A Randomized Method for Simulating Lindblad Equations and Thermal State Preparation

Hongrui Chen; Bowen Li; Jianfeng Lu; Lexing Ying

doi:10.22331/q-2025-11-20-1917

A Randomized Method for Simulating Lindblad Equations and Thermal State Preparation

Hongrui Chen¹, Bowen Li², Jianfeng Lu³, and Lexing Ying⁴

¹Department of Mathematics, Stanford University, Stanford, CA 94305
²Department of Mathematics, City University of Hong Kong, Kowloon Tong, Hong Kong SAR
³Departments of Mathematics, Physics, and Chemistry, Duke University, Durham, NC 27708
⁴Department of Mathematics and ICME, Stanford University, Stanford, CA 94305

Published:	2025-11-20, volume 9, page 1917
Editor:	Jin-Peng Liu
Eprint:	arXiv:2407.06594v5
Doi:	https://doi.org/10.22331/q-2025-11-20-1917
Citation:	Quantum 9, 1917 (2025).

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

Abstract

We study a qDRIFT-type randomized method to simulate Lindblad dynamics by decomposing its generator into an ensemble of Lindbladians, $\mathcal{L} = \sum_{a \in \mathcal{A}} \mathcal{L}_a$, where each $\mathcal{L}_a$ comprises a simple Hamiltonian and a single jump operator. Assuming an efficient quantum simulation is available for the Lindblad evolution $e^{t\mathcal{L}_a}$, we implement $e^{t\mathcal{L}_a}$ for a randomly sampled $\mathcal{L}_a$ at each time step according to a probability distribution $\mu$ over the ensemble $\{\mathcal{L}_a\}_{a \in \mathcal{A}}$. This randomized strategy reduces the quantum cost of simulating Lindblad dynamics, particularly in quantum many-body systems with a large or even infinite number of jump operators.
Our contributions are two-fold. First, we provide a detailed convergence analysis of the proposed randomized method, covering both average and typical algorithmic realizations. This analysis extends the known results for the random product formula from closed systems to open systems, ensuring rigorous performance guarantees. Second, based on the random product approximation, we derive a new quantum Gibbs sampler algorithm that utilizes jump operators sampled from a Clifford-random circuit. This generator (i) can be efficiently implemented using our randomized algorithm, and (ii) exhibits a spectral gap lower bound that depends on the spectrum of the Hamiltonian. Our results present a new instance of a class of Hamiltonians for which the thermal states can be efficiently prepared using a quantum Gibbs sampling algorithm.

A talk at Simons Institute

► BibTeX data

@article{Chen2025randomizedmethod,
  doi = {10.22331/q-2025-11-20-1917},
  url = {https://doi.org/10.22331/q-2025-11-20-1917},
  title = {A {R}andomized {M}ethod for {S}imulating {L}indblad {E}quations and {T}hermal {S}tate {P}reparation},
  author = {Chen, Hongrui and Li, Bowen and Lu, Jianfeng and Ying, Lexing},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {9},
  pages = {1917},
  month = nov,
  year = {2025}
}

► References

[1] Heinz-Peter Breuer and Francesco Petruccione. ``The theory of open quantum systems''. Oxford University Press. (2007).
https://doi.org/10.1093/acprof:oso/9780199213900.001.0001

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

[3] E. B. Davies. ``Markovian master equations''. Communications in Mathematical Physics 39, 91 – 110 (1974). 10.1007/BF01608389.
https://doi.org/10.1007/BF01608389

[4] Upendra Harbola, Massimiliano Esposito, and Shaul Mukamel. ``Quantum master equation for electron transport through quantum dots and single molecules''. Physical Review B 74 (2006).
https://doi.org/10.1103/PhysRevB.74.235309

[5] Mario Bertolotti. ``An introduction to quantum optics: an open system approach''. Contemporary Physics 62, 57–58 (2021).
https://doi.org/10.1080/00107514.2021.1924281

[6] T. Pellizzari, S. A. Gardiner, J. I. Cirac, and P. Zoller. ``Decoherence, continuous observation, and quantum computing: A cavity qed model''. Phys. Rev. Lett. 75, 3788–3791 (1995). 10.1103/PhysRevLett.75.3788.
https://doi.org/10.1103/PhysRevLett.75.3788

