Efficient thermalization and universal quantum computing with quantum Gibbs samplers

Nature Physics (2026)Cite this article The preparation of thermal states of matter is a crucial task in quantum simulation. Here we prove that recently introduced, efficiently implementable dissipative evolution thermalizes to the Gibbs state in time scaling polynomially with system size at high-enough temperatures for any Hamiltonian that satisfies a Lieb–Robinson bound, such as local Hamiltonians on a lattice. Furthermore, we show the efficient adiabatic preparation of the associated purifications or ‘thermofield double’ states. These results establish the efficient preparation of high-temperature Gibbs states and their purifications. In the low-temperature regime, we show that implementing this family of dissipative evolutions for inverse temperature polynomial in the system’s size is computationally equivalent to polynomial-time quantum computations. On a technical level, for high temperatures, our proof makes use of the mapping of the generator of the evolution into a Hamiltonian, and then connecting its convergence to that of the infinite temperature limit. For low temperature, we instead perform a perturbation at zero temperature and resort to circuit-to-Hamiltonian mappings akin to the proof of universality of quantum adiabatic computing. Taken together, our results show that a family of quasi-local dissipative evolutions efficiently prepares a large class of quantum many-body states of interest, and has the potential to mirror the success of classical Monte Carlo methods for quantum many-body systems.This is a preview of subscription content, access via your institution Access Nature and 54 other Nature Portfolio journals Get Nature+, our best-value online-access subscription $32.99 / 30 days cancel any timeSubscribe to this journal Receive 12 print issues and online access $259.00 per yearonly $21.58 per issueBuy this articleUSD 39.95Prices may be subject to local taxes which are calculated during checkoutLevin, D. A. & Peres, Y. Markov Chains and Mixing Times (American Mathematical Society, 2017).Brooks, S., Gelman, A., Jones, G. & Meng, X.-L. Handbook of Markov Chain Monte Carlo (CRC Press, 2011).Martinelli, F. Lectures on Glauber dynamics for discrete spin models. In Lectures on Probability Theory and Statistics 93–191 (Springer, 1999).Temme, K., Osborne, T. J., Vollbrecht, K. G., Poulin, D. & Verstraete, F. Quantum Metropolis sampling. Nature 471, 87–90 (2011).Article ADS Google Scholar Brandão, F. G. S. L. & Kastoryano, M. J. Finite correlation length implies efficient preparation of quantum thermal states. Commun. Math. Phys. 365, 1–16 (2019).Article MathSciNet Google Scholar Kastoryano, M. J. & Brandão, F. G. S. L. Quantum Gibbs samplers: the commuting case. Commun. Math. Phys. 344, 915–957 (2016).Article MathSciNet Google Scholar Bardet, I., Capel, A., Lucia, A., Pérez-García, D. & Rouzé, C. On the modified logarithmic Sobolev inequality for the heat-bath dynamics for 1D systems. J. Math. Phys. 62, 061901 (2021).Article ADS MathSciNet Google Scholar Capel, Á., Rouzé, C. & França, D. S. The modified logarithmic Sobolev inequality for quantum spin systems: classical and commuting nearest neighbour interactions. Preprint at https://arxiv.org/abs/2009.11817 (2021).Bardet, I. et al. Entropy decay for Davies semigroups of a one dimensional quantum lattice. Commun. Math. Phys. 405, 42 (2024).Article ADS MathSciNet Google Scholar Bardet, I. et al. Rapid thermalization of spin chain commuting Hamiltonians. Phys. Rev. Lett. 130, 060401 (2023).Article ADS MathSciNet Google Scholar Zhang, D., Bosse, J. L. & Cubitt, T. Dissipative quantum Gibbs sampling. Preprint at https://arxiv.org/abs/2304.04526 (2023).Shtanko, O. & Movassagh, R. Preparing thermal states on noiseless and noisy programmable quantum processors. Preprint at https://arxiv.org/abs/2112.14688 (2023).Chen, C.-F., Kastoryano, M., Brandão, F. G. S. L. & Gilyén, A. Efficient quantum thermal simulation. Nature 646, 561 (2025).Article ADS Google Scholar Nachtergaele, B. & Sims, R. Lieb-Robinson bounds and the exponential clustering theorem. Commun. Math. Phys. 265, 119–130 (2006).Article ADS MathSciNet Google Scholar Cottrell, W., Freivogel, B., Hofman, D. M. & Lokhande, S. F. How to build the thermofield double state. J.

High Energy Phys. 2019, 58 (2019).Article MathSciNet Google Scholar Sundar, B., Elben, A., Joshi, L. K. & Zache, T. V. Proposal for measuring out-of-time-ordered correlators at finite temperature with coupled spin chains. New J. Phys. 24, 023037 (2022).Article ADS Google Scholar Brown, A. R. et al. Quantum gravity in the lab. I. Teleportation by size and traversable wormholes. PRX Quantum 4, 010320 (2023).Article ADS Google Scholar Chen, C.-F., Huang, H.-Y., Preskill, J. & Zhou, L. Local minima in quantum systems. Nat. Phys. 21, 654 (2025).Article Google Scholar Michalakis, S. & Zwolak, J. P. Stability of frustration-free Hamiltonians. Commun. Math. Phys. 322, 277 (2013).Article ADS MathSciNet Google Scholar Caputo, P., Martinelli, F., Simenhaus, F. & Toninelli, F. L. Zero temperature stochastic 3D ising model and dimer covering fluctuations: a first step towards interface mean curvature motion. Commun. Pure Appl. Math. 64, 778–831 (2011).Article MathSciNet Google Scholar Feiguin, A. E. & Klich, I. Hermitian and non-Hermitian thermal Hamiltonians. Preprint at https://arxiv.org/abs/1308.0756 (2013).Ambainis, A. & Regev, O. An elementary proof of the quantum adiabatic theorem. Preprint at https://arxiv.org/abs/quant-ph/0411152 (2006).Haah, J., Hastings, M. B., Kothari, R. & Low, G. H. Quantum algorithm for simulating real time evolution of lattice Hamiltonians. SIAM J. Comput. 52, FOCS18 (2021).MathSciNet Google Scholar Nezami, S. et al. Quantum gravity in the lab: teleportation by size and traversable wormholes, part II. PRX Quantum 4, 010321 (2023).Article ADS Google Scholar Wu, J. & Hsieh, T. H. Variational thermal quantum simulation via thermofield double states. Phys. Rev. Lett. 123, 220502 (2019).Article ADS Google Scholar Martyn, J. & Swingle, B. Product spectrum ansatz and the simplicity of thermal states. Phys. Rev. A 100, 032107 (2019).Article ADS MathSciNet Google Scholar Chowdhury, A. N., Low, G. H. & Wiebe, N. A variational quantum algorithm for preparing quantum Gibbs states. Preprint at https://arxiv.org/abs/2002.00055 (2020).Su, V. P. Variational preparation of the thermofield double state of the Sachdev-Ye-Kitaev model. Phys. Rev. A 104, 012427 (2021).Article ADS MathSciNet Google Scholar Faílde, D., Santos-Suárez, J., Herrera-Martí, D. A. & Mas, J. Hamiltonian forging of a thermofield double. Phys. Rev. A 111, 012432 (2025).Article ADS MathSciNet Google Scholar Zhu, D. et al. Generation of thermofield double states and critical ground states with a quantum computer. Proc. Natl Acad. Sci. USA 117, 25402–25406 (2020).Article ADS MathSciNet Google Scholar Aharonov, D. et al. Adiabatic quantum computation is equivalent to standard quantum computation. SIAM Rev. 50, 755–787 (2008).Article ADS MathSciNet Google Scholar Oliveira, R. & Terhal, B. M. The complexity of quantum spin systems on a two-dimensional square lattice. Quantum Inf. Comput. 8, 900–924 (2008).MathSciNet Google Scholar Kempe, J., Kitaev, A. & Regev, O. The complexity of the local Hamiltonian problem. SIAM J. Comput. 35, 1070–1097 (2006).Article MathSciNet Google Scholar Ding, Z., Li, B. & Lin, L. Efficient quantum Gibbs samplers with Kubo-Martin-Schwinger detailed balance condition. Commun. Math. Phys. 406, – (2025).Article MathSciNet Google Scholar Scandi, M. & Alhambra, Á. M. Thermalization in open many-body systems and KMS detailed balance. Phys. Rev. X 16, 011040 (2026).

Google Scholar Kashyap, V., Styliaris, G., Mouradian, S., Cirac, J. I. & Trivedi, R. Accuracy guarantees and quantum advantage in analog open quantum simulation with and without noise. Phys. Rev. X 15, 021017 (2025).

Google Scholar Poulin, D. & Wocjan, P. Sampling from the thermal quantum Gibbs state and evaluating partition functions with a quantum computer. Phys. Rev. Lett. 103, 220502 (2009).Article ADS MathSciNet Google Scholar Chowdhury, A. N. & Somma, R. D. Quantum algorithms for Gibbs sampling and hitting-time estimation. Quantum Inf. Comput. 17, 41–68 (2017).MathSciNet Google Scholar Holmes, Z., Muraleedharan, G., Somma, R. D., Subasi, Y. & Şahinoğlu, B. Quantum algorithms from fluctuation theorems: thermal-state preparation. Quantum 6, 825 (2022).Article Google Scholar Wang, Y., Li, G. & Wang, X. Variational quantum gibbs state preparation with a truncated Taylor series. Phys. Rev. Appl. 16, 054035 (2021).Article ADS Google Scholar Chen, C.-F. & Brandão, F. G. S. L. Fast thermalization from the eigenstate thermalization hypothesis. Preprint at https://arxiv.org/abs/2112.07646 (2023).Li, X. & Wang, C. Simulating Markovian open quantum systems using higher-order series expansion. In Proc. 50th International Colloquium on Automata, Languages, and Programming Vol. 261 87:1–87:20 (Schloss Dagstuhl, 2023).Rall, P., Wang, C. & Wocjan, P. Thermal state preparation via rounding promises. Quantum 7, 1132 (2023).Article Google Scholar Bilgin, E. & Boixo, S. Preparing thermal states of quantum systems by dimension reduction. Phys. Rev. Lett. 105, 170405 (2010).Article ADS Google Scholar Ge, Y., Molnár, A. & Cirac, J. I. Rapid adiabatic preparation of injective projected entangled pair states and Gibbs states. Phys. Rev. Lett. 116, 080503 (2016).Article ADS Google Scholar Helmuth, T., Perkins, W. & Regts, G. Algorithmic Pirogov–Sinai theory. Probab. Theory Relat. Fields 176, 851–895 (2020).Article MathSciNet Google Scholar Mann, R. L. & Helmuth, T. Efficient algorithms for approximating quantum partition functions. J. Math. Phys. 62, – (2021).Article MathSciNet Google Scholar Haah, J., Kothari, R. & Tang, E. Learning quantum Hamiltonians from high-temperature Gibbs states and real-time evolutions. Nat. Phys. 20, 1027–1031 (2024).Article Google Scholar Mozgunov, E. & Lidar, D. Completely positive master equation for arbitrary driving and small level spacing. Quantum 4, 227 (2020).Article Google Scholar Nathan, F. & Rudner, M. S. Universal Lindblad equation for open quantum systems. Phys. Rev. B 102, 115109 (2020).Article ADS Google Scholar Soret, A., Cavina, V. & Esposito, M. Thermodynamic consistency of quantum master equations. Phys. Rev. A 106, 062209 (2022).Article ADS MathSciNet Google Scholar Coser, A. & Pérez-García, D. Classification of phases for mixed states via fast dissipative evolution. Quantum 3, 174 (2019).Article Google Scholar Rakovszky, T., Gopalakrishnan, S. & von Keyserlingk, C. Defining stable phases of open quantum systems. Phys. Rev. X 14, 041031 (2024).

Google Scholar Bakshi, A., Liu, A., Moitra, A. & Tang, E. High-temperature Gibbs states are unentangled and efficiently preparable. In Proc. 65th Annual Symposium on Foundations of Computer Science 1027–1036 (IEEE, 2024).Tran, M. C. et al. Lieb-Robinson light cone for power-law interactions. Phys. Rev. Lett. 127, 160401 (2021).Article ADS MathSciNet Google Scholar Verstraete, F., Wolf, M. M. & Ignacio Cirac, J. Quantum computation and quantum-state engineering driven by dissipation. Nat. Phys. 5, 633–636 (2009).Article Google Scholar Kraus, B. et al. Preparation of entangled states by quantum Markov processes. Phys. Rev. A 78, 042307 (2008).Article ADS Google Scholar Cubitt, T. S., Lucia, A., Michalakis, S. & Perez-Garcia, D. Stability of local quantum dissipative systems. Commun. Math. Phys. 337, 1275–1315 (2015).Article ADS MathSciNet Google Scholar Borregaard, J., Christandl, M. & Stilck França, D. Noise-robust exploration of many-body quantum states on near-term quantum devices. npj Quantum Inf. 7, 45 (2021).Article ADS Google Scholar Kim, I. H. & Swingle, B. Robust entanglement renormalization on a noisy quantum computer. Preprint at https://arxiv.org/abs/1711.07500 (2017).Childs, A. M., Farhi, E. & Preskill, J. Robustness of adiabatic quantum computation. Phys. Rev. A 65, 012322 (2001).Article ADS Google Scholar Young, K. C., Sarovar, M. & Blume-Kohout, R. Error suppression and error correction in adiabatic quantum computation: techniques and challenges. Phys. Rev. X 3, 041013 (2013).

Google Scholar Davies, E. B. Markovian master equations. Commun. Math. Phys. 39, 91–110 (1974).Article ADS MathSciNet Google Scholar Bravyi, S., Hastings, M. B. & Michalakis, S. Topological quantum order: stability under local perturbations. J. Math. Phys. 51, 093512 (2010).Article ADS MathSciNet Google Scholar Hastings, M. B. & Koma, T. Spectral gap and exponential decay of correlations. Commun. Math. Phys. 265, 781–804 (2006).Article ADS MathSciNet Google Scholar Anthony Chen, C.-F., Lucas, A. & Yin, C. Speed limits and locality in many-body quantum dynamics. Rep. Progr. Phys. 86, 116001 (2023).Article ADS MathSciNet Google Scholar Rouzé, C., França, D. S. & Alhambra, Á. M. Efficient thermalization and universal quantum computing with quantum Gibbs samplers. In Proc. 57th Annual ACM Symposium on Theory of Computing 1488–1495 (ACM, 2025).Download referencesWe acknowledge useful discussions with A. Chen, and thank him for finding a gap in a previous version of Theorem 3. Á.M.A. acknowledges useful discussions with T. Hsieh and D. Gosset. C.R. would like to thank S. Warzel for fruitful discussions on Hamiltonian gaps, and acknowledges financial support from the ANR project QTraj (ANR-20-CE40-0024-01) of the French National Research Agency (ANR). D.S.F. acknowledges funding from the European Union under grant agreement number 101080142 and the project EQUALITY and from the Novo Nordisk Foundation (grant number NNF20OC0059939 Quantum for Life). Á.M.A. acknowledges support from the Spanish Agencia Estatal de Investigacion through the grants ‘IFT Centro de Excelencia Severo Ochoa CEX2020-001007-S’, ‘Proyecto de Colaboración Internacional PCI2024-153448’, ‘PID2023-150847NA-I00’ and ‘Ramón y Cajal RyC2021-031610-I’, financed by MCIN/AEI/10.13039/501100011033 and the European Union NextGenerationEU/PRTR. This project was funded within the QuantERA II Programme that has received funding from the EU’s H2020 research and innovation programme under grant agreement number 101017733. Part of the paper was previously published as an STOC abstract67.Inria, Télécom Paris—LTCI, Institut Polytechnique de Paris, Palaiseau, FranceCambyse RouzéUniv Lyon, ENS Lyon, UCBL, CNRS, Inria, Lyon Cedex 07, FranceDaniel Stilck FrançaDepartment of Mathematical Sciences, University of Copenhagen, Copenhagen, DenmarkDaniel Stilck FrançaInstituto de Física Téorica UAM/CSIC, Madrid, SpainÁlvaro M. AlhambraSearch author on:PubMed Google ScholarSearch author on:PubMed Google ScholarSearch author on:PubMed Google ScholarAll authors contributed equally to the research and preparation of the paper.Correspondence to Cambyse Rouzé, Daniel Stilck França or Álvaro M. Alhambra.The authors declare no competing interests.Nature Physics thanks Roberto Bondesan and Anirban Chowdhury for their contribution to the peer review of this work.Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.Supplementary Sections A–C.Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.Reprints and permissionsRouzé, C., Stilck França, D. & Alhambra, Á.M. Efficient thermalization and universal quantum computing with quantum Gibbs samplers. Nat. Phys. (2026). https://doi.org/10.1038/s41567-026-03246-yDownload citationReceived: 24 February 2025Accepted: 05 March 2026Published: 15 April 2026Version of record: 15 April 2026DOI: https://doi.org/10.1038/s41567-026-03246-yAnyone you share the following link with will be able to read this content:Sorry, a shareable link is not currently available for this article. Provided by the Springer Nature SharedIt content-sharing initiative