- Physical Review Journals
All JournalsPhysics Magazine

PRX Quantum

Reuse & Permissions

It is not necessary to obtain permission to reuse this article or its components as it is available under the terms of the Creative Commons Attribution 4.0 International license. This license permits unrestricted use, distribution, and reproduction in any medium, provided attribution to the author(s) and the published article's title, journal citation, and DOI are maintained. Please note that some figures may have been included with permission from other third parties. It is your responsibility to obtain the proper permission from the rights holder directly for these figures.

Export citation

Export citation

Choose format for download:

Download Citation
  • Open Access

Abelian State Hidden Subgroup Problem: Learning Stabilizer Groups and Beyond

Marcel Hinsche*, Jens Eisert, and Jose Carrasco†

  • *Contact author: m.hinsche@fu-berlin.de
  • †Contact author: jose.carrasco@fu-berlin.de
PDF

PRX Quantum 7, 020337 – Published 26 May, 2026

DOI: https://doi.org/10.1103/6frk-891j

Abstract

Identifying the symmetry properties of quantum states is a central theme in quantum information theory and quantum many-body physics. In this work, we investigate quantum learning problems in which the goal is to identify a hidden symmetry of an unknown quantum state. Building on the recent formulation of the state hidden subgroup problem (StateHSP), we focus on abelian groups and develop an efficient quantum algorithm that learns any hidden symmetry subgroup using a generalized form of Fourier sampling. We showcase the versatility of the approach in three concrete applications: These are learning (i) qubit and qudit stabilizer groups, (ii) cuts along which a state is unentangled, and (iii) hidden translation symmetries. Through these applications, we reveal that well-known quantum learning primitives, such as Bell sampling and Bell difference sampling, are, in fact, special cases of Fourier sampling. Our results highlight the broad potential of the StateHSP framework for symmetry-based quantum learning tasks and provide protocols that are easier to implement on near-term quantum devices.

View figure in article

Physics Subject Headings (PhySH)

Popular Summary

Quantum states often exhibit symmetries—patterns that remain unchanged under certain transformations—and identifying them is key to understanding complex quantum systems and designing quantum algorithms. We study how a quantum computer can learn such hidden symmetries when given access to unknown quantum states. Our approach uses a generalized form of Fourier sampling to efficiently identify hidden symmetry subgroups for a broad class of problems involving finite abelian groups. This unified framework captures several important tasks in quantum information and many-body physics, including determining stabilizer structures of qubit and qudit states, identifying partitions of a system that are not entangled, and detecting hidden translation symmetries. It also shows that techniques such as Bell sampling and Bell difference sampling arise as special cases. Importantly, our methods are substantially simpler and more experimentally accessible than prior approaches, potentially making symmetry-learning protocols easier to implement on near-term quantum devices.

Article Text

References (36)

