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  • Open Access

Quantum Gibbs Sampling Through the Detectability Lemma

Di Fang1,2, Jianfeng Lu1,3,4, Yu Tong1,2,5, and Chu Zhao5

  • 1Department of Mathematics, Duke University, Durham, North Carolina 27708, USA
  • 2Duke Quantum Center, Duke University, Durham, North Carolina 27701, USA
  • 3Department of Physics, Duke University, Durham, North Carolina 27708, USA
  • 4Department of Chemistry, Duke University, Durham, North Carolina 27708, USA
  • 5Department of Electrical and Computer Engineering, Duke University, Durham, North Carolina 27708, USA

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PRX Quantum 7, 043025 – Published 5 November, 2026

DOI: https://doi.org/10.1103/gfv4-qc3s

Abstract

Gibbs state preparation is an important subroutine in quantum computing. In this work we use the detectability lemma to improve Gibbs state preparation. Specifically, we design new Gibbs state preparation methods that do not rely on simulating Lindbladian evolution, thus avoiding the overhead from it. For local Lindbladians consisting of M terms, this approach reduces the cost by a factor of O(M). We also combine the detectability lemma operator and quantum singular value transformation to implement ground state projection operators of frustration-free Hamiltonians, resulting in a quadratic speedup in the spectral gap dependence. Applying this method to Lindbladians for the Gibbs state of local commuting Hamiltonians, we achieve quadratically better dependence on the Lindbladian spectral gap.

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Popular Summary

Preparing quantum states at a chosen temperature is essential for simulating materials, but it can demand substantial computing resources. We use a mathematical tool called the detectability lemma to design algorithms that prepare these states through repeated updates to small parts of a system. This avoids simulating the full relaxation process and can reduce computational overhead as the system grows. For systems with commuting local interactions, we also develop an annealing-based algorithm that improves the cost’s dependence on the relaxation timescale from linear to square root. These results turn a mathematical tool for understanding interacting quantum systems into a method for designing more efficient quantum algorithms to study their thermal behavior.

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References (69)

  1. F. G. Brandão, A. Kalev, T. Li, C. Y.-Y. Lin, K. M. Svore, and X. Wu, Quantum SDP solvers: Large speed-ups, optimality, and applications to quantum learning, arXiv:1710.02581.
  2. J. Van Apeldoorn, A. Gilyén, S. Gribling, and R. de Wolf, Quantum SDP-solvers: Better upper and lower bounds, in 2017 IEEE 58th Annual Symposium on Foundations of Computer Science (FOCS) (IEEE, New York, 2017), pp. 403–414.
  3. M. Motta, C. Sun, A. T. K. Tan, M. J. O’Rourke, E. Ye, A. J. Minnich, F. G. S. L. Brandão, and G. K..-L. Chan, Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution, Nat. Phys. 16, 205 (2020).
  4. C.-F. Chen, M. J. Kastoryano, F. G. Brandão, and A. Gilyén, Quantum thermal state preparation, arXiv:2303.18224.
  5. J. Lloyd and D. A. Abanin, Quantum thermal state preparation for near-term quantum processors, arXiv:2506.21318.
  6. C.-F. Chen, M. J. Kastoryano, F. G. S. L. Brandao, and A. Gilyen, Quantum thermal state preparation, arXiv:2303.18224.
  7. P. Rall, C. Wang, and P. Wocjan, Thermal state preparation via rounding promises, Quantum 7, 1132 (2023).
  8. Z. Ding, B. Li, and L. Lin, Efficient quantum Gibbs samplers with Kubo–Martin–Schwinger detailed balance condition, Commun. Math. Phys. 406, 67 (2025).
  9. R. Cleve and C. Wang, Efficient quantum algorithms for simulating Lindblad evolution, in 44th International Colloquium on Automata, Languages, and Programming (ICALP 2017) (Schloss-Dagstuhl-Leibniz Zentrum für Informatik, Dagstuhl, Germany, 2017).
  10. P. Wocjan and K. Temme, Szegedy walk unitaries for quantum maps, Commun. Math. Phys. 402, 3201 (2023).
  11. C.-F. Chen, M. J. Kastoryano, and A. Gilyen, An efficient and exact noncommutative quantum Gibbs sampler, arXiv:2311.09207.
  12. X. Li and C. Wang, Simulating Markovian open quantum systems using higher-order series expansion, in 50th International Colloquium on Automata, Languages, and Programming (ICALP 2023) (Schloss-Dagstuhl-Leibniz Zentrum für Informatik, Dagstuhl, Germany, 2023).
  13. Z. Ding, X. Li, and L. Lin, Simulating open quantum systems using Hamiltonian simulations, PRX Quantum 5, 020332 (2024).
  14. M. Pocrnic, D. Segal, and N. Wiebe, Quantum simulation of Lindbladian dynamics via repeated interactions, arXiv:2312.05371.
  15. C.-F. Chen, M. Kastoryano, F. G. S. L. Brandão, and A. Gilyén, Efficient quantum thermal simulation, Nature (London) 646, 561 (2025).
  16. K. Temme, M. J. Kastoryano, M. B. Ruskai, M. M. Wolf, and F. Verstraete, Hypercontractivity of quasi-free quantum semigroups, J. Math. Phys. 47, 405303 (2014).
  17. Z. Ding, B. Li, L. Lin, and R. Zhang, Polynomial-time preparation of low-temperature Gibbs states for 2d toric code, arXiv:2410.01206.
  18. I. Bardet, Á. Capel, L. Gao, A. Lucia, D. Pérez-García, and C. Rouzé, Entropy decay for Davies semigroups of a one dimensional quantum lattice, Commun. Math. Phys. 405, 42 (2024).
  19. R. Alicki, M. Fannes, and M. Horodecki, On thermalization in Kitaev’s 2D model, J. Phys. A 42, 065303 (2009).
  20. A. Ramkumar and M. Soleimanifar, Mixing time of quantum Gibbs sampling for random sparse Hamiltonians, arXiv:2411.04454.
  21. J. Kochanowski, A. M. Alhambra, A. Capel, and C. Rouzé, Rapid thermalization of dissipative many-body dynamics of commuting Hamiltonians, Commun. Math. Phys. 406, 176 (2025).
  22. I. Bardet, Á. Capel, L. Gao, A. Lucia, D. Pérez-García, and C. Rouzé, Rapid thermalization of spin chain commuting Hamiltonians, Phys. Rev. Lett. 130, 060401 (2023).
  23. C. Rouzé, D. S. França, and Á. M. Alhambra, Optimal quantum algorithm for Gibbs state preparation, Phys. Rev. Lett. 136, 060601 (2026).
  24. Á. Capel, P. Gondolf, J. Kochanowski, and C. Rouzé, Quasi-optimal sampling from Gibbs states via non-commutative optimal transport metrics, Ann. Henri Poincaré (2025).
  25. M. J. Kastoryano and K. Temme, Quantum logarithmic Sobolev inequalities and rapid mixing, J. Math. Phys. 54, 052202 (2013).
  26. K. Temme and M. J. Kastoryano, How fast do stabilizer Hamiltonians thermalize?, arXiv:1505.07811.
  27. H. Chen, B. Li, J. Lu, and L. Ying, A randomized method for simulating Lindblad equations and thermal state preparation, Quantum 9, 1917 (2025).
  28. T. Rakovszky, B. Placke, N. P. Breuckmann, and V. Khemani, Bottlenecks in quantum channels and finite temperature phases of matter, arXiv:2412.09598.
  29. B. Li and J. Lu, Quantum space-time Poincare inequality for Lindblad dynamics, arXiv:2406.09115.
  30. T. Barthel and Y. Zhang, Solving quasi-free and quadratic Lindblad master equations for open fermionic and bosonic systems, J. Stat. Mech.:Theory Exp. 2022, 113101 (2022).
  31. D. Fang, J. Lu, and Y. Tong, Mixing time of open quantum systems via hypocoercivity, Phys. Rev. Lett. 134, 140405 (2025).10.1103/PhysRevLett.134.140405
  32. Z. Ding, C.-F. Chen, and L. Lin, Single-ancilla ground state preparation via Lindbladians, Phys. Rev. Res. 6, 033147 (2024).
  33. Y. Tong and Y. Zhan, Fast mixing of weakly interacting fermionic systems at any temperature, PRX Quantum 6, 030301 (2025).
  34. A. Bakshi, A. Liu, A. Moitra, and E. Tang, A Dobrushin condition for quantum Markov chains: Rapid mixing and conditional mutual information at high temperature, arXiv:2510.08542.
  35. A. Gilyén, C.-F. Chen, J. F. Doriguello, and M. J. Kastoryano, Quantum generalizations of Glauber and metropolis dynamics, arXiv:2405.20322.
  36. J. Jiang and S. Irani, Quantum metropolis sampling via weak measurement, arXiv:2406.16023.
  37. M. Szegedy, Quantum speed-up of Markov chain based algorithms, in 45th Annual IEEE Symposium on Foundations of Computer Science (IEEE, New York, 2004), pp. 32–41.
  38. J. Leng, Z. Ding, Z. Chen, and L. Lin, Operator-level quantum acceleration of non-Logconcave sampling, Proc. Natl. Acad. Sci. U. S. A. 123, e2512789123 (2026).
  39. D. Aharonov, I. Arad, Z. Landau, and U. Vazirani, Quantum Hamiltonian complexity and the detectability lemma, arXiv:1011.3445.
  40. I. Arad, A. Kitaev, Z. Landau, and U. Vazirani, An area law and sub-exponential algorithm for 1d systems, arXiv:1301.1162.
  41. D. Aharonov, I. Arad, Z. Landau, and U. Vazirani, The detectability lemma and quantum gap amplification, in Proceedings of the forty-first annual ACM symposium on Theory of computing (Association for Computing Machinery, New York, NY, USA, 2009), pp. 417–426.
  42. F. Ascolani, H. Lavenant, and G. Zanella, Entropy contraction of the Gibbs sampler under log-concavity, arXiv:2410.00858.
  43. J. S. Liu, W. H. Wong, and A. Kong, Covariance structure and convergence rate of the Gibbs sampler with various scans, J. R. Stat. Soc.: Series B (Methodological) 57, 157 (1995).
  44. H. Narayanan and P. Srivastava, On the mixing time of coordinate hit-and-run, Comb. Probab. Comput. 31, 320 (2022).
  45. A. Gilyén, Y. Su, G. H. Low, and N. Wiebe, Quantum singular value transformation and beyond: Exponential improvements for quantum matrix arithmetics, in Proceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing, STOC ’19 (ACM, New York, NY, USA, 2019), pp. 193–204.
  46. L. Lin and Y. Tong, Near-optimal ground state preparation, Quantum 4, 372 (2020).
  47. Y. Ge, J. Tura, and J. I. Cirac, Faster ground state preparation and high-precision ground energy estimation with fewer qubits, J. Math. Phys. 60, 022202 (2019).

  48. Here we use  † to denote conjugation under the Hilbert-Schmidt inner product.

  49. A. Anshu, I. Arad, and T. Vidick, Simple proof of the detectability lemma and spectral gap amplification, Phys. Rev. B 93, 205142 (2016).
  50. G. H. Low and N. Wiebe, Hamiltonian simulation in the interaction picture, arXiv:1805.00675.
  51. D. Fang, L. Lin, and Y. Tong, Time-marching based quantum solvers for time-dependent linear differential equations, Quantum 7, 955 (2023).
  52. M. Thibodeau and B. K. Clark, Nearly-frustration-free ground state preparation, Quantum 7, 1084 (2023).
  53. L. Lin and Y. Tong, Optimal polynomial based quantum eigenstate filtering with application to solving quantum linear systems, Quantum 4, 361 (2020).

  54. Strictly speaking, this state is the top eigenstate of the parent Hamiltonian, but they are equivalent up to flipping sign.

  55. M. J. Kastoryano and F. G. S. L. Brandão, Quantum Gibbs samplers: The commuting case, Commun. Math. Phys. 344, 915 (2016).
  56. Y. Hwang and J. Jiang, Gibbs state preparation for commuting Hamiltonian: Mapping to classical Gibbs sampling, Quantum 10, 2209 (2026).
  57. T.-C. Lu, T. H. Hsieh, and T. Grover, Detecting topological order at finite temperature using entanglement negativity, Phys. Rev. Lett. 125, 116801 (2020).
  58. E. B. Davies, Markovian master equations, Commun. Math. Phys. 39, 91 (1974).
  59. E. B. Davies, Markovian master equations. ii, Math. Ann. 219, 147 (1976).
  60. C.-F. Chen, M. J. Kastoryano, and A. Gilyén, An efficient and exact noncommutative quantum Gibbs sampler, arXiv:2311.09207.
  61. E. H. Lieb and D. W. Robinson, The finite group velocity of quantum spin systems, Commun. Math. Phys. 28, 251 (1972).
  62. A. Bakshi, A. Liu, A. Moitra, and E. Tang, High-temperature Gibbs states are unentangled and efficiently preparable, arXiv:2403.16850.
  63. B. T. McDonough, C. Yin, A. Lucas, and C. Zhang, Lieb-Robinson bounds with exponential-in-volume tails, PRX Quantum 6, 040322 (2025).
  64. A. Ramkumar, Y. Cai, Y. Tong, and J. Jiang, High-temperature fermionic Gibbs states are mixtures of Gaussian states, arXiv:2505.09730.
  65. H. Chen, C. Rouzé, J. Chen, J. Jiang, S. O. Scalet, Y. Zhan, G. K. Chan, L. Ying, and Y. Tong, Convergence of the cumulant expansion and polynomial-time algorithm for weakly interacting fermions, arXiv:2512.12010.
  66. A. Zlokapa and B. T. Kiani, SYK thermal expectations are classically easy at any temperature, arXiv:2602.22619.
  67. X. Wang, S. Zhou, X. Wang, Y.-C. Zheng, S. Zhang, and T. Li, Lindbladian simulation with commutator bounds, arXiv:2603.28602.
  68. J. Haah, M. B. Hastings, R. Kothari, and G. H. Low, Quantum algorithm for simulating real time evolution of lattice Hamiltonians, SIAM J. Comput. 52, FOCS18 (2021).
  69. A. M. Childs, Y. Su, M. C. Tran, N. Wiebe, and S. Zhu, Theory of trotter error with commutator scaling, Phys. Rev. X 11, 011020 (2021).

Outline

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PRX Quantum 7, 043025– Published 5 November, 2026
  • Received 1 May 2026
  • Revised 25 August 2026
  • Accepted 7 October 2026
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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.

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