CoChat
Sign Up
Cited by 21Quantum Computing Algorithms and ArchitectureQuantum many-body systemsQuantum Information and Cryptography

The DOI matches its Crossref record; 8 of the 8 references it cites were found in a registry or an index.

Machine learning on quantum experimental data toward solving quantum many-body problems

Open access
Cho, Gyungmin, Kim, Dohun·2024·Nature Communications
DOI 10.1038/s41467-024-51932-3PMC PMC11364692

Published version of a preprint

Open this paper in CoChat — free account

Abstract

Abstract Advancements in the implementation of quantum hardware have enabled the acquisition of data that are intractable for emulation with classical computers. The integration of classical machine learning (ML) algorithms with these data holds potential for unveiling obscure patterns. Although this hybrid approach extends the class of efficiently solvable problems compared to using only classical computers, this approach has been realized for solving restricted problems because of the prevalence of noise in current quantum computers. Here, we extend the applicability of the hybrid approach to problems of interest in many-body physics, such as predicting the properties of the ground state of a given Hamiltonian and classifying quantum phases. By performing experiments with various error-reducing procedures on superconducting quantum hardware with 127 qubits, we managed to acquire refined data from the quantum computer. This enabled us to demonstrate the successful implementation of classical ML algorithms for systems with up to 44 qubits. Our results verify the scalability and effectiveness of the classical ML algorithms for processing quantum experimental data.

Overview

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.

A few free checks a day · papers already verified open instantly · results are public

Integrity signals

Registry

Crossref · 10.1038/s41467-024-51932-3

Cites

8 of 8 references verified0 citation errors · 0 not checkable

Statistics

Figures

Fig. 1 Classical machine learning (ML) on quantum experimental data. a Conversion of the information represented in a quantum state into a classical form for the application of classical ML. The training data were the expectation values measured via the quantum computer or the classical shadows of the quantum state. b Predicting the properties of the ground state of a given Hamiltonian casts to regression in classical ML. The objective of the regression is to approximate the target function f(x) = Tr(Oρ(x)) as closely as possible by trained ML model using training data ({()xi,Tr(Oρ(xi))})i=1^Ndata. After training, when a new input xnew is given, the trained ML model outputs f^(xnew) as a guess for f(xnew). c In classification, the objective is to distinguish a particular quantum phase from others by training a ML model that gives rise to the phase boundary on a relevant space. After training, when a new state is given, the trained ML model assigns a phase of the state by considering its position relative to the phase boundary on the space.
Fig. 1 Classical machine learning (ML) on quantum experimental data. a Conversion of the information represented in a quantum state into a classical form for the application of classical ML. The training data were the expectation values measured via the quantum computer or the classical shadows of the quantum state. b Predicting the properties of the ground state of a given Hamiltonian casts to regression in classical ML. The objective of the regression is to approximate the target function f(x) = Tr(Oρ(x)) as closely as possible by trained ML model using training data ({()xi,Tr(Oρ(xi))})i=1^Ndata. After training, when a new input xnew is given, the trained ML model outputs f^(xnew) as a guess for f(xnew). c In classification, the objective is to distinguish a particular quantum phase from others by training a ML model that gives rise to the phase boundary on a relevant space. After training, when a new state is given, the trained ML model assigns a phase of the state by considering its position relative to the phase boundary on the space.CC BY-NC-ND · PubMed Central
Fig. 2 Predicting ground state properties. a Hamiltonian for random hopping system and Su–Schrieffer–Heeger (SSH) system. xi in Hhop(x) is sampled uniformly from [0, 2]. b Ground state preparation circuit. Gates labeled G and P represent a Givens rotation and a basis rotation for parity measurement, respectively. c Correlation matrix for random hopping system. Results from ML prediction using the Gaussian kernel and exact values from exact diagonalization (ED) are shown on the left and right, respectively. d Prediction error. Model performance is measured by the root-mean-square error (RMSE). The error bars represent the standard deviation of the RMSEs of each observable. The dotted red line means the average RMSE for the training data. Inset shows the relationship between the log(Ndata) and 1/RMSE. The solid lines indicate a linear fit to the data points. e Correlation matrix for 1D SSH system (v = 1, w = 1). Results from ML predictions using Gaussian kernel and exact values from ED are shown on the left and right, respectively. f Prediction error. RMSE* (=√(2/n(n−1)(∑)i≥jⁿ(⟨ai^(†)aj⟩−⟨ai^(†)aj⟩ML)²)) from different kernels while keeping v = 1 and varying w in 0.25 intervals from 0.25 to 1.75. g Edge correlation. The edge correlation 〈a1†an〉 is measured between sites located at either end of a chain. The dotted red line is the known phase boundary between the trivial and topological phases. The black curve indicates exact values obtained by ED. In d, f, g the green, blue, and orange points each indicate the results predicted by the ML model trained by a (modified) Dirichlet kernel, neural tangent kernel, and Gaussian kernel, respectively.
Fig. 2 Predicting ground state properties. a Hamiltonian for random hopping system and Su–Schrieffer–Heeger (SSH) system. xi in Hhop(x) is sampled uniformly from [0, 2]. b Ground state preparation circuit. Gates labeled G and P represent a Givens rotation and a basis rotation for parity measurement, respectively. c Correlation matrix for random hopping system. Results from ML prediction using the Gaussian kernel and exact values from exact diagonalization (ED) are shown on the left and right, respectively. d Prediction error. Model performance is measured by the root-mean-square error (RMSE). The error bars represent the standard deviation of the RMSEs of each observable. The dotted red line means the average RMSE for the training data. Inset shows the relationship between the log(Ndata) and 1/RMSE. The solid lines indicate a linear fit to the data points. e Correlation matrix for 1D SSH system (v = 1, w = 1). Results from ML predictions using Gaussian kernel and exact values from ED are shown on the left and right, respectively. f Prediction error. RMSE* (=√(2/n(n−1)(∑)i≥jⁿ(⟨ai^(†)aj⟩−⟨ai^(†)aj⟩ML)²)) from different kernels while keeping v = 1 and varying w in 0.25 intervals from 0.25 to 1.75. g Edge correlation. The edge correlation 〈a1†an〉 is measured between sites located at either end of a chain. The dotted red line is the known phase boundary between the trivial and topological phases. The black curve indicates exact values obtained by ED. In d, f, g the green, blue, and orange points each indicate the results predicted by the ML model trained by a (modified) Dirichlet kernel, neural tangent kernel, and Gaussian kernel, respectively.CC BY-NC-ND · PubMed Central
Fig. 3 Experimental results for distinguishing between symmetry protected topological (SPT) and trivial phases. a Trivial and SPT phases. When considering a set of gapped symmetric Hamiltonians only, the two phases cannot be connected by a constant depth symmetric local unitary (Path I). However, without consideration of the symmetry, there exists a constant depth local unitary circuit (Path II) connecting them. b Generation of data. The process involves applying two layers of symmetric local random unitary to a fixed-point state associated with a particular phase, followed by randomized measurement for classical shadows. Random gates used can be found in the Supplementary Note 5. c String order parameter (SOP) (Sab = ZaXa+1Xa+3…Xb-3Xb-1Zb) of various lengths measured by direct measurements on the quantum computer. The blue and orange dots correspond to the SPT and trivial phases, respectively. The error bars represent the standard deviation of 300 SOPs. d Classification errors. The chart shows the classification error of the ML model for cases with ℤ2⊗ℤ2, TRS, and without symmetry constraints. The error bars represent the standard deviation from 10 instances. e, f 2D representation of the test data. Each figure corresponds to ℤ2⊗ℤ2 (e) and time-reversal symmetry (TRS) (f). g Classification results. The background represents the expectation values of a SOP measured from the ground states of the HCI, and these values are used to determine the ground truth of the quantum phase. For the 40 randomly sampled test data (marked by circles), each phase predicted by the trained ML model is colored in blue for the SPT phase and orange for the trivial phase. The red dot represents a point where the model incorrectly predicted the phase.
Fig. 3 Experimental results for distinguishing between symmetry protected topological (SPT) and trivial phases. a Trivial and SPT phases. When considering a set of gapped symmetric Hamiltonians only, the two phases cannot be connected by a constant depth symmetric local unitary (Path I). However, without consideration of the symmetry, there exists a constant depth local unitary circuit (Path II) connecting them. b Generation of data. The process involves applying two layers of symmetric local random unitary to a fixed-point state associated with a particular phase, followed by randomized measurement for classical shadows. Random gates used can be found in the Supplementary Note 5. c String order parameter (SOP) (Sab = ZaXa+1Xa+3…Xb-3Xb-1Zb) of various lengths measured by direct measurements on the quantum computer. The blue and orange dots correspond to the SPT and trivial phases, respectively. The error bars represent the standard deviation of 300 SOPs. d Classification errors. The chart shows the classification error of the ML model for cases with ℤ2⊗ℤ2, TRS, and without symmetry constraints. The error bars represent the standard deviation from 10 instances. e, f 2D representation of the test data. Each figure corresponds to ℤ2⊗ℤ2 (e) and time-reversal symmetry (TRS) (f). g Classification results. The background represents the expectation values of a SOP measured from the ground states of the HCI, and these values are used to determine the ground truth of the quantum phase. For the 40 randomly sampled test data (marked by circles), each phase predicted by the trained ML model is colored in blue for the SPT phase and orange for the trivial phase. The red dot represents a point where the model incorrectly predicted the phase.CC BY-NC-ND · PubMed Central
Fig. 4 Experimental results for distinguishing between topologically ordered and trivial phases. a Schematic diagram for measurement-assisted state preparation method on the heavy-hexagonal lattice. The application of a sequence of unitary transformations and adaptive gates depending on the measurement results of some ancilla qubits was followed by randomized measurements for classical shadows. The blue shaded areas represent the X-plaquettes. b Generation of data in the trivial phase. States in this phase are generated by applying local random unitary to the product state at varying circuit depth (dLU). Detailed information about the local random unitary that was used is provided in the Supplementary Note 5. c 1D projection of data. Using kernel principal component analysis with the shadow kernel in Eq. (3), each data point was projected onto the first principal axis. d 1D projection of data without topological order.
Fig. 4 Experimental results for distinguishing between topologically ordered and trivial phases. a Schematic diagram for measurement-assisted state preparation method on the heavy-hexagonal lattice. The application of a sequence of unitary transformations and adaptive gates depending on the measurement results of some ancilla qubits was followed by randomized measurements for classical shadows. The blue shaded areas represent the X-plaquettes. b Generation of data in the trivial phase. States in this phase are generated by applying local random unitary to the product state at varying circuit depth (dLU). Detailed information about the local random unitary that was used is provided in the Supplementary Note 5. c 1D projection of data. Using kernel principal component analysis with the shadow kernel in Eq. (3), each data point was projected onto the first principal axis. d 1D projection of data without topological order.CC BY-NC-ND · PubMed Central
Fig. 5 Extracting a phase classifier from a trained ML model. a Illustrative diagram of ML procedure. In the first step, given a quantum state ρ, we used feature mapping φ(ρ) to reduce the dimensionality of the data. To achieve an error-resilient phase classifier fML(ρ), training data are generated by applying random unitary transformations to fixed-point states. b Classification error. The orange bars represent the results from the ML model trained on raw data without measurement error mitigation (MEM). The blue bars represent the classification error of the ML model trained by the data with MEM and the Renyi-2 Topological entanglement entropy (TEE). The error bar denotes the standard deviation of 100 instances of the test data set.
Fig. 5 Extracting a phase classifier from a trained ML model. a Illustrative diagram of ML procedure. In the first step, given a quantum state ρ, we used feature mapping φ(ρ) to reduce the dimensionality of the data. To achieve an error-resilient phase classifier fML(ρ), training data are generated by applying random unitary transformations to fixed-point states. b Classification error. The orange bars represent the results from the ML model trained on raw data without measurement error mitigation (MEM). The blue bars represent the classification error of the ML model trained by the data with MEM and the Renyi-2 Topological entanglement entropy (TEE). The error bar denotes the standard deviation of 100 instances of the test data set.CC BY-NC-ND · PubMed Central