[7] Chi-Fang Chen, Michael J. Kastoryano, Fernando G. S. L. Brandão, and András Gilyén. ``Efficient quantum thermal simulation''. Nature (2025).
https://doi.org/10.1038/s41586-025-09583-x

[8] Zhiyan Ding, Chi-Fang Chen, and Lin Lin. ``Single-ancilla ground state preparation via Lindbladians''. Physical Review Research 6, 033147 (2024).
https://doi.org/10.1103/PhysRevResearch.6.033147

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

[10] 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

[11] 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

[12] Yu Cao and Jianfeng Lu. ``Structure-preserving numerical schemes for lindblad equations''. J. Sci. Comput. 102 (2024).
https://doi.org/10.1007/s10915-024-02707-x

[13] Andrew M Childs and Tongyang Li. ``Efficient simulation of sparse Markovian quantum dynamics''. Quantum Information and Computation (2017).
https://doi.org/10.26421/qic17.11-12

[14] Richard Cleve and Chunhao Wang. ``Efficient quantum algorithms for simulating Lindblad evolution'' (2016).
arXiv:1612.09512

[15] Xiantao Li and Chunhao Wang. ``Simulating markovian open quantum systems using higher-order series expansion'' (2023). arXiv:2212.02051.
arXiv:2212.02051

[16] Earl Campbell. ``Random compiler for fast Hamiltonian simulation''. Physical Review Letters 123 (2019).
https://doi.org/10.1103/physrevlett.123.070503

[17] Chi-Fang Chen, Hsin-Yuan Huang, Richard Kueng, and Joel A. Tropp. ``Concentration for random product formulas''. PRX Quantum 2 (2021).
https://doi.org/10.1103/prxquantum.2.040305

[18] Kristan Temme, Michael James Kastoryano, Mary Beth Ruskai, Michael Marc Wolf, and Frank Verstraete. ``The $\chi^2$-divergence and mixing times of quantum Markov processes''. Journal of Mathematical Physics 51 (2010).
https://doi.org/10.1063/1.3511335

[19] De Huang, Jonathan Niles-Weed, Joel A. Tropp, and Rachel Ward. ``Matrix concentration for products'' (2020). arXiv:2003.05437.
arXiv:2003.05437

[20] Christoph Dankert, Richard Cleve, Joseph Emerson, and Etera Livine. ``Exact and approximate unitary 2-designs and their application to fidelity estimation''. Physical Review A 80 (2009).
https://doi.org/10.1103/physreva.80.012304

[21] Alexei Kitaev. ``A simple model of quantum holography'' (2015). Talks at KITP, April 7 and May 27, 2015.

[22] Renjie Feng, Gang Tian, and Dongyi Wei. ``Spectrum of SYK model''. Peking Mathematical Journal 2, 41–70 (2019).
https://doi.org/10.1007/s42543-018-0007-1

[23] O. Bohigas, M. J. Giannoni, and C. Schmit. ``Characterization of chaotic quantum spectra and universality of level fluctuation laws''. Phys. Rev. Lett. 52, 1–4 (1984).
https://doi.org/10.1103/PhysRevLett.52.1

[24] Jan Kochanowski, Alvaro M. Alhambra, Angela Capel, and Cambyse Rouzé. ``Rapid thermalization of dissipative many-body dynamics of commuting Hamiltonians'' (2024). arXiv:2404.16780.
arXiv:2404.16780

[25] Cambyse Rouzé, Daniel Stilck Franca, and Alvaro M. Alhambra. ``Efficient thermalization and universal quantum computing with quantum gibbs samplers'' (2024). arXiv:2403.12691.
arXiv:2403.12691

[26] Thiago Bergamaschi, Chi-Fang Chen, and Yunchao Liu. ``Quantum computational advantage with constant-temperature gibbs sampling'' (2024). arXiv:2404.14639.
https://doi.org/10.1109/FOCS61266.2024.00071
arXiv:2404.14639

[27] Chi-Fang Chen and Fernando G. S. L. Brandão. ``Fast thermalization from the eigenstate thermalization hypothesis'' (2023). arXiv:2112.07646.
arXiv:2112.07646

[28] Vittorio Gorini, Andrzej Kossakowski, and Ennackal Chandy George Sudarshan. ``Completely positive dynamical semigroups of n-level systems''. J. Math. Phys. 17, 821–825 (1976).
https://doi.org/10.1063/1.522979

[29] Goran Lindblad. ``On the generators of quantum dynamical semigroups''. Commun. Math. Phys. 48, 119–130 (1976).
https://doi.org/10.1007/BF01608499

[30] Alexander Müller-Hermes and Daniel Stilck Franca. ``Sandwiched Rényi convergence for quantum evolutions''. Quantum 2, 55 (2018).
https://doi.org/10.22331/q-2018-02-27-55

[31] Dénes Petz and Csaba Sudár. ``Geometries of quantum states''. Journal of Mathematical Physics 37, 2662–2673 (1996).
https://doi.org/10.1017/CBO9780511535048

[32] Andrew Lesniewski and Mary Beth Ruskai. ``Monotone Riemannian metrics and relative entropy on noncommutative probability spaces''. Journal of Mathematical Physics 40, 5702–5724 (1999).
https://doi.org/10.1063/1.533053

[33] Eric A. Carlen and Jan Maas. ``Non-commutative calculus, optimal transport and functional inequalities in dissipative quantum systems''. Journal of Statistical Physics 178, 319–378 (2019).
https://doi.org/10.1007/s10955-019-02434-w

[34] Zhiyan Ding, Xiantao Li, and Lin Lin. ``Simulating open quantum systems using hamiltonian simulations''. PRX Quantum 5, 020332 (2024).
https://doi.org/10.1103/PRXQuantum.5.020332

[35] Andrew M. Childs, Yuan Su, Minh C. Tran, Nathan Wiebe, and Shuchen Zhu. ``Theory of trotter error with commutator scaling''. Physical Review X 11 (2021).
https://doi.org/10.1103/physrevx.11.011020

[36] M. Kliesch, T. Barthel, C. Gogolin, M. Kastoryano, and J. Eisert. ``Dissipative quantum church-turing theorem''. Physical Review Letters 107 (2011).
https://doi.org/10.1103/physrevlett.107.120501

[37] Albert H Werner, Daniel Jaschke, Pietro Silvi, Martin Kliesch, Tommaso Calarco, Jens Eisert, and Simone Montangero. ``Positive tensor network approach for simulating open quantum many-body systems''. Physical review letters 116, 237201 (2016).
https://doi.org/10.1103/PhysRevLett.116.237201

[38] Evan Borras and Milad Marvian. ``Quantum algorithm to simulate Lindblad master equations''. Physical Review Research 7, 023076 (2025).
https://doi.org/10.1103/PhysRevResearch.7.023076

[39] Sirui Peng, Xiaoming Sun, Qi Zhao, and Hongyi Zhou. ``Quantum-trajectory-inspired lindbladian simulation''. PRX Quantum 6, 030358 (2025).
https://doi.org/10.1103/ssrs-8x32

[40] I. J. David, I. Sinayskiy, and F. Petruccione. ``Faster quantum simulation of markovian open quantum systems via randomisation'' (2024). arXiv:2408.11683.
arXiv:2408.11683

[41] Andrew M. Childs, Aaron Ostrander, and Yuan Su. ``Faster quantum simulation by randomization''. Quantum 3, 182 (2019).
https://doi.org/10.22331/q-2019-09-02-182

[42] Michel Ledoux and Michel Talagrand. ``Probability in Banach spaces: isoperimetry and processes''. Springer Science & Business Media. (2013).
https://doi.org/10.1007/978-3-642-20212-4

[43] Sinho Chewi. ``Log-concave sampling''. Book draft available at https://chewisinho.github.io (2023).
https://chewisinho.github.io/main.pdf

[44] James D. Whitfield, Jacob Biamonte, and Alán Aspuru-Guzik. ``Simulation of electronic structure Hamiltonians using quantum computers''. Molecular Physics 109, 735–750 (2011).
https://doi.org/10.1080/00268976.2011.552441

[45] Persi Diaconis and Daniel Stroock. ``Geometric Bounds for Eigenvalues of Markov Chains''. The Annals of Applied Probability 1, 36 – 61 (1991).
https://doi.org/10.1214/aoap/1177005980

[46] Luca D’Alessio, Yariv Kafri, Anatoli Polkovnikov, and Marcos Rigol. ``From quantum chaos and eigenstate thermalization to statistical mechanics and thermodynamics''. Advances in Physics 65, 239–362 (2016).
https://doi.org/10.1080/00018732.2016.1198134

[47] Chi-Fang Chen, Alexander M. Dalzell, Mario Berta, Fernando G. S. L. Brandão, and Joel A. Tropