippoliti, M. & Khemani, V. Learnability transitions in monitored quantum dynamics via eavesdropper’s classical shadows. PRX Quantum 5, 020304 (2024).

- Akhtar, A. A., Hu, H.-Y. & You, Y.-Z. Measurement-induced criticality is tomographically optimal. Phys. Rev. B 109, 094209 (2024).

- Agrawal, U., Lopez-Piqueres, J., Vasseur, R., Gopalakrishnan, S. & Potter, A. C. Observing quantum measurement collapse as a learnability phase transition. Phys. Rev. X 14, 041012 (2024).

- Dehghani, H., Lavasani, A., Hafezi, M. & Gullans, M. J. Neural-network decoders for measurement induced phase transitions. Nat. Commun. 14, 2918 (2023).

- Kim, H. et al. Learning measurement-induced phase transitions using attention. Preprint at https://doi.org/10.48550/arXiv.2508.15895 (2025).

- Hou, W. et al. Machine learning the effects of many quantum measurements. Preprint at https://doi.org/10.48550/arXiv.2509.08890 (2025).

- Garratt, S. J. & Altman, E. Probing postmeasurement entanglement without postselection. PRX Quantum 5, 030311 (2024).

- Aaronson, S. Shadow tomography of quantum states. In Proc. 50th Annual ACM SIGACT Symposium on Theory of Computing (STOC ’18) 325–338 (ACM, 2018).

- Bengio, Y., Courville, A. & Vincent, P. Representation learning: a review and new perspectives. IEEE Trans. Pattern Anal. Mach. Intell. 35, 1798–1828 (2013).

- Xiao, T., Huang, J., Li, H., Fan, J. & Zeng, G. Intelligent certification for quantum simulators via machine learning. npj Quantum Inf. 8, 138 (2022).

- Mohseni, N., Marquardt, F. & Schmidt, P. Transfer learning in predicting quantum many-body dynamics: from physical observables to entanglement entropy. Quantum Sci. Tech. 10, 025038 (2025).

- Van Engelen, J. E. & Hoos, H. H. A survey on semi-supervised learning. Mach. Learn. 109, 373–440 (2020).

- Tang, Y., Yang, N., Long, M. & Yan, J. Ssl4q: semi-supervised learning of quantum data with application to quantum state classification. In Proc. Forty-First International Conference on Machine Learning (ICML ’24) (PMLR, 2024).

- Wu, D., Wang, L. & Zhang, P. Solving statistical mechanics using variational autoregressive networks. Phys. Rev. Lett. 122, 080602 (2019).

- Sharir, O., Levine, Y., Wies, N., Carleo, G. & Shashua, A. Deep autoregressive models for the efficient variational simulation of many-body quantum systems. Phys. Rev. Lett. 124, 020503 (2020).

- Zhong, L., Guo, C. & Wang, X. Quantum state tomography inspired by language modeling. Preprint at https://doi.org/10.48550/arXiv.2212.04940 (2022).

- Zhang, Z. & You, Y.-Z. Observing Schrödinger’s cat with artificial intelligence: emergent classicality from information bottleneck. Mach. Learn. Sci. Technol. 5, 015051 (2024).

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

- Czischek, S., Moss, M. S., Radzihovsky, M., Merali, E. & Melko, R. G. Data-enhanced variational Monte Carlo simulations for Rydberg atom arrays. Phys. Rev. B 105, 205108 (2022).

- Moss, M. S. et al. Enhancing variational Monte Carlo simulations using a programmable quantum simulator. Phys. Rev. A 109, 032410 (2024).

- Lange, H. et al. Transformer neural networks and quantum simulators: a hybrid approach for simulating strongly correlated systems. Quantum 9, 1675 (2025).

- Torlai, G. et al. Quantum process tomography with unsupervised learning and tensor networks. Nat. Commun. 14, 2858 (2023).

- Ahmed, S., Quijandría, F. & Kockum, A. F. Gradient-descent quantum process tomography by learning Kraus operators. Phys. Rev. Lett. 130, 150402 (2023).

- Zhu, Y., Wu, Y.-D., Liu, Q., Wang, Y. & Chiribella, G. Predictive modelling of quantum process with neural networks. Preprint at https://doi.org/10.48550/arXiv.2308.08815 (2023).

- Hartmann, M. J. & Carleo, G. Neural-network approach to dissipative quantum many-body dynamics. Phys. Rev. Lett. 122, 250502 (2019).

- Carrasquilla, J. Machine learning for quantum matter. Adv. Phys. X 5, 1797528 (2020).

- Harrington, P. M., Mueller, E. J. & Murch, K. W. Engineered dissipation for quantum information science. Nat. Rev. Phys. 4, 660–671 (2022).

- Dugan, O. M., Lu, P. Y., Dangovski, R., Luo, D. & Soljacic, M. Q-Flow: generative modeling for differential equations of open quantum dynamics with normalizing flows. In Proc. 40th International Conference on Machine Learning 8879–8901 (PMLR, 2023).

- Campaioli, F., Cole, J. H. & Hapuarachchi, H. Quantum master equations: tips and tricks for quantum optics, quantum computing, and beyond. PRX Quantum 5, 020202 (2024).

- Valenti, A., van Nieuwenburg, E., Huber, S. & Greplova, E. Hamiltonian learning for quantum error correction. Phys. Rev. Res. 1, 033092 (2019).

- Valenti, A., Jin, G., Léonard, J., Huber, S. D. & Greplova, E. Scalable Hamiltonian learning for large-scale out-of-equilibrium quantum dynamics. Phys. Rev. A 105, 023302 (2022).

- Nandy, S., Schmitt, M., Bukov, M. & Lenarčič, Z. Reconstructing effective Hamiltonians from nonequilibrium thermal and prethermal steady states. Phys. Rev. Res. 6, 023160 (2024).

- Che, L. et al. Learning quantum Hamiltonians from single-qubit measurements. Phys. Rev. Res. 3, 023246 (2021).

- Proctor, T., Young, K., Baczewski, A. D. & Blume-Kohout, R. Benchmarking quantum computers. Nat. Rev. Phys. 7, 1–14 (2025).

- Hothem, D., Miller, A. & Proctor, T. What is my quantum computer good for? Quantum capability learning with physics-aware neural networks. In Proc. The Thirty-Eighth Annual Conference on Neural Information Processing Systems (Curran Associates, 2024).

- Hartnett, G. S. et al. Learning to rank quantum circuits for hardware-optimized performance enhancement. Quantum 8, 1542 (2024).

- Cai, Z. et al. Quantum error mitigation. Rev. Mod. Phys. 95, 045005 (2023).

- Kim, C., Park, K. D. & Rhee, J.-K. Quantum error mitigation with artificial neural network. IEEE Access 8, 188853–188860 (2020).

- Sack, S. H. & Egger, D. J. Large-scale quantum approximate optimization on nonplanar graphs with machine learning noise mitigation. Phys. Rev. Res. 6, 013223 (2024).

- Bao, T. et al. Beyond circuit connections: a non-message passing graph transformer approach for quantum error mitigation. In Proc. The Thirteenth International Conference on Learning Representations (ICLR ’25) (OpenReview, 2025).

- Liao, H. et al. Machine learning for practical quantum error mitigation. Nat. Mach. Intell. 6, 1478–1486 (2024).

- Liao, M., Zhu, Y., Chiribella, G. & Yang, Y. Noise-agnostic quantum error mitigation with data augmented neural models. npj Quantum Inf. 11, 8 (2025).

- Sweke, R., Kesselring, M. S., van Nieuwenburg, E. P. L. & Eisert, J. Reinforcement learning decoders for fault-tolerant quantum computation. Mach. Learn. Sci. Technol. 2, 025005 (2018).

- Zhou, Y. et al. Learning to decode logical circuits. Nat. Comput. Sci. 5, 1158–1167 (2025).

- Wang, H. et al. Transformer-QEC: quantum error correction code decoding with transferable transformers. Preprint at https://doi.org/10.48550/arXiv.2311.16082 (2023).

- Bausch, J. et al. Learning high-accuracy error decoding for quantum processors. Nature 635, 834–840 (2024).

- Cao, H., Pan, F., Feng, D., Wang, Y. & Zhang, P. Generative decoding for quantum error-correcting codes. Preprint at https://doi.org/10.48550/arXiv.2503.21374 (2025).

- Olle, J., Zen, R., Puviani, M. & Marquardt, F. Simultaneous discovery of quantum error correction codes and encoders with a noise-aware reinforcement learning agent. npj Quantum Inf. 10, 126 (2024).

- Zeng, Y., Qin, W., Chen, Y.-H., Gneiting, C. & Nori, F. Neural-network-based design of approximate Gottesman–Kitaev–Preskill code. Phys. Rev. Lett. 134, 060601 (2025).

- Cervera-Lierta, A., Kottmann, J. S. & Aspuru-Guzik, A. Meta-variational quantum eigensolver: learning energy profiles of parameterized Hamiltonians for quantum simulation. PRX Quantum 2, 020329 (2021).

- Jain, N., Coyle, B., Kashefi, E. & Kumar, N. Graph neural network initialisation of quantum approximate optimisation. Quantum 6, 861 (2022).

- Luo, D., Shen, J., Dangovski, R. & Soljacic, M. Quack: accelerating gradient-based quantum optimization with Koopman operator learning. In Proc. Advances in Neural Information Processing Systems 25662–25692 (Curran Associates, 2023).

- Lee, J., Cho, J. & Kim, S. Q-MAML: quantum model-agnostic meta-learning for variational quantum algorithms. In Proc. AAAI Conference on Artificial Intelligence Vol. 39, 18137–18144 (AAAI, 2025).

- Zhang, S.-X., Hsieh, C.-Y., Zhang, S. & Yao, H. Neural predictor based quantum architecture search. Mach. Learn. Sci. Technol. 2, 045027 (2021).

- He, Z. et al. A GNN-based predictor for quantum architecture search. Quantum Inf. Process. 22, 128 (2023).

- Qian, Y., Wang, X., Du, Y., Luo, Y. & Tao, D. MG-Net: learn to customize QAOA with circuit depth awareness. In Proc. Advances in Neural Information Processing Systems 33691–33725 (Curran Associates, 2024).

- Fösel, T., Niu, M. Y., Marquardt, F. & Li, L. Quantum circuit optimization with deep reinforcement learning. Preprint at https://doi.org/10.48550/arXiv.2103.07585 (2021).

- Fürrutter, F., Muñoz-Gil, G. & Briegel, H. J. Quantum circuit synthesis with diffusion models. Nat. Mach. Intell. 6, 515–524 (2024).

- Ruiz, F. J. et al. Quantum circuit optimization with AlphaTensor. Nat. Mach. Intell. 7, 374–385 (2025).

- Arnold, J. & Schäfer, F. Replacing neural networks by optimal analytical predictors for the detection of phase transitions. Phys. Rev. X 12, 031044 (2022).

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

- Sharir, O., Shashua, A. & Carleo, G. Neural tensor contractions and the expressive power of deep neural quantum states. Phys. Rev. B 106, 205136 (2022).

- Zhao, L., Guo, N., Luo, M.-X. & Rebentrost, P. Provable learning of quantum states with graphical models. Phys. Rev. Res. 8, 013109 (2026).

- Yang, T.-H., Soleimanifar, M., Bergamaschi, T. & Preskill, J. When can classical neural networks represent quantum states? Preprint at https://doi.org/10.48550/arXiv.2410.23152 (2024).

- Iten, R., Metger, T., Wilming, H., del Rio, L. & Renner, R. Discovering physical concepts with neural networks. Phys. Rev. Lett. 124, 010508 (2020).

- Miles, C. et al. Correlator convolutional neural networks as an interpretable architecture for image-like quantum matter data. Nat. Commun. 12, 3905 (2021).

- Flam-Shepherd, D. et al. Learning interpretable representations of entanglement in quantum optics experiments using deep generative models. Nat. Mach. Intell. 4, 544–554 (2022).

- Frohnert, F. & van Nieuwenburg, E. Explainable representation learning of small quantum states. Mach. Learn. Sci. Technol. 5, 015001 (2024).

- Cybiński, K. et al. Characterizing out-of-distribution generalization of neural networks: application to the disordered Su–Schrieffer–Heeger model. Mach. Learn. Sci. Technol. 6, 015014 (2025).

- de Schoulepnikoff, P., Muñoz-Gil, G., Nautrup, H. P. & Briegel, H. J. Interpretable representation learning of quantum data enabled by probabilistic variational autoencoders. Preprint at https://doi.org/10.48550/arXiv.2506.11982 (2025).

- Cao, Y. et al. A comprehensive survey of AI-generated content (AIGC): a history of generative AI from GAN to ChatGPT. Preprint at https://doi.org/10.48550/arXiv.2303.04226 (2023).

- Chang, Y. et al. A survey on evaluation of large language models. ACM Trans. Intell. Syst. Technol. 15, 1–45 (2024).

- Brown, T. et al. Language models are few-shot learners. In Proc. Advances in Neural Information Processing Systems 1877–1901 (Curran Associates, 2020).

- Kaplan, J. et al. Scaling laws for neural language models. Preprint at https://doi.org/10.48550/arXiv.2001.08361 (2020).

- Wang, H., Weber, M., Izaac, J. & Lin, C. Y.-Y. Predicting properties of quantum systems with conditional generative models. Preprint at https://doi.org/10.48550/arXiv.2211.16943 (2022). Trains a conditional generative transformer on data from a family of quantum states, enabling property prediction for both unseen states and new observables without retraining.

- Zhang, Y.-H. & Di Ventra, M. Transformer quantum state: a multipurpose model for quantum many-body problems. Phys. Rev. B 107, 075147 (2023). A single transformer-based neural quantum state that covers entire phase diagrams within one model and transfers to new Hamiltonians via fine-tuning.

- An, Z., Wu, J., Yang, M., Zhou, D. L. & Zeng, B. Unified quantum state tomography and Hamiltonian learning: a language-translation-like approach for quantum systems. Phys. Rev. Appl. 21, 014037 (2024).

- Yao, J. & You, Y.-Z. ShadowGPT: learning to solve quantum many-body problems from randomized measurements. Preprint at https://doi.org/10.48550/arXiv.2411.03285 (2024).

- Fitzek, D. et al. RydbergGPT. Mach. Learn. Sci. Technol. 6, 045057 (2025).

- Tang, Y., Xiong, H., Yang, N., Xiao, T. & Yan, J. Towards LLM4QPE: unsupervised pretraining of quantum property estimation and a benchmark. In Proc. The Twelfth International Conference on Learning Representations (ICLR ’24) (OpenReview, 2024).

- Chen, A. & Heyl, M. Empowering deep neural quantum states through efficient optimization. Nat. Phys. 20, 1476–1481 (2024).

- Rende, R. et al. Foundation neural-networks quantum states as a unified ansatz for multiple Hamiltonians. Nat. Commun. 16, 7213 (2025). The first foundation neural-network quantum state: taking spin configurations and Hamiltonian couplings as multimodal inputs, and generalizing to Hamiltonians beyond training.

- Ho, J., Jain, A. & Abbeel, P. Denoising diffusion probabilistic models. In Proc. Advances in Neural Information Processing Systems 6840–6851 (Curran Associates, 2020).

- Song, Y. et al. Score-based generative modeling through stochastic differential equations. In Proc. 9th International Conference on Learning Representations (ICLR ’21) (OpenReview, 2021).

- Yang, L. et al. Diffusion models: a comprehensive survey of methods and applications. ACM Comput. Surv. 56, 1–39 (2023).

- Tang, Y., Long, M. & Yan, J. QuaDiM: a conditional diffusion model for quantum state property estimation. In Proc. The Thirteenth International Conference on Learning Representations (OpenReview, 2025).

- Dupuis, N. et al. Qiskit code assistant: training LLMs for generating quantum computing code. In Proc. 2024 IEEE LLM Aided Design Workshop (LAD) 1–4 (IEEE, 2024).

- Yang, R., Wang, Z., Gu, Y., Liang, Y. & Li, T. QCircuitBench: a large-scale dataset for benchmarking quantum algorithm design. In Proc. The Thirty-Ninth Annual Conference on Neural Information Processing Systems Datasets and Benchmarks Track (OpenReview, 2026).

- Campbell, C., Chen, H. M., Luk, W. & Fan, H. Enhancing LLM-based quantum code generation with multi-agent optimization and quantum error correction. Preprint at https://doi.org/10.48550/arXiv.2504.14557 (2025).

- Liang, Z. et al. Unleashing the potential of LLMs for quantum computing: a study in quantum architecture design. Preprint at https://doi.org/10.48550/arXiv.2307.08191 (2023).

- Nakaji, K. et al. The generative quantum eigensolver (GQE) and its application for ground state search. Preprint at https://doi.org/10.48550/arXiv.2401.09253 (2024).

- Minami, S., Nakaji, K., Suzuki, Y., Aspuru-Guzik, A. & Kadowaki, T. Generative quantum combinatorial optimization by means of a novel conditional generative quantum eigensolver. Digit. Discov. 4, 2229–2243 (2026).

- Zhao, Y., Zhang, C. & Du, Y. Rethink the role of deep learning towards large-scale quantum systems. In Proc. Forty-Second International Conference on Machine Learning (ICML ’25) (OpenReview, 2025).

- Wang, H. et al. Scientific discovery in the age of artificial intelligence. Nature 620, 47–60 (2023).

- Romera-Paredes, B. et al. Mathematical discoveries from program search with large language models. Nature 625, 468–475 (2024).

- Boiko, D. A., MacKnight, R., Kline, B. & Gomes, G. Autonomous chemical research with large language models. Nature 624, 570–578 (2023).

- Ataides, J., Gu, A., Yelin, S. F. & Lukin, M. D. Neural decoders for universal quantum algorithms. Preprint at https://doi.org/10.48550/arXiv.2509.11370 (2025).

- Ge, Y. et al. Quantum circuit synthesis and compilation optimization: overview and prospects. Preprint at https://doi.org/10.48550/arXiv.2407.00736 (2024).

- Wang, L. et al. A survey on large language model based autonomous agents. Front. Comput. Sci. 18, 186345 (2024).

- Hafner, D., Pasukonis, J., Ba, J. & Lillicrap, T. Mastering diverse control tasks through world models. Nature 640, 647–653 (2025).

- Baltrušaitis, T., Ahuja, C. & Morency, L.-P. Multimodal machine learning: a survey and taxonomy. IEEE Trans. Pattern Anal. Mach. Intell. 41, 423–443 (2018).

- Han, Z., Gao, C., Liu, J., Zhang, J. & Zhang, S. Q. Parameter-efficient fine-tuning for large models: a comprehensive survey. Preprint at https://doi.org/10.48550/arXiv.2403.14608 (2024).

- Bertoni, C. et al. Shallow shadows: expectation estimation using low-depth random Clifford circuits. Phys. Rev. Lett. 133, 020602 (2024).

- Schuster, T., Haferkamp, J. & Huang, H.-Y. Random unitaries in extremely low depth. Science 389, 92–96 (2025).

- Hu, H.-Y. et al. Demonstration of robust and efficient quantum property learning with shallow shadows. Nat. Commun. 16, 2943 (2025).

- Schölkopf, B. & Smola, A. J. Learning with Kernels: Support Vector Machines, Regularization,Optimization, and Beyond (MIT Press, 2001).

Acknowledgements

Y.D. acknowledges the funding from A*STAR (H25-MRO3488) and NTU SUG (025257-00001). Y.-D.W. acknowledges funding from the National Natural Science Foundation of China through grant no. 12405022. J.E. acknowledges funding from the German BMFTR, Berlin Quantum, the Munich Quantum Valley, the Quantum Flagship (Millenion and PasQuans2), and the European Research Council. G.C. acknowledges support from the Hong Kong Research Grant Council through grant nos. SRFS2021-7S02, R7035-21F and 17310725. D.T. acknowledges the funding from NRF-P2024-001. Research at the Perimeter Institute is supported 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 Research, Innovation and Science.

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Contributions

Y.D. and Y.-D.W. initiated the writing of this Technical Review and completed its first draft. M.-H.H., P.R. and W.G. contributed to the editing of the ‘Machine learning paradigm’ section, Y.Z. and Y.-Z.Y. contributed to the ‘Deep learning paradigm’ section, and Y.-H.Z. contributed to the ‘Language model paradigm’ section. J.E., G.C., D.T. and B.C.S. contributed to the overall editing of this Technical Review.

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Nature Reviews Physics thanks Hans J. Briegel and Uman Khalid for their contribution to the peer review of this work.

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Glossary

- Attention mechanisms

-

Neural-network operations that assign different weights to different parts of the input when constructing a representation, allowing the model to focus on the most relevant information for a given task; they are a key ingredient of transformer architectures and enables the efficient modelling of long-range dependencies and contextual relationships in complex data.

- Clifford circuits

-

Quantum circuits composed entirely of Clifford gates, that is, gates whose conjugation maps Pauli operators to Pauli operators; such circuits can be efficiently simulated classically using the Gottesman–Knill theorem.

- Convolutional neural network

-

(CNN). A deep learning architecture that uses convolutional layers to learn spatial and hierarchical features from grid-structured data, such as images.

- Fully connected neural network

-

(FCNN). A type of artificial neural network wherein each neuron in one layer is connected to every neuron in the adjacent layer.

- Graph neural networks

-

(GNNs). A class of neural networks designed to operate directly on graph-structured data by iteratively aggregating information from the neighbours of a node to learn representations that capture both node features and topological structure.

- Kernel machine

-

A class of machine learning models that performs prediction through a kernel function, which quantifies the similarity between data points in a feature space; by replacing explicit feature construction with kernel evaluation, it enables nonlinear learning while retaining a simple linear structure in the induced space; common examples include the Gaussian kernel, polynomial kernel and Dirichlet kernel, each suited to different structural properties of the target function.

- Kraus operators

-

Operators that provide a general mathematical framework for describing quantum channels, by representing any completely positive trace-preserving map as a sum of operators acting on a density matrix in the form \({\sum }_{i}{K}_{i}\rho {K}_{i}^{\dagger }\) with \({\sum }_{i}{K}_{i}^{\dagger }{K}_{i}={\mathbb{I}}\).

- Language models

-

A class of generative models, typically based on transformer architectures, that learn the statistical structure of sequences by modelling token probabilities conditioned on context; although originally developed for natural language, language models can also be applied to quantum systems.

- Long short-term memory

-

(LSTM). A specialized type of RNN that addresses the vanishing gradient problem through a gated cell structure, enabling it to learn long-range dependencies and remember information over extended sequences.

- Megaquop quantum computer

-

An error-corrected quantum computer capable of executing on the order of 1 million coherent quantum operations.

- Non-Clifford gates

-

Quantum gates that cannot be generated by Clifford circuits alone and are necessary for achieving universal quantum computing; examples are T gate and Toffoli gate.

- Positive operator-valued measure

-

(POVM). A set of positive semidefinite operators that sum to the identity, providing the most general mathematical description of a quantum measurement.

- Recurrent neural networks

-

(RNNs). A class of neural networks with recurrent connections and hidden states, designed to process sequential or time-dependent data.

- Su–Schrieffer–Heeger system

-

A 1D tight-binding model of a chain with alternating hopping amplitudes, which serves as a prototypical example of a topological insulator.

- Transformers

-

Neural-network architectures built on self-attention mechanisms, designed to model long-range dependencies in sequential or structured data; they serve as a foundation for modern large language models and have become a powerful tool for representation learning and generative modelling.

- Transverse-field Ising model

-

A model that describes a spin system with Ising interactions along one spin direction and an external field applied along a transverse direction; it is a paradigmatic model for studying quantum phase transitions, criticality and non-equilibrium quantum dynamics.

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Du, Y., Zhu, Y., Zhang, YH. et al. Artificial intelligence for representing and characterizing quantum systems. Nat Rev Phys (2026). https://doi.org/10.1038/s42254-026-00962-5

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