Verification

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.

A few free checks a day · papers already verified open instantly · results are public

Self-check

Percentages, sums and reported differences re-computed from the paper’s own numbers.

The self-check needs the paper’s full text.

Citations

Cites

Show all 8 references

Research works printed but not linked: 55

  1. Philips, S. G. J. et al. Universal control of a six-qubit quantum processor in silicon. Nature 609, 919–924 (2022). · DOI · Paper page
  2. Moses, S. A. et al. A race-track trapped-ion quantum processor. Phys. Rev. X 13, 041052 (2023).
  3. Bravyi, S., Dial, O., Gambetta, J. M., Gil, D. & Nazario, Z. The future of quantum computing with superconducting qubits. J. Appl. Phys. 132, 160902 (2022). · DOI · Paper page
  4. Abobeih, M. H. et al. Fault-tolerant operation of a logical qubit in a diamond quantum processor. Nature 606, 884–889 (2022). · DOI
  5. Bluvstein, D. et al. A quantum processor based on coherent transport of entangled atom arrays. Nature 604, 451–456 (2022). · DOI · Paper page
  6. Alipanahi, B., Delong, A., Weirauch, M. T. & Frey, B. J. Predicting the sequence specificities of DNA- and RNA-binding proteins by deep learning. Nat. Biotechnol. 33, 831–838 (2015). · DOI · Paper page
  7. Hohenberg, P. & Kohn, W. Inhomogeneous Electron Gas. Phys. Rev. 136, B864–B871 (1964). · DOI · Paper page
  8. White, S. R. Density matrix formulation for quantum renormalization groups. Phys. Rev. Lett. 69, 2863–2866 (1992). · DOI · Paper page
  9. Knill, E., Laflamme, R. & Zurek, W. H. Resilient Quantum Computation. Science 279, 342–345 (1998). · DOI · Paper page
  10. Google Quantum, A. I. et al. Suppressing quantum errors by scaling a surface code logical qubit. Nature 614, 676–681 (2023). · DOI · Paper page
  11. Huang, H.-Y., Kueng, R. & Preskill, J. Predicting many properties of a quantum system from very few measurements. Nat. Phys. 16, 1050–1057 (2020). · DOI · Paper page
  12. Elben, A. et al. The randomized measurement toolbox. Nat. Rev. Phys. 5, 9–24 (2022). · DOI
  13. Zhang, T. et al. Experimental Quantum State Measurement with Classical Shadows. Phys. Rev. Lett. 127, 200501 (2021). · DOI
  14. 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). · DOI
  15. Lewis, L. et al. Improved machine learning algorithm for predicting ground state properties. Nat. Commun. 15, 895 (2024). · DOI
  16. 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). · DOI
  17. Onorati, E., Rouzé, C., França, D. S. & Watson, J. D. Efficient learning of ground & thermal states within phases of matter. Preprint at http://arxiv.org/abs/2301.12946 (2023). · arXiv
  18. Jung, K. et al. Deep learning enhanced individual nuclear-spin detection. npj Quantum Inf. 7, 41 (2021). · DOI
  19. Miles, C. et al. Machine learning discovery of new phases in programmable quantum simulator snapshots. Phys. Rev. Res. 5, 013026 (2023). · DOI
  20. Herrmann, J. et al. Realizing quantum convolutional neural networks on a superconducting quantum processor to recognize quantum phases. Nat. Commun. 13, 4144 (2022). · DOI
  21. Smith, A., Jobst, B., Green, A. G. & Pollmann, F. Crossing a topological phase transition with a quantum computer. Phys. Rev. Res. 4, L022020 (2022). · DOI
  22. Pollmann, F. & Turner, A. M. Detection of symmetry-protected topological phases in one dimension. Phys. Rev. B 86, 125441 (2012). · DOI
  23. Haah, J., Harrow, A. W., Ji, Z., Wu, X. & Yu, N. Sample-optimal tomography of quantum states. IEEE Trans. Inform. Theory 1–1 https://doi.org/10.1109/TIT.2017.2719044 (2017). · DOI
  24. Aaronson, S. Shadow tomography of quantum states. SIAM J. Comput. 49, STOC18-368–STOC18-394 (2020). · DOI
  25. Paini, M., Kalev, A., Padilha, D., & Ruck, B. Estimating expectation values using approximate quantum states. Quantum 5, 413 (2021). · DOI
  26. Bishop, C. M. Pattern recognition and machine learning (Springer, 2006).
  27. Google AI Quantum and Collaborators et al. Hartree-Fock on a superconducting qubit quantum computer. Science 369, 1084–1089 (2020). · DOI
