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Low-Density Parity-Check Stabilizer Codes as Gapped Quantum Phases: Stability under Graph-Local Perturbations

Wojciech De Roeck1, Vedika Khemani2, Yaodong Li2, Nicholas O’Dea2,*, and Tibor Rakovszky3,4

  • *Contact author: nickodea@stanford.edu
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PRX Quantum 6, 030330 – Published 19 August, 2025

DOI: https://doi.org/10.1103/7x71-8j7k

Abstract

We generalize the proof of stability of topological order, due to Bravyi, Hastings, and Michalakis, to stabilizer Hamiltonians corresponding to low-density parity-check (LDPC) codes without the restriction of geometric locality in Euclidean space. We consider Hamiltonians H0 defined by ⟦N,K,d⟧ LDPC codes, which obey certain topological quantum order conditions: (i) code distance d≥clog(N), implying local indistinguishability of ground states, and (ii) a mild condition on local and global compatibility of ground states—these include good quantum LDPC codes and the toric code on a hyperbolic lattice, among others. We consider stability under weak perturbations that are quasilocal on the interaction graph defined by H0 and that can be represented as sums of bounded-norm terms. As long as the local perturbation strength is smaller than a finite constant, we show that the perturbed Hamiltonian has well-defined spectral bands originating from the O(1) smallest eigenvalues of H0. The band originating from the smallest eigenvalue has 2K states, is separated from the rest of the spectrum by a finite energy gap, and has exponentially narrow bandwidth δ=CNe−Θ(d), which is tighter than the best-known bounds even in the Euclidean case. We also obtain that the new ground-state subspace is related to the initial-code subspace by a quasilocal unitary, allowing one to relate their physical properties. Our proof uses an iterative procedure that performs successive rotations to eliminate non-frustration-free terms in the Hamiltonian. Our results extend to quantum Hamiltonians built from classical LDPC codes, which give rise to stable symmetry-breaking phases. These results show that LDPC codes very generally define stable gapped quantum phases, even in the non-Euclidean setting, initiating a systematic study of such phases of matter.

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Popular Summary

Phases of matter provide a key organizing concept throughout physics. A major breakthrough in modern condensed matter has been the notion of topological phases that are stable to all local perturbations, making them extremely robust. Our work rigorously proves robustness for a wide family of models that live on graphs rather than in conventional Euclidean space, proving that the concept of phase still exists in this much more general setting.

In quantum information theory, there is a similar notion of robustness given by quantum error correction: by encoding quantum information into entangled states, one can track and correct errors to protect the information from local noise. Our proof applies to stabilizer Hamiltonians, which can be associated with the most widely studied family of error-correcting codes. These Hamiltonians include the very best codes, called “good” codes, which have recently been constructed and are an active area of research. Our proof is a generalization of one by Bravyi, Hastings, and Michalakis, where the perturbation is rotated into a simpler form. One of our breakthroughs is a method for stronger control over the locality of the rotated perturbation.

We have proven stability of a vast number of phases. But how should we characterize these unusual phases? For example, we have proven that many ground-state spectral properties are maintained under perturbation, but some of these quantum codes are known to have an extremely strict form of topological order that extends to finite energy density. Is this property also maintained under perturbation?

See Also

Low-Density Parity-Check Codes as Stable Phases of Quantum Matter

Chao Yin and Andrew Lucas
PRX Quantum 6, 030329 (2025)

Article Text

References (67)

