Efficient Learning of Long-Range and Equivariant Quantum Systems

Štěpán Šmíd; Roberto Bondesan

doi:10.22331/q-2025-01-15-1597

Efficient Learning of Long-Range and Equivariant Quantum Systems

Štěpán Šmíd and Roberto Bondesan

Department of Computing, Imperial College London, London SW7 2AZ, United Kingdom

Published:	2025-01-15, volume 9, page 1597
Editor:	Álvaro Alhambra
Eprint:	arXiv:2312.17019v3
Doi:	https://doi.org/10.22331/q-2025-01-15-1597
Citation:	Quantum 9, 1597 (2025).

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Abstract

In this work, we consider a fundamental task in quantum many-body physics – finding and learning ground states of quantum Hamiltonians and their properties. Recent works have studied the task of predicting the ground state expectation value of sums of geometrically local observables by learning from data. For short-range gapped Hamiltonians, a sample complexity that is logarithmic in the number of qubits and quasipolynomial in the error was obtained. Here we extend these results beyond the local requirements on both Hamiltonians and observables, motivated by the relevance of long-range interactions in molecular and atomic systems. For interactions decaying as a power law with exponent greater than twice the dimension of the system, we recover the same efficient logarithmic scaling with respect to the number of qubits, but the dependence on the error worsens to exponential. Further, we show that learning algorithms equivariant under the automorphism group of the interaction hypergraph achieve a sample complexity reduction, leading in particular to a constant number of samples for learning sums of local observables in systems with periodic boundary conditions. We demonstrate the efficient scaling in practice by learning from DMRG simulations of $1$D long-range and disordered systems with up to $128$ qubits. Finally, we provide an analysis of the concentration of expectation values of global observables stemming from the central limit theorem, resulting in increased prediction accuracy.

Featured image: Overview of the efficient machine learning algorithm.

Popular summary

The application of quantum computers to molecule and material discovery requires computing electronic ground states for many atomic configurations. In this work, motivated by this application, we derive provably efficient learning algorithms that predict properties of quantum systems with long-range interactions. We also investigate how observables in these systems change under geometric transformations and introduce machine learning models that transform nicely – in technical terms, that are equivariant – under those, which leads to a further reduction of the number of samples required for accurate predictions. We verify our theoretical guarantees with large scale numerical studies.

► BibTeX data

@article{Smid2025efficientlearningof,
  doi = {10.22331/q-2025-01-15-1597},
  url = {https://doi.org/10.22331/q-2025-01-15-1597},
  title = {Efficient {L}earning of {L}ong-{R}ange and {E}quivariant {Q}uantum {S}ystems},
  author = {{\v{S}}m{\'{i}}d, {\v{S}}t{\v{e}}p{\'{a}}n and Bondesan, Roberto},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {9},
  pages = {1597},
  month = jan,
  year = {2025}
}

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Cited by

[1] Yuxuan Du, Yan Zhu, Yuan-Hang Zhang, Min-Hsiu Hsieh, Patrick Rebentrost, Weibo Gao, Yi-Zhuang You, Jens Eisert, Giulio Chiribella, Dacheng Tao, Barry C. Sanders, and Ya-Dong Wu, "Artificial intelligence for representing and characterizing quantum systems", Nature Reviews Physics (2026).

[2] Yanming Che, Clemens Gneiting, Xiaoguang Wang, and Franco Nori, "Quantum circuit complexity and unsupervised machine learning of topological order", Nature Communications 17 1, 5179 (2026).

[3] Štěpán Šmíd and Roberto Bondesan, "Accurate Learning of Equivariant Quantum Systems from a Single Ground State", arXiv:2405.12309, (2024).

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