  1. Barbara M. Terhal, Quantum error correction for quantum memories, Rev. Mod. Phys. 87, 307 (2015).
  2. Zhenyu Cai, Ryan Babbush, Simon C. Benjamin, Suguru Endo, William J. Huggins, Ying Li, Jarrod R. McClean, and Thomas E. O’Brien, Quantum error mitigation, Rev. Mod. Phys. 95, 045005 (2023).
  3. Jens Eisert, Dominik Hangleiter, Nathan Walk, Ingo Roth, Damian Markham, Rhea Parekh, Ulysse Chabaud, and Elham Kashefi, Quantum certification and benchmarking, Nat. Rev. Phys. 2, 382 (2020).
  4. Martin Kliesch and Ingo Roth, Theory of quantum system certification, PRX Quantum 2, 010201 (2021).
  5. Bei Zeng, Xie Chen, Duan-Lu Zhou, and Xiao-Gang Wen, Quantum Information Meets Quantum Matter: From Quantum Entanglement to Topological Phases of Many-Body Systems (Springer, New York, 2019).
  6. Andrew M. Childs, Lecture notes on quantum algorithms, 2025. URL: www.cs.umd.edu/amchilds/qa/qa.pdf.
  7. Peter W. Shor, in Proceedings 35th Annual Symposium on Foundations of Computer Science (IEEE Computer Society, Santa Fe, New Mexico, USA, 1994), pp. 124–134.
  8. Ashley Montanaro and Ronald de Wolf, A Survey of Quantum Property Testing. Graduate Surveys: Theory of Computing Library (Theory of Computing, 2016).
  9. Anurag Anshu and Srinivasan Arunachalam, A survey on the complexity of learning quantum states, Nat. Rev. Phys. 6, 59 (2024).
  10. Srinivasan Arunachalam and Ronald de Wolf, Guest column: A survey of quantum learning theory, ACM SIGACT News 48, 41 (2017).
  11. Adam Bouland, Tudor Giurgică-Tiron, and John Wright, in Proceedings of the 57th Annual ACM Symposium on Theory of Computing, STOC ’25 (Association for Computing Machinery, New York, NY, USA, 2025), pp. 463–470. doi:10.1145/3717823.3718118.
  12. Aram W. Harrow, Applications of coherent classical communication and the Schur transform to quantum information theory, Ph.D. thesis, Massachusetts Institute of Technology, Cambridge, MA, 2005.
  13. Jens Eisert and John Preskill, Mind the gaps: The fraught road to quantum advantage, arXiv:2510.19928.
  14. Dominik Hangleiter and Michael J. Gullans, Bell sampling from quantum circuits, Phys. Rev. Lett. 133, 020601 (2024).
  15. Hsin-Yuan Huang, Richard Kueng, and John Preskill, Information-theoretic bounds on quantum advantage in machine learning, Phys. Rev. Lett. 126, 190505 (2021).
  16. Ashley Montanaro, Learning stabilizer states by Bell sampling, arXiv:1707.04012.
  17. Sitan Chen, Weiyuan Gong, Qi Ye, and Zhihan Zhang, in Proceedings of the 57th Annual ACM Symposium on Theory of Computing, STOC ’25 (Association for Computing Machinery, New York, NY, USA, 2025), pp. 429–438. doi:10.1145/3717823.3718191
  18. Sabee Grewal, Vishnu Iyer, William Kretschmer, and Daniel Liang, Efficient learning of quantum states prepared with few non-Clifford gates, Quantum 9, 1907 (2025).
  19. David Gross, Sepehr Nezami, and Michael Walter, Schur–Weyl duality for the Clifford group with applications: Property testing, a robust Hudson theorem, and de Finetti representations, Commun. Math. Phys. 385, 1325 (2021).
  20. Andrew M. Childs, Aram W. Harrow, and Paweł Wocjan, in Proceedings of the 24th Annual Conference on Theoretical Aspects of Computer Science, STACS’07 (Springer-Verlag, Berlin, Heidelberg, 2007), pp. 598–609.
  21. Andrew M. Childs and Wim van Dam, Quantum algorithms for algebraic problems, Rev. Mod. Phys. 82, 1 (2010).
  22. Marcel Hinsche and Jonas Helsen, in Proceedings of the 57th Annual ACM Symposium on Theory of Computing, STOC ’25 (Association for Computing Machinery, New York, NY, USA, 2025), pp. 439–450. doi:10.1145/3717823.3718169.
  23. Scott Aaronson and Daniel Gottesman, Identifying Stabilizer States (Perimeter Institute, 2008).
  24. Sabee Grewal, Vishnu Iyer, William Kretschmer, and Daniel Liang, in Proceedings of the 56th Annual ACM Symposium on Theory of Computing, STOC 2024 (Association for Computing Machinery, New York, NY, USA, 2024), pp. 1352–1363.
  25. Jonathan Allcock, Joao F. Doriguello, Gábor Ivanyos, and Miklos Santha, Beyond Bell sampling: Stabilizer state learning and quantum pseudorandomness lower bounds on qudits, arXiv:2405.06357.
  26. Lorenzo Leone, Salvatore F. E. Oliviero, and Alioscia Hamma, Learning t-doped stabilizer states, Quantum 8, 1361 (2024).
  27. Dominic V. Else, Stephen D. Bartlett, and Andrew C. Doherty, Symmetry protection of measurement-based quantum computation in ground states, New J. Phys. 14, 113016 (2012).
  28. Margarite L. LaBorde, Soorya Rethinasamy, and Mark M. Wilde, Testing symmetry on quantum computers, Quantum 7, 1120 (2023).
  29. Soorya Rethinasamy, Margarite L. LaBorde, and Mark M. Wilde, Quantum computational complexity and symmetry, Can. J. Phys. 103, 215 (2025).
  30. Victor M. Bastidas, Nathan Fitzpatrick, K. J. Joven, Zane M. Rossi, Shariful Islam, Troy Van Voorhis, Isaac L. Chuang, and Yuan Liu, Unification of finite symmetries in the simulation of many-body systems on quantum computers, Phys. Rev. A 111, 052433 (2025).
  31. David Wakeham and Maria Schuld, Inference, interference and invariance: How the quantum Fourier transform can help to learn from data, arXiv:2409.00172.
  32. Erik Hostens, Jeroen Dehaene, and Bart De Moor, Stabilizer states and Clifford operations for systems of arbitrary dimensions, and modular arithmetic, Phys. Rev. A 71, 042315 (2005).
  33. J. Ignacio Cirac, David Pérez-García, Norbert Schuch, and Frank Verstraete, Matrix product states and projected entangled pair states: Concepts, symmetries, theorems, Rev. Mod. Phys. 93, 045003 (2021).
  34. Dominik Hangleiter and Jens Eisert, Computational advantage of quantum random sampling, Rev. Mod. Phys. 95, 035001 (2023).
  35. Hsin-Yuan Huang, Soonwon Choi, Jarrod R. McClean, and John Preskill, The vast world of quantum advantage, arXiv:2508.05720.
  36. Nai-Hui Chia, Ching-Yi Lai, and Han-Hsuan Lin, Efficient learning of t-doped stabilizer states with single-copy measurements, Quantum 8, 1250 (2024).

Outline

Information

PRX Quantum 7, 020337– Published 26 May, 2026
  • Received 29 October 2025
  • Revised 19 February 2026
  • Accepted 25 February 2026
Crossmark - Check for updates button
Creative Commons Logo

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

Published by the American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation