A little bit of self-correction

Michael J. Kastoryano; Lasse B. Kristensen; Chi-Fang Chen; Andras Gilyén

doi:10.22331/q-2025-08-04-1820

A little bit of self-correction

Michael J. Kastoryano^1,2, Lasse B. Kristensen¹, Chi-Fang Chen³, and Andras Gilyén⁴

¹Department of Computer Science, University of Copenhagen, Denmark
²AWS Center for Quantum Computing, Pasadena, CA
³Institute for Quantum Information and Matter, California Institute of Technology, Pasadena, CA, USA
⁴HUN-REN Alfréd Rényi Institute of Mathematics, Budapest, Hungary

Published:	2025-08-04, volume 9, page 1820
Editor:	Álvaro Alhambra
Eprint:	arXiv:2408.14970v2
Doi:	https://doi.org/10.22331/q-2025-08-04-1820
Citation:	Quantum 9, 1820 (2025).

Find this paper interesting or want to discuss? Scite or leave a comment on SciRate.

Abstract

We investigate the emergence of stable subspaces in the low-temperature quantum thermal dynamics of finite spin chains. Our analysis reveals the existence of effective decoherence-free qudit subspaces, persisting for timescales exponential in $\beta$. Surprisingly, the appearance of metastable subspaces is not directly related to the entanglement structure of the ground state(s). Rather, they arise from symmetry relations in low-lying excited states. Despite their stability within a 'phase', practical realization of stable qubits is hindered by susceptibility to symmetry-breaking perturbations. This work highlights that there can be non-trivial quantum behavior in the thermal dynamics of noncommuting many body models, and opens the door to more extensive studies of self-correction in such systems.

Featured image: Illustration of the J1 − J2 spin chain, with reflection R and translation T symmetries. These discrete symmetries give to approximate decoherence subspaces at low temperatures, under Gibbs sampling dynamics.

Popular summary

Most quantum systems are susceptible to external disturbances, rendering their quantum properties fragile. A search for inherently robust quantum properties has therefore been ongoing, with the hope of discovering robust quantum memories to be used for quantum technologies. Until recently, most of these studies focused on systems governed by so-called commuting Hamiltonians. However, recent progress has made it possible to study also the behaviour of more general quantum systems put in contact with a thermal environment.
In this work, we use these new methods to study the emergence of thermally robust subspaces in quantum spin chains. Specifically, we identify subspaces in a J1-J2 spin-1/2 ring that are robust to low-temperature thermal noise. Interestingly, the protection is not related only to the ground state properties, but also the symmetry properties of low-energy excited states. Furthermore, the form of the coupling to the surroundings plays a key role, in contrast to intuition from similar classical systems.
While it is uncertain whether the symmetries are too fragile to allow implementation of practically protected quantum memories, the study highlights how non-trivial quantum effects can impact thermalization in low-temperature quantum systems. In addition, it shows how protected subspaces may be more prevalent than previously thought, paving the way for further studies of quantum memories governed by non-commuting Hamiltonians.

► BibTeX data

@article{Kastoryano2025littlebitofself,
  doi = {10.22331/q-2025-08-04-1820},
  url = {https://doi.org/10.22331/q-2025-08-04-1820},
  title = {A little bit of self-correction},
  author = {Kastoryano, Michael J. and Kristensen, Lasse B. and Chen, Chi-Fang and Gily{\'{e}}n, Andras},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {9},
  pages = {1820},
  month = aug,
  year = {2025}
}

► References

[1] Benjamin J Brown, Daniel Loss, Jiannis K Pachos, Chris N Self, and James R Wootton. ``Quantum memories at finite temperature''. Reviews of Modern Physics 88, 045005 (2016).
https://doi.org/10.1103/RevModPhys.88.045005

[2] A Yu Kitaev. ``Fault-tolerant quantum computation by anyons''. Annals of physics 303, 2–30 (2003).
https://doi.org/10.1016/S0003-4916(02)00018-0

[3] Héctor Bombín. ``Gauge color codes: optimal transversal gates and gauge fixing in topological stabilizer codes''. New Journal of Physics 17, 083002 (2015).
https://doi.org/10.1088/1367-2630/17/8/083002

[4] Dave Bacon. ``Operator quantum error-correcting subsystems for self-correcting quantum memories''. Physical Review A 73, 012340 (2006).
https://doi.org/10.1103/PhysRevA.73.012340

[5] Jeongwan Haah. ``Local stabilizer codes in three dimensions without string logical operators''. Physical Review A 83, 042330 (2011).
https://doi.org/10.1103/PhysRevA.83.042330

[6] Michael J Kastoryano and Fernando GSL Brandao. ``Quantum Gibbs samplers: The commuting case''. Communications in Mathematical Physics 344, 915–957 (2016).
https://doi.org/10.1007/s00220-016-2641-8

[7] Héctor Bombín. ``Single-shot fault-tolerant quantum error correction''. Physical Review X 5, 031043 (2015).
https://doi.org/10.1103/PhysRevX.5.031043

[8] Frederik Nathan, Liam O'Brien, Kyungjoo Noh, Matthew H Matheny, Arne L Grimsmo, Liang Jiang, and Gil Refael. ``Self-correcting GKP qubit and gates in a driven-dissipative circuit'' (2024). arXiv:2405.05671.
arXiv:2405.05671

[9] Robert Alicki, Mark Fannes, and Michal Horodecki. ``On thermalization in Kitaev's 2d model''. Journal of Physics A: Mathematical and Theoretical 42, 065303 (2009).
https://doi.org/10.1088/1751-8113/42/6/065303

[10] Robert Alicki, Michal Horodecki, Pawel Horodecki, and Ryszard Horodecki. ``On thermal stability of topological qubit in Kitaev's 4d model''. Open Systems & Information Dynamics 17, 1–20 (2010).
https://doi.org/10.1142/S1230161210000023

[11] Ainesh Bakshi, Allen Liu, Ankur Moitra, and Ewin Tang. ``High-temperature Gibbs states are unentangled and efficiently preparable''. In 2024 IEEE 65th Annual Symposium on Foundations of Computer Science (FOCS). Pages 1027–1036. IEEE (2024).
https://doi.org/10.1109/FOCS61266.2024.00068

[12] Cambyse Rouzé, Daniel Stilck França, and Álvaro M Alhambra. ``Efficient thermalization and universal quantum computing with quantum Gibbs samplers'' (2024). arXiv:2403.12691.
arXiv:2403.12691

[13] Fabio Martinelli. ``Lectures on Glauber dynamics for discrete spin models''. In Lectures on probability theory and statistics. Pages 93–191. Springer (1999).
https://doi.org/10.1007/978-3-540-48115-7_2

[14] Ángela Capel, Cambyse Rouzé, and Daniel Stilck França. ``The modified logarithmic Sobolev inequality for quantum spin systems: classical and commuting nearest neighbour interactions'' (2021) arXiv:2009.11817.
arXiv:2009.11817

[15] Chi-Fang Chen, Michael J Kastoryano, Fernando GSL Brandão, and András Gilyén. ``Quantum thermal state preparation'' (2023). arXiv:2303.18224.
arXiv:2303.18224

[16] Chi-Fang Chen, Michael J Kastoryano, and András Gilyén. ``An efficient and exact noncommutative quantum Gibbs sampler'' (2023). arXiv:2311.09207.
arXiv:2311.09207

[17] András Gilyén, Chi-Fang Chen, Joao F Doriguello, and Michael J Kastoryano. ``Quantum generalizations of Glauber and Metropolis dynamics'' (2024). arXiv:2405.20322.
arXiv:2405.20322

[18] Zhiyan Ding, Bowen Li, and Lin Lin. ``Efficient quantum Gibbs samplers with Kubo–Martin–Schwinger detailed balance condition''. Communications in Mathematical Physics 406, 67 (2025).
https://doi.org/10.1007/s00220-025-05235-3

[19] Subir Sachdev. ``Quantum phase transitions''. Physics world 12, 33 (1999).
https://doi.org/10.1088/2058-7058/12/4/23

[20] Bei Zeng, Xie Chen, Duan-Lu Zhou, Xiao-Gang Wen, et al. ``Quantum information meets quantum matter''. Springer. (2019).
https://doi.org/10.1007/978-1-4939-9084-9

[21] Ivan Bardet, Ángela Capel, Li Gao, Angelo Lucia, David Pérez-García, and Cambyse Rouzé. ``Rapid thermalization of spin chain commuting Hamiltonians''. Physical Review Letters 130, 060401 (2023).
https://doi.org/10.1103/PhysRevLett.130.060401

[22] Crispin Gardiner and Peter Zoller. ``Quantum noise: a handbook of Markovian and non-Markovian quantum stochastic methods with applications to quantum optics''. Springer Science & Business Media. (2004).
https://doi.org/10.1007/978-3-662-04103-1

[23] Frederik Nathan and Mark S Rudner. ``Universal Lindblad equation for open quantum systems''. Physical Review B 102, 115109 (2020).
https://doi.org/10.1103/PhysRevB.102.115109

[24] Noam Berger, Claire Kenyon, Elchanan Mossel, and Yuval Peres. ``Glauber dynamics on trees and hyperbolic graphs''. Probability Theory and Related Fields 131, 311–340 (2005).
https://doi.org/10.1007/s00440-004-0369-4

[25] Emanuel Knill, Raymond Laflamme, and Lorenza Viola. ``Theory of quantum error correction for general noise''. Physical Review Letters 84, 2525 (2000).
https://doi.org/10.1103/PhysRevLett.84.2525

[26] Paolo Zanardi. ``Stabilizing quantum information''. Physical Review A 63, 012301 (2000).
https://doi.org/10.1103/PhysRevA.63.012301

[27] Barbara Kraus, Hans P Büchler, Sebastian Diehl, Adrian Kantian, Andrea Micheli, and Peter Zoller. ``Preparation of entangled states by quantum Markov processes''. Physical Review A 78, 042307 (2008).
https://doi.org/10.1103/PhysRevA.78.042307

[28] Daniel A Lidar and K Birgitta Whaley. ``Decoherence-free subspaces and subsystems''. In Irreversible quantum dynamics. Pages 83–120. Springer (2003).
https://doi.org/10.1007/3-540-44874-8_5

[29] Stefano Chesi, Daniel Loss, Sergey Bravyi, and Barbara M Terhal. ``Thermodynamic stability criteria for a quantum memory based on stabilizer and subsystem codes''. New Journal of Physics 12, 025013 (2010).
https://doi.org/10.1088/1367-2630/12/2/025013

[30] J Robert Johansson, Paul D Nation, and Franco Nori. ``QuTiP: An open-source Python framework for the dynamics of open quantum systems''. Computer physics communications 183, 1760–1772 (2012).
https://doi.org/10.1016/j.cpc.2012.02.021

[31] Kiyomi Okamoto and Kiyohide Nomura. ``Fluid-dimer critical point in S = 12 antiferromagnetic Heisenberg chain with next nearest neighbor interactions''. Physics Letters A 169, 433–437 (1992).
https://doi.org/10.1016/0375-9601(92)90823-5

[32] WJ Caspers and W Magnus. ``Some exact excited states in a linear antiferromagnetic spin system''. Physics letters A 88, 103–105 (1982).
https://doi.org/10.1016/0375-9601(82)90603-x

[33] M. Hamermesh. ``Group theory and its application to physical problems''. Dover Books on Physics. Dover Publications. (2012).
https://doi.org/10.1063/1.3050758

[34] B.H. Bransden and C.J. Joachain. ``Physics of atoms and molecules''. Pearson Education. Prentice Hall. (2003). url: https://books.google.dk/books?id=i5IPWXDQlcIC.
https://books.google.dk/books?id=i5IPWXDQlcIC

[35] Jorge A Campos-Gonzalez-Angulo, Raphael F Ribeiro, and Joel Yuen-Zhou. ``Generalization of the Tavis–Cummings model for multi-level anharmonic systems''. New Journal of Physics 23, 063081 (2021).
https://doi.org/10.1088/1367-2630/ac00d7

[36] Daniel P Arovas. ``Two exact excited states for the S = 1 AKLT chain''. Physics Letters A 137, 431–433 (1989).
https://doi.org/10.1016/0375-9601(89)90921-3

[37] Lyudvig Dmitrievich Faddeev and Leon Armenovich Takhtadzhyan. ``Spectrum and scattering of excitations in the one-dimensional isotropic Heisenberg model''. Journal of Soviet Mathematics 24, 241–267 (1984).
https://doi.org/10.1007/BF01087245

[38] Anastasia Doikou and Rafael I Nepomechie. ``Parity and charge conjugation symmetries and S matrix of the XXZ chain''. Statistical Physics on the Eve of the Twenty-First Century, M. Batchelor and L. Wille, edsPages 391–411 (1999). arXiv:hep-th/9810034.
arXiv:hep-th/9810034

[39] Xiao-Gang Wen. ``Quantum orders in an exact soluble model''. Physical review letters 90, 016803 (2003).
https://doi.org/10.1103/PhysRevLett.90.016803

[40] Anne Matthies, Mark Rudner, Achim Rosch, and Erez Berg. ``Programmable adiabatic demagnetization for systems with trivial and topological excitations''. Quantum 8, 1505 (2024).
https://doi.org/10.22331/q-2024-10-23-1505

[41] Gilad Kishony, Mark S Rudner, Achim Rosch, and Erez Berg. ``Gauged cooling of topological excitations and emergent fermions on quantum simulators''. Physical Review Letters 134, 086503 (2025).
https://doi.org/10.1103/PhysRevLett.134.086503

[42] Trond I Andersen, Nikita Astrakhantsev, Amir H Karamlou, Julia Berndtsson, Johannes Motruk, Aaron Szasz, Jonathan A Gross, Alexander Schuckert, Tom Westerhout, Yaxing Zhang, et al. ``Thermalization and criticality on an analogue–digital quantum simulator''. Nature 638, 79–85 (2025).
https://doi.org/10.1038/s41586-024-08460-3

[43] Jerome Lloyd, Alexios A Michailidis, Xiao Mi, Vadim Smelyanskiy, and Dmitry A Abanin. ``Quasiparticle cooling algorithms for quantum many-body state preparation''. PRX Quantum 6, 010361 (2025).
https://doi.org/10.1103/PRXQuantum.6.010361

[44] Masaaki Nakamura and Synge Todo. ``Order parameter to characterize valence-bond-solid states in quantum spin chains''. Physical review letters 89, 077204 (2002).
https://doi.org/10.1103/PhysRevLett.89.077204

[45] Dong-Sheng Wang. ``Classes of topological qubits from low-dimensional quantum spin systems''. Annals of Physics 412, 168015 (2020).
https://doi.org/10.1016/j.aop.2019.168015

Cited by

[1] Chi-Fang Chen, Michael Kastoryano, Fernando G. S. L. Brandão, and András Gilyén, "Efficient quantum thermal simulation", Nature 646 8085, 561 (2025).

[2] Cambyse Rouzé, Daniel Stilck França, and Álvaro M. Alhambra, "Optimal Quantum Algorithm for Gibbs State Preparation", Physical Review Letters 136 6, 060601 (2026).

[3] Zhiyan Ding, Zeph Landau, Bowen Li, Lin Lin, and Ruizhe Zhang, "Polynomial-time preparation of low-temperature Gibbs states for two-dimensional toric code", Journal of Mathematical Physics 67 5, 052201 (2026).

[4] Jordan Cotler, Weiyuan Gong, and Ishaan Kannan, "Noisy quantum learning theory", Nature Communications 17 1, 6979 (2026).

[5] Zhiyan Ding, Zeph Landau, Bowen Li, Lin Lin, and Ruizhe Zhang, "Polynomial-Time Preparation of Low-Temperature Gibbs States for 2D Toric Code", arXiv:2410.01206, (2024).

[6] Matteo Scandi and Álvaro M. Alhambra, "Thermalization in Open Many-Body Systems and KMS Detailed Balance", Physical Review X 16 1, 011040 (2026).

[7] Max Geier and Frederik Nathan, "Self-correcting GKP qubit in a superconducting circuit with an oscillating voltage bias", arXiv:2412.03650, (2024).

[8] Victor V. Albert and Philippe Faist, "Handbook of Error-Correcting Codes", arXiv:2606.11484, (2026).

The above citations are from Crossref's cited-by service (last updated successfully 2026-08-17 21:28:19) and SAO/NASA ADS (last updated successfully 2026-08-17 21:28:20). The list may be incomplete as not all publishers provide suitable and complete citation data.

This Paper is published in Quantum under the Creative Commons Attribution 4.0 International (CC BY 4.0) license. Copyright remains with the original copyright holders such as the authors or their institutions.