Thermal State Preparation via Rounding Promises

Patrick Rall; Chunhao Wang; Pawel Wocjan

doi:10.22331/q-2023-10-10-1132

Thermal State Preparation via Rounding Promises

Patrick Rall¹, Chunhao Wang², and Pawel Wocjan³

¹IBM Quantum, MIT-IBM Watson AI Lab, Cambridge, Massachusetts 02142, USA
²Department of Computer Science and Engineering, Pennsylvania State University
³IBM Quantum, Thomas J Watson Research Center, Yorktown Heights, New York 10598, USA

Published:	2023-10-10, volume 7, page 1132
Eprint:	arXiv:2210.01670v2
Doi:	https://doi.org/10.22331/q-2023-10-10-1132
Citation:	Quantum 7, 1132 (2023).

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Abstract

A promising avenue for the preparation of Gibbs states on a quantum computer is to simulate the physical thermalization process. The Davies generator describes the dynamics of an open quantum system that is in contact with a heat bath. Crucially, it does not require simulation of the heat bath itself, only the system we hope to thermalize. Using the state-of-the-art techniques for quantum simulation of the Lindblad equation, we devise a technique for the preparation of Gibbs states via thermalization as specified by the Davies generator.
In doing so, we encounter a severe technical challenge: implementation of the Davies generator demands the ability to estimate the energy of the system unambiguously. That is, each energy of the system must be deterministically mapped to a unique estimate. Previous work showed that this is only possible if the system satisfies an unphysical 'rounding promise' assumption. We solve this problem by engineering a random ensemble of rounding promises that simultaneously solves three problems: First, each rounding promise admits preparation of a 'promised' thermal state via a Davies generator. Second, these Davies generators have a similar mixing time as the ideal Davies generator. Third, the average of these promised thermal states approximates the ideal thermal state.

► BibTeX data

@article{Rall2023thermalstate,
  doi = {10.22331/q-2023-10-10-1132},
  url = {https://doi.org/10.22331/q-2023-10-10-1132},
  title = {Thermal {S}tate {P}reparation via {R}ounding {P}romises},
  author = {Rall, Patrick and Wang, Chunhao and Wocjan, Pawel},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {7},
  pages = {1132},
  month = oct,
  year = {2023}
}

► References

[1] Álvaro M Alhambra. Quantum many-body systems in thermal equilibrium. arXiv:2204.08349, 2022.
https://doi.org/10.48550/arXiv.2204.08349
arXiv:2204.08349

[2] Sergey Bravyi, Anirban Chowdhury, David Gosset, and Pawel Wocjan. On the complexity of quantum partition functions. arXiv:2110.15466, 2021.
https://doi.org/10.1038/s41567-022-01742-5
arXiv:2110.15466

[3] Fernando G. S. L. Brandão, Amir Kalev, Tongyang Li, Cedric Yen-Yu Lin, Krysta M. Svore, and Xiaodi Wu. Quantum SDP solvers: Large speed-ups, optimality, and applications to quantum learning. In 46th International Colloquium on Automata, Languages, and Programming (ICALP 2017), volume 132, page 27, 2019.
https://doi.org/10.4230/LIPIcs.ICALP.2019.27

[4] Heinz-Peter Breuer and Francesco Petruccione. The Theory of Open Quantum Systems. Oxford University Press, 2002.
https://doi.org/10.1093/acprof:oso/9780199213900.001.0001

[5] Chi-Fang Chen and Fernando GSL Brandão. Fast thermalization from the eigenstate thermalization hypothesis. arXiv:2112.07646, 2021.
https://doi.org/10.48550/arXiv.2112.07646
arXiv:2112.07646

[6] Chi-Fang Chen, Michael J. Kastoryano, Fernando G. S. L. Brandão, and András Gilyén. Quantum thermal state preparation. arXiv:2303.18224, 2023.
https://doi.org/10.48550/arXiv.2212.02051
arXiv:2303.18224

[7] Anirban Narayan Chowdhury and Rolando D Somma. Quantum algorithms for Gibbs sampling and hitting-time estimation. Quantum Information & Computation, 17(1-2):41–64, 2017.
https://doi.org/10.26421/QIC17.1-2-3

[8] Richard Cleve and Chunhao Wang. Efficient quantum algorithms for simulating Lindblad evolution. arXiv:1612.09512 Proceedings of the 44th International Colloquium on Automata, Languages, and Programming (ICALP 2017), 2017.
https://doi.org/10.48550/arXiv.1612.09512
arXiv:1612.09512

[9] Edward Brian Davies. Quantum Theory of Open Systems. Academic Press, 1976.

[10] Edward Brian Davies. Generators of dynamical semigroups. Journal of Functional Analysis, 34(3):421–432, 1979.
https://doi.org/10.1016/0022-1236(79)90085-5

[11] András Gilyén, Yuan Su, Guang Hao Low, and Nathan 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 2019), pages 193–204, 2019.
https://doi.org/10.1145/3313276.3316366

[12] Zoe Holmes, Gopikrishnan Muraleedharan, Rolando D. Somma, Yigit Subasi, and Burak Şahinoğlu. Quantum algorithms from fluctuation theorems: Thermal-state preparation. Quantum, 6:825, October 2022.
https://doi.org/10.22331/q-2022-10-06-825

[13] Mária Kieferová and Nathan Wiebe. Tomography and generative training with quantum Boltzmann machines. Physical Review A, 96(6):062327, 2017.
https://doi.org/10.1103/PhysRevA.96.062327

[14] Guang Hao Low and Isaac L Chuang. Hamiltonian simulation by uniform spectral amplification. arXiv:1707.05391, 2017.
https://doi.org/10.48550/arXiv.1707.05391
arXiv:1707.05391

[15] Goran Lindblad. On the generators of quantum dynamical semigroups. Communications in Mathematical Physics, 48(2):119–130, 1976.
https://doi.org/10.1007/BF01608499

[16] Xiantao Li and Chunhao Wang. Simulating Markovian open quantum systems using higher-order series expansion. 2212.02051, 2022.
https://doi.org/10.48550/arXiv.2212.02051

[17] John M Martyn, Zane M Rossi, Andrew K Tan, and Isaac L Chuang. Grand unification of quantum algorithms. PRX Quantum, 2(4):040203, 2021.
https://doi.org/10.1103/PRXQuantum.2.040203

[18] Davide Nigro. On the uniqueness of the steady-state solution of the Lindblad–gorini–Kossakowski–Sudarshan equation. Journal of Statistical Mechanics: Theory and Experiment, 2019(4):043202, 2019.
https://doi.org/10.1088/1742-5468/ab0c1c

[19] David Poulin and Pawel Wocjan. Sampling from the thermal quantum Gibbs state and evaluating partition functions with a quantum computer. Physical Review Letters, 103(22):220502, 2009.
https://doi.org/10.1103/PhysRevLett.103.220502

[20] Patrick Rall. Faster coherent quantum algorithms for phase, energy, and amplitude estimation. Quantum, 5:566, 2021.
https://doi.org/10.22331/q-2021-10-19-566

[21] S. Slezak and E. Crosson. Eigenstate thermalization and quantum Metropolis sampling, 2022. Presentation at QIP 2022. https://youtu.be/by4rvu7RMtY.
https://youtu.be/by4rvu7RMtY

[22] Herbert Spohn. An algebraic condition for the approach to equilibrium of an open $n$-level system. Letters in Mathematical Physics, 2(1):33–38, 1977.
https://doi.org/10.1007/BF00420668

[23] Kristan Temme, Tobias J Osborne, Karl G Vollbrecht, David Poulin, and Frank Verstraete. Quantum Metropolis sampling. Nature, 471(7336):87–90, 2011.
https://doi.org/10.1038/nature09770

[24] Joran van Apeldoorn and András Gilyén. Improvements in quantum SDP-solving with applications. In Proceedings of the 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019), 2019.
https://doi.org/10.4230/LIPIcs.ICALP.2019.99

[25] Joran Van Apeldoorn, András Gilyén, Sander Gribling, and Ronald de Wolf. Quantum SDP-solvers: Better upper and lower bounds. Quantum, 4:230, 2020.
https://doi.org/10.22331/q-2020-02-14-230

[26] John Watrous. The Theory of Quantum Information. Cambridge University Press, 2018.
https://doi.org/10.5555/3240076

[27] Pawel Wocjan and Kristan Temme. Szegedy walk unitaries for quantum maps. Commun. Math. Phys., 2023.
https://doi.org/10.1007/s00220-023-04797-4

Cited by

[1] Shi-Xin Zhang, Jiaqi Miao, and Chang-Yu Hsieh, "Variational post-selection for ground states and thermal states simulation", Quantum Science and Technology 10 1, 015028 (2025).

[2] Alev Orfi and Dries Sels, "Bounding the speedup of the quantum-enhanced Markov-chain Monte Carlo algorithm", Physical Review A 110 5, 052414 (2024).

[3] Hao-En Li, Yongtao Zhan, and Lin Lin, "Dissipative ground state preparation in ab initio electronic structure theory", npj Quantum Information 11 1, 183 (2025).

[4] Štěpán Šmíd, Richard Meister, Mario Berta, and Roberto Bondesan, "Polynomial-time quantum Gibbs sampling for the weak and strong coupling regime of the Fermi-Hubbard model at any temperature", Nature Communications 16 1, 10736 (2025).

[5] Di Fang, Jianfeng Lu, and Yu Tong, "Mixing Time of Open Quantum Systems via Hypocoercivity", Physical Review Letters 134 14, 140405 (2025).

[6] Zhiyan Ding, Bowen Li, and Lin Lin, "Efficient Quantum Gibbs Samplers with Kubo–Martin–Schwinger Detailed Balance Condition", Communications in Mathematical Physics 406 3, 67 (2025).

[7] Ángela Capel, Paul Gondolf, Jan Kochanowski, and Cambyse Rouzé, "Quasi-optimal Sampling from Gibbs States via Non-commutative Optimal Transport Metrics", Annales Henri Poincaré (2025).

[8] Cambyse Rouzé, Daniel Stilck França, and Álvaro M. Alhambra, Proceedings of the 57th Annual ACM Symposium on Theory of Computing 1488 (2025) ISBN:9798400715105.

[9] Thiago Bergamaschi, Chi-Fang Chen, and Yunchao Liu, 2024 IEEE 65th Annual Symposium on Foundations of Computer Science (FOCS) 1063 (2024) ISBN:979-8-3315-1674-1.

[10] Zhiyan Ding, Xiantao Li, and Lin Lin, "Simulating Open Quantum Systems Using Hamiltonian Simulations", PRX Quantum 5 2, 020332 (2024).

[11] Peter J. Eder, Jernej Rudi Finžgar, Sarah Braun, and Christian B. Mendl, "Quantum dissipative search via Lindbladians", Physical Review A 111 4, 042430 (2025).

[12] Jinkang Guo, Oliver Hart, Chi-Fang Chen, Aaron J. Friedman, and Andrew Lucas, "Designing open quantum systems with known steady states: Davies generators and beyond", Quantum 9, 1612 (2025).

[13] Lin Lin, "Dissipative preparation of many-body quantum states: Toward practical quantum advantage", APL Computational Physics 1 1, 010901 (2025).

[14] Nikita Guseynov, Xiajie Huang, and Nana Liu, "Quantum framework for simulating linear PDEs with Robin boundary conditions", Quantum Science and Technology 11 1, 015057 (2026).

[15] Cambyse Rouzé, Daniel Stilck França, and Álvaro M. Alhambra, "Efficient thermalization and universal quantum computing with quantum Gibbs samplers", Nature Physics (2026).

[16] Annie Ray, Esha Swaroop, Ningping Cao, Michael Vasmer, and Anirban Chowdhury, "Quasiprobabilistic Imaginary-Time Evolution on Quantum Computers", Quantum Information & Computation 26 1, 89 (2026).

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

[18] Dominik Hahn, S. A. Parameswaran, and Benedikt Placke, "Efficient quantum thermal state preparation via local driving: Lindbladian simulation with provable guarantees", Physical Review B 114 1, 014302 (2026).

[19] Longfei Chang, Zhendong Li, and Wei-Hai Fang, "Quantum-Assisted Variational Monte Carlo", Precision Chemistry 3 9, 541 (2025).

[20] Elias Zapusek, Kristina Kirova, Walter Hahn, Michael Marthaler, and Florentin Reiter, "Variational quantum thermalizers based on weakly-symmetric nonunitary multi-qubit operations", Quantum Science and Technology 11 2, 025006 (2026).

[21] Etienne Granet and Henrik Dreyer, "Adiabatic preparation of thermal states and entropy-noise relation on noisy quantum computers", npj Quantum Information 12 1, 105 (2026).

[22] Chi-Fang Chen, Hsin-Yuan Huang, John Preskill, and Leo Zhou, "Local minima in quantum systems", Nature Physics 21 4, 654 (2025).

[23] Zhiyan Ding, Lin Lin, Yilun Yang, and Ruizhe Zhang, "Quantum Filtering and Analysis of Multiplicities in Eigenvalue Spectra", PRX Quantum 7 2, 020318 (2026).

[24] Anas Ait Chlih, Samira Elghaayda, Nabil Habiballah, and Mostafa Mansour, "Unveiling non-classical correlations, quantum coherence, and steering measurement uncertainty in two-qubit $$XY-\Gamma$$ spin model", Optical and Quantum Electronics 56 8, 1370 (2024).

[25] Jiaqi Leng, Zhiyan Ding, Zherui Chen, and Lin Lin, "Operator-level quantum acceleration of non-logconcave sampling", Proceedings of the National Academy of Sciences 123 8, e2512789123 (2026).

[26] Hamza Fawzi, Omar Fawzi, and Samuel O. Scalet, "Certified algorithms for equilibrium states of local quantum Hamiltonians", Nature Communications 15 1, 7394 (2024).

[27] Thiago Bergamaschi and Chi-Fang Chen, Proceedings of the 58th Annual ACM Symposium on Theory of Computing 835 (2026) ISBN:9798400725364.

[28] Shi-Kang Sun and Shu Chen, "Emergence of the Gibbs ensemble as a steady state in Lindbladian dynamics", Physical Review B 110 13, 134301 (2024).

[29] Tongyang Li, "Quantum computers quickly find local minima", Nature Physics 21 4, 512 (2025).

[30] 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 1, 010361 (2025).

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

[32] Eric Brunner, Luuk Coopmans, Gabriel Matos, Matthias Rosenkranz, Frederic Sauvage, and Yuta Kikuchi, "Lindblad engineering for quantum Gibbs state preparation under the eigenstate thermalization hypothesis", Quantum 9, 1843 (2025).

[33] 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).

[34] Alev Orfi and Dries Sels, "Barriers to efficient mixing of quantum-enhanced Markov chains", Physical Review A 110 5, 052434 (2024).

[35] Yuri Alexeev, Maximilian Amsler, Marco Antonio Barroca, Sanzio Bassini, Torey Battelle, Daan Camps, David Casanova, Young Jay Choi, Frederic T. Chong, Charles Chung, Christopher Codella, Antonio D. Córcoles, James Cruise, Alberto Di Meglio, Ivan Duran, Thomas Eckl, Sophia Economou, Stephan Eidenbenz, Bruce Elmegreen, Clyde Fare, Ismael Faro, Cristina Sanz Fernández, Rodrigo Neumann Barros Ferreira, Keisuke Fuji, Bryce Fuller, Laura Gagliardi, Giulia Galli, Jennifer R. Glick, Isacco Gobbi, Pranav Gokhale, Salvador de la Puente Gonzalez, Johannes Greiner, Bill Gropp, Michele Grossi, Emanuel Gull, Burns Healy, Matthew R. Hermes, Benchen Huang, Travis S. Humble, Nobuyasu Ito, Artur F. Izmaylov, Ali Javadi-Abhari, Douglas Jennewein, Shantenu Jha, Liang Jiang, Barbara Jones, Wibe Albert de Jong, Petar Jurcevic, William Kirby, Stefan Kister, Masahiro Kitagawa, Joel Klassen, Katherine Klymko, Kwangwon Koh, Masaaki Kondo, Dog̃a Murat Kürkçüog̃lu, Krzysztof Kurowski, Teodoro Laino, Ryan Landfield, Matt Leininger, Vicente Leyton-Ortega, Ang Li, Meifeng Lin, Junyu Liu, Nicolas Lorente, Andre Luckow, Simon Martiel, Francisco Martin-Fernandez, Margaret Martonosi, Claire Marvinney, Arcesio Castaneda Medina, Dirk Merten, Antonio Mezzacapo, Kristel Michielsen, Abhishek Mitra, Tushar Mittal, Kyungsun Moon, Joel Moore, Sarah Mostame, Mario Motta, Young-Hye Na, Yunseong Nam, Prineha Narang, Yu-ya Ohnishi, Daniele Ottaviani, Matthew Otten, Scott Pakin, Vincent R. Pascuzzi, Edwin Pednault, Tomasz Piontek, Jed Pitera, Patrick Rall, Gokul Subramanian Ravi, Niall Robertson, Matteo A.C. Rossi, Piotr Rydlichowski, Hoon Ryu, Georgy Samsonidze, Mitsuhisa Sato, Nishant Saurabh, Vidushi Sharma, Kunal Sharma, Soyoung Shin, George Slessman, Mathias Steiner, Iskandar Sitdikov, In-Saeng Suh, Eric D. Switzer, Wei Tang, Joel Thompson, Synge Todo, Minh C. Tran, Dimitar Trenev, Christian Trott, Huan-Hsin Tseng, Norm M. Tubman, Esin Tureci, David García Valiñas, Sofia Vallecorsa, Christopher Wever, Konrad Wojciechowski, Xiaodi Wu, Shinjae Yoo, Nobuyuki Yoshioka, Victor Wen-zhe Yu, Seiji Yunoki, Sergiy Zhuk, and Dmitry Zubarev, "Quantum-centric supercomputing for materials science: A perspective on challenges and future directions", Future Generation Computer Systems 160, 666 (2024).

[36] Zhiyan Ding, Marius Junge, Philipp Schleich, and Peixue Wu, "Lower Bound for Simulation Cost of Open Quantum Systems: Lipschitz Continuity Approach", Communications in Mathematical Physics 406 3, 60 (2025).

[37] Tong Jiang, Jinghong Zhang, Moritz K. A. Baumgarten, Meng-Fu Chen, Hieu Q. Dinh, Aadithya Ganeshram, Nishad Maskara, Anton Ni, and Joonho Lee, "Walking through Hilbert Space with Quantum Computers", Chemical Reviews 125 9, 4569 (2025).

[38] Koray Aydoğan, Anthony W. Schlimgen, and Kade Head-Marsden, "Stabilizing steady-state properties of open quantum systems with parameter engineering", Physical Review Research 7 2, 023057 (2025).

[39] Kristan Temme and Pawel Wocjan, "Quantized Markov Chain Couplings that Prepare Qsamples", Quantum 9, 1951 (2025).

[40] Dominik Hahn, Ryan Sweke, Abhinav Deshpande, and Oles Shtanko, "Efficient Quantum Gibbs Sampling with Local Circuits", PRX Quantum 7 2, 020314 (2026).

[41] Zhiyan Ding, Chi-Fang Chen, and Lin Lin, "Single-ancilla ground state preparation via Lindbladians", Physical Review Research 6 3, 033147 (2024).

[42] Yongtao Zhan, Zhiyan Ding, Jakob Huhn, Johnnie Gray, John Preskill, Garnet Kin-Lic Chan, and Lin Lin, "Rapid Quantum Ground State Preparation via Dissipative Dynamics", Physical Review X 16 1, 011004 (2026).

[43] Norhan M. Eassa, Mahmoud M. Moustafa, Arnab Banerjee, and Jeffrey Cohn, "Gibbs state sampling via cluster expansions", npj Quantum Information 10 1, 97 (2024).

[44] Mirko Consiglio, Jacopo Settino, Andrea Giordano, Carlo Mastroianni, Francesco Plastina, Salvatore Lorenzo, Sabrina Maniscalco, John Goold, and Tony J. G. Apollaro, "Variational Gibbs state preparation on noisy intermediate-scale quantum devices", Physical Review A 110 1, 012445 (2024).

[45] Anas Ait Chlih and Atta ur Rahman, "Nonclassicality and teleportation fidelity probes in amplitude-tailored superconducting charge qubits", Physica A: Statistical Mechanics and its Applications 650, 129986 (2024).

[46] Chi-Fang Chen, Hsin-Yuan Huang, John Preskill, and Leo Zhou, Proceedings of the 56th Annual ACM Symposium on Theory of Computing 1323 (2024) ISBN:9798400703836.

[47] Pablo Páez-Velasco, Niclas Schilling, Samuel O. Scalet, Frank Verstraete, and Ángela Capel, "Efficient and simple Gibbs state preparation of the two-dimensional toric code via duality to classical Ising chains", Physical Review Research 8 2, 023277 (2026).

[48] Alexander M. Dalzell, Sam McArdle, Mario Berta, Przemyslaw Bienias, Chi-Fang Chen, András Gilyén, Connor T. Hann, Michael J. Kastoryano, Emil T. Khabiboulline, Aleksander Kubica, Grant Salton, Samson Wang, and Fernando G. S. L. Brandão, "Quantum algorithms: A survey of applications and end-to-end complexities", arXiv:2310.03011, (2023).

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

[50] Mirko Consiglio, "Variational Quantum Algorithms for Many-Body Systems", arXiv:2502.11985, (2025).

[51] Yudai Suzuki, Marek Gluza, Jeongrak Son, Bi Hong Tiang, Nelly H. Y. Ng, and Zoë Holmes, "Grover's algorithm is an approximation of imaginary-time evolution", arXiv:2507.15065, (2025).

[52] Xiantao Li and Chunhao Wang, "Simulating Markovian open quantum systems using higher-order series expansion", arXiv:2212.02051, (2022).

[53] Mirko Consiglio, "Variational Quantum Algorithms for Gibbs State Preparation", arXiv:2305.17713, (2023).

[54] Maria Demidik, Cenk Tüysüz, Michele Grossi, and Karl Jansen, "Sample-based training of quantum generative models", arXiv:2511.11802, (2025).

[55] Arthur J. Parzygnat and Andrew Vlasic, "Quantum encodings that preserve persistent homology", arXiv:2605.28927, (2026).

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