  28. Jiang, Z., Sung, K. J., Kechedzhi, K., Smelyanskiy, V. N. & Boixo, S. Quantum Algorithms to Simulate Many-Body Physics of Correlated Fermions. Phys. Rev. Appl. 9, 044036 (2018). · DOI · Paper page
  29. Kim, Y. et al. Scalable error mitigation for noisy quantum circuits produces competitive expectation values. Nat. Phys. 19, 752–759 (2023). · DOI
  30. Sung, K. J., Rančić, M. J., Lanes, O. T. & Bronn, N. T. Simulating Majorana zero modes on a noisy quantum processor. Quantum Sci. Technol. 8, 025010 (2023). · DOI
  31. Zeng, B. & Wen, X.-G. Gapped quantum liquids and topological order, stochastic local transformations and emergence of unitarity. Phys. Rev. B 91, 125121 (2015). · DOI
  32. Liu, Y.-J., Smith, A., Knap, M. & Pollmann, F. Model-Independent Learning of Quantum Phases of Matter with Quantum Convolutional Neural Networks. Phys. Rev. Lett. 130, 220603 (2023). · DOI
  33. Chen, X., Zeng, B., Wen, X.-G., & Zhou, D.-L. (eds) Quantum Information Meets Quantum Matter: From Quantum Entanglement to Topological Phases of Many-Body Systems (Springer, 2019).
  34. Elben, A. et al. Many-body topological invariants from randomized measurements. Sci. Adv. 6, eaaz3666 (2020). · DOI
  35. Caro, M. C. et al. Generalization in quantum machine learning from few training data. Nat. Commun. 13, 4919 (2022). · DOI · Paper page
  36. Cong, I., Choi, S. & Lukin, M. D. Quantum convolutional neural networks. Nat. Phys. 15, 1273–1278 (2019). · DOI · Paper page
  37. Ferris, A. J. & Vidal, G. Perfect sampling with unitary tensor networks. Phys. Rev. B 85, 165146 (2012). · DOI
  38. Schollwöck, U. The density-matrix renormalization group in the age of matrix product states. Ann. Phys. 326, 96–192 (2011). · DOI · Paper page
  39. Levin, M. & Wen, X.-G. Detecting Topological Order in a Ground State Wave Function. Phys. Rev. Lett. 96, 110405 (2006). · DOI
  40. Kitaev, A. & Preskill, J. Topological Entanglement Entropy. Phys. Rev. Lett. 96, 110404 (2006). · DOI
  41. Satzinger, K. J. et al. Realizing topologically ordered states on a quantum processor. Science 374, 1237–1241 (2021). · DOI
  42. Perez, L. & Wang, J. The Effectiveness of Data Augmentation in Image Classification using Deep Learning. Preprint at http://arxiv.org/abs/1712.04621 (2017). · arXiv
  43. Kim, Y. et al. Evidence for the utility of quantum computing before fault tolerance. Nature 618, 500–505 (2023). · DOI · Paper page
  44. Chen, S., Yu, W., Zeng, P. & Flammia, S. T. Robust Shadow Estimation. PRX Quantum 2, 030348 (2021). · DOI
  45. Koh, D. E. & Grewal, S. Classical Shadows With Noise. Quantum 6, 776 (2022). · DOI
  46. Jnane, H., Steinberg, J., Cai, Z., Nguyen, H. C., & Koczor, B. Quantum error mitigated classical shadows. PRX Quantum 5, 010324 (2024). · DOI
  47. Brieger, R., Heinrich, M., Roth, I. & Kliesch, M. Stability of classical shadows under gate-dependent noise. Preprint at http://arxiv.org/abs/2310.19947 (2023). · arXiv
  48. Zhao, A. & Miyake, A. Group-theoretic error mitigation enabled by classical shadows and symmetries. npj Quantum Inf. 10, 57 (2024). · DOI
  49. Wu, B. & Koh, D. E. Error-mitigated fermionic classical shadows on noisy quantum devices. npj Quantum Inf. 10, 39 (2024). · DOI
  50. Huang, H.-Y., Kueng, R. & Preskill, J. Efficient estimation of Pauli observables by derandomization. Phys. Rev. Lett. 127, 030503 (2021). · DOI
  51. Hu, H.-Y. & You, Y.-Z. Hamiltonian-driven shadow tomography of quantum states. Phys. Rev. Res. 4, 013054 (2022). · DOI
  52. Bu, K., Koh, D. E., Garcia, R. J. & Jaffe, A. Classical shadows with Pauli-invariant unitary ensembles. npj Quantum Inf. 10, 6 (2024). · DOI
  53. Akhtar, A. A., Hu, H.-Y. & You, Y.-Z. Scalable and Flexible Classical Shadow Tomography with Tensor Networks. Quantum 7, 1026 (2023). · DOI
  54. Zhou, Y. & Liu, Q. Performance analysis of multi-shot shadow estimation. Quantum 7, 1044 (2023). · DOI
  55. Zhou, Y. & Liu, Z. A hybrid framework for estimating nonlinear functions of quantum states. npj Quantum Inf. 10, 62 (2024). · DOI

Metadata

Authors
Cho, G., & Kim, D.
Affiliations
Seoul National University (KR)
Corresponding author
  • Cho, Gyungmin
  • Kim, Dohun
ORCID
Venue
Nature Communications
Publisher
Springer Science and Business Media LLC
Published
2024-08-30
Volume
15
Issue
1
Pages
7552-7552
ISSN
2041-1723 (ISSN-L)
DOI
10.1038/s41467-024-51932-3
PMC
PMC11364692
OpenAlex
W4402039629
Cited by
21
References
8
Citations by year
2026: 11 · 2025: 7 · 2024: 3
FWCI
3.10
Type
journal article
Language
en
Open access
Licence
Locations
  • Nature Communications (publishedVersion, cc-by-nc-nd)
  • Research Square (acceptedVersion, cc-by)
  • PubMed Central (publishedVersion, cc-by-nc-nd)
  • PubMed (publishedVersion)
  • arXiv (Cornell University) (submittedVersion, cc-by)
  • RePEc: Research Papers in Economics (submittedVersion)
  • DOAJ (DOAJ: Directory of Open Access Journals) (submittedVersion, cc-by-sa)
  • arXiv (Cornell University) (cc-by)
Funders
  • National Research Foundation of Korea (2019M3E4A1080145, 2019R1A5A1027055, RS-2023-00283291, RS-2023-00207732, 2020M3H3A1110365, 2023R1A2C2005809, 2019M3E4A1080144, RS-2024-00413957)
  • Korea Basic Science Institute (2021R1A6C101B418)
Topics
  • Physical Sciences › Computer Science › Artificial Intelligence › Quantum Computing Algorithms and Architecture
  • Physical Sciences › Physics and Astronomy › Atomic and Molecular Physics, and Optics › Quantum many-body systems
  • Physical Sciences › Physics and Astronomy › Atomic and Molecular Physics, and Optics › Quantum Information and Cryptography
Keywords
hybrid quantum-classical machine learning, quantum many-body problem, superconducting quantum processors, error mitigation, scalability

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

How to cite

Cite CoChat for the checks on this page, and the paper for its own findings.

This page
CoChat. (2026). Citation check: Machine learning on quantum experimental data toward solving quantum many-body problems. https://app.cochat.ai/papers/10.1038/s41467-024-51932-3
The paper
Cho, G., & Kim, D. (2024). Machine learning on quantum experimental data toward solving quantum many-body problems. Nature Communications. https://doi.org/10.1038/s41467-024-51932-3