  1. B. Zeng, X. Chen, D.-L. Zhou, and X.-G. WenQuantum Information Meets Quantum Matter: From Quantum Entanglement to Topological Phases of Many-Body Systems, 1st ed. (Quantum Science and Technology, Springer New York, New York, 2019).
  2. A. R. Calderbank and P. W. Shor, Good quantum error-correcting codes exist, Phys. Rev. A 54, 1098 (1996).
  3. A. Steane, Multiple-particle interference and quantum error correction, Proc. R. Soc. London, A 452, 2551 (1996).
  4. D. Gottesman, Stabilizer Codes and Quantum Error Correction (California Institute of Technology, Pasadena, California, 1997).
  5. The code rate of a code is K/N.
  6. E. Knill and R. Laflamme, Theory of quantum error-correcting codes, Phys. Rev. A 55, 900 (1997).
  7. A. Y. Kitaev, Fault-tolerant quantum computation by anyons, Ann. Phys. (NY) 303, 2 (2003).
  8. E. Fradkin and S. H. Shenker, Phase diagrams of lattice gauge theories with Higgs fields, Phys. Rev. D 19, 3682 (1979).
  9. C. Chamon, Quantum glassiness in strongly correlated clean systems: An example of topological overprotection, Phys. Rev. Lett. 94, 040402 (2005).
  10. J. Haah, Local stabilizer codes in three dimensions without string logical operators, Phys. Rev. A 83, 042330 (2011).
  11. S. Bravyi, M. B. Hastings, and S. Michalakis, Topological quantum order: Stability under local perturbations, J. Math. Phys. 51, 093512 (2010).
  12. I. Klich, On the stability of topological phases on a lattice, Ann. Phys. (NY) 325, 2120 (2010).
  13. S. Bravyi and M. B. Hastings, A short proof of stability of topological order under local perturbations, Commun. Math. Phys. 307, 609 (2011).
  14. S. Michalakis and J. P. Zwolak, Stability of frustration-free Hamiltonians, Commun. Math. Phys. 322, 277 (2013).
  15. M. B. Hastings and X.-G. Wen, Quasiadiabatic continuation of quantum states: The stability of topological ground-state degeneracy and emergent gauge invariance, Phys. Rev. B 72, 045141 (2005).
  16. X. Chen, Z.-C. Gu, and X.-G. Wen, Local unitary transformation long-range quantum entanglement, wave function renormalization, and topological order, long-range quantum entanglement, wave function renormalization, and topological order, Phys. Rev. B 82, 155138 (2010).
  17. S. Bravyi and M. B. Hastings, A short proof of stability of topological order under local perturbations, Commun. Math. Phys. 307, 609 (2011).
  18. B. Nachtergaele, R. Sims, and A. Young, Quasi-locality bounds for quantum lattice systems. Part Ii. Perturbations of frustration-free spin models with gapped ground states, Ann. Henri Poincaré 23, 393 (2021).
  19. M. Sipser and D. A. Spielman, Expander codes, IEEE Trans. Inf. Theory 42, 1710 (1996).
  20. M. B. Hastings, J. Haah, and R. O’Donnell, in Proceedings of the 53rd Annual ACM SIGACT Symposium on Theory of Computing (Association for Computing Machinery, New York, New York, 2021), p. 1276.
  21. P. Panteleev and G. Kalachev, Degenerate quantum LDPC codes with good finite length performance, Quantum 5, 585 (2021).
  22. P. Panteleev and G. Kalachev, Quantum LDPC codes with almost linear minimum distance, IEEE Trans. Inf. Theory 68, 213 (2021).
  23. P. Panteleev and G. Kalachev, in Proceedings of the 54th Annual ACM SIGACT Symposium on Theory of Computing (Association for Computing Machinery, New York, New York, 2022), p. 375.
  24. N. P. Breuckmann and J. N. Eberhardt, Balanced product quantum codes, IEEE Trans. Inf. Theory 67, 6653 (2021).
  25. A. Leverrier and G. Zémor, in 2022 IEEE 63rd Annual Symposium on Foundations of Computer Science (FOCS) (IEEE, Denver, Colorado, 2022), p. 872, https://ieeexplore.ieee.org/xpl/conhome/9996589/proceeding.
  26. I. Dinur, M.-H. Hsieh, T.-C. Lin, and T. Vidick, in Proceedings of the 55th Annual ACM Symposium on Theory of Computing (Association for Computing Machinery, New York, NY, USA, 2023), p. 905.
  27. T.-C. Lin and M.-H. Hsieh, Good quantum LDPC codes with linear time decoder from lossless expanders, ArXiv:2203.03581.
  28. S. Bravyi, D. Poulin, and B. Terhal, Tradeoffs for reliable quantum information storage in 2D systems, Phys. Rev. Lett. 104, 050503 (2010).
  29. Embedding non-Euclidean LDPC models within Euclidean geometries will necessarily produce nonlocal and/or nonplanar interactions.
  30. A. J. Kollár, M. Fitzpatrick, and A. A. Houck, Hyperbolic lattices in circuit quantum electrodynamics, Nature 571, 45 (2019).
  31. A. Periwal, E. S. Cooper, P. Kunkel, J. F. Wienand, E. J. Davis, and M. Schleier-Smith, Programmable interactions and emergent geometry in an array of atom clouds, Nature 600, 630 (2021).
  32. D. Bluvstein, H. Levine, G. Semeghini, T. T. Wang, S. Ebadi, M. Kalinowski, A. Keesling, N. Maskara, H. Pichler, M. Greiner, V. Vuletić, and M. D. Lukin, A quantum processor based on coherent transport of entangled atom arrays, Nature 604, 451 (2022).
  33. Q. Xu, J. P. B. Ataides, C. A. Pattison, N. Raveendran, D. Bluvstein, J. Wurtz, B. Vasic, M. D. Lukin, L. Jiang, and H. Zhou, Constant-overhead fault-tolerant quantum computation with reconfigurable atom arrays, ArXiv:2308.08648.
  34. T. Rakovszky and V. Khemani, The physics of (good) LDPC codes I. Gauging and dualities, ArXiv:2310.16032.
  35. T. Rakovszky and V. Khemani, The physics of (good) LDPC codes II. Product constructions, ArXiv:2402.16831.
  36. Y. Li, N. O’Dea, and V. Khemani, Perturbative stability and error correction thresholds of quantum codes, ArXiv:2406.15757.
  37. M. H. Freedman and M. B. Hastings, Quantum systems on non-k-hyperfinite complexes: A generalization of classical statistical mechanics on expander graphs, ArXiv:1301.1363.
  38. A. Anshu, N. P. Breuckmann, and C. Nirkhe, in Proceedings of the 55th Annual ACM Symposium on Theory of Computing, STOC ’23 (ACM, Orlando, Florida, 2023), https://dl.acm.org/doi/proceedings/10.1145/3564246.
  39. M. B. Hastings and T. Koma, Spectral gap and exponential decay of correlations, Commun. Math. Phys. 265, 781 (2006).
  40. D. Aharonov and M. Ben-Or, Fault-tolerant quantum computation with constant error rate, SIAM J. Comput. 38, 1207 (2008).
  41. I. Dumer, A. A. Kovalev, and L. P. Pryadko, Thresholds for correcting errors, erasures, and faulty syndrome measurements in degenerate quantum codes, Phys. Rev. Lett. 115, 050502 (2015).
  42. A. Lavasani, M. J. Gullans, V. V. Albert, and M. Barkeshli, On stability of k-local quantum phases of matter, ArXiv:2405.19412.
  43. This might seem alarming, but we remind the reader that such violations would occur on expander graphs in which bulk and boundary scale proportionally, which is outside the purview of standard treatments in statistical mechanics.
  44. N. P. Breuckmann and B. M. Terhal, Constructions and noise threshold of hyperbolic surface codes, IEEE Trans. Inf. Theory 62, 3731 (2016).
  45. I. Dinur, S. Evra, R. Livne, A. Lubotzky, and S. Mozes, Locally testable codes with constant rate, distance, and locality, ArXiv:2111.04808.
  46. P. Panteleev and G. Kalachev, Asymptotically good quantum and locally testable classical LDPC codes, ArXiv:2111.03654.
  47. I. Dinur, M.-H. Hsieh, T.-C. Lin, and T. Vidick, Good quantum LDPC codes with linear time decoders, ArXiv:2206.07750.
  48. A. N. Kolmogorov, On the conservation of conditionally periodic motions for a small change in Hamilton’s function, Dokl. Akad. Nauk SSSR 98, 527 (1954), English translation available in: Los Alamos Scientific Laboratory Translation LA-TR-71-67.
  49. J. Moser, On invariant curves of area-preserving mappings of an annulus, Nachr. Akad. Wiss. Gottingen Math.-Phys. Kl. II 1962, 1 (1962), Reprinted in: Moser, J. (1973). Stable and Random Motions in Dynamical Systems. Princeton University Press.
  50. V. I. Arnold, Small denominators and problems of stability of motion in classical and celestial mechanics, Usp. Mat. Nauk. 18, 91 (1963), English translation in: Russian Mathematical Surveys, 1964, 18(6):85–191.
  51. W. D. Roeck and M. Schütz, An exponentially local spectral flow for possibly non-self-adjoint perturbations of non-interacting quantum spins, inspired by KAM theory, Lett. Math. Phys. 107, 505 (2017).
  52. D. A. Yarotsky, Ground states in relatively bounded quantum perturbations of classical lattice systems, Commun. Math. Phys. 261, 799 (2006).
  53. C. Albanese, On the spectrum of the Heisenberg Hamiltonian, J. Stat. Phys. 55, 297 (1989).
  54. J. Fröhlich and A. Pizzo, Lie-Schwinger block-diagonalization and gapped quantum chains, Commun. Math. Phys. 375, 2039 (2020).
  55. The use of the word “string” here is historical and does not imply a one-dimensional geometry.
  56. T. Kato, Perturbation Theory for Linear Operators (Springer Science & Business Media, New York, New York, 2013), Vol. 132, https://link.springer.com/book/10.1007/978-3-642-66282-9.
  57. S. Bravyi, M. B. Hastings, and S. Michalakis, Topological quantum order: Stability under local perturbations, J. Math. Phys. 51, 093512 (2010).
  58. S. Bravyi and B. Terhal, A no-go theorem for a two-dimensional self-correcting quantum memory based on stabilizer codes, New J. Phys. 11, 043029 (2009).
  59. D. Gaiotto, A. Kapustin, N. Seiberg, and B. Willett, Generalized global symmetries, J. High Energy Phys. 2015, 1 (2015).
  60. J. McGreevy, Generalized symmetries in condensed matter, Annu. Rev. Condens. Matter Phys. 14, 57 (2023).
  61. J.-P. Tillich and G. Zémor, Quantum LDPC codes with positive rate and minimum distance proportional to the square root of the blocklength, IEEE Trans. Inf. Theory 60, 1193 (2013).
  62. Y. Hong, J. Guo, and A. Lucas, Quantum memory at nonzero temperature in a thermodynamically trivial system, ArXiv:2403.10599.
  63. The argument is more straightforward for classical codes and it is worth spelling out. In that case, the probe operator O, standing in for local energy density, should be chosen to be symmetric under all the classical logicals. All such operators are generated by the checks Cα together with single-site Pauli-X operators. Dα commutes with all elements of this generating set with the exception of Cα. Thus, the only question is whether or not O contains this check. Then, by TQO-IIc, the answer is no if O is supported sufficiently far away from Cα.
  64. M. B. Hastings, Quantum belief propagation: An algorithm for thermal quantum systems, Phys. Rev. B: Condens. Matter Mater. Phys. 76, 201102 (2007).
  65. Á. Capel, M. Moscolari, S. Teufel, and T. Wessel, From decay of correlations to locality and stability of the Gibbs state, ArXiv:2310.09182.
  66. C. Yin and A. Lucas, Low-density parity-check codes as stable phases of quantum matter, PRX Quantum 6, 030329 (2025).
  67. A. Kubica and B. Yoshida, Ungauging quantum error-correcting codes, ArXiv:1805.01836.

Outline

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PRX Quantum 6, 030330– Published 19 August, 2025
  • Received 19 December 2024
  • Revised 6 June 2025
  • Accepted 2 July 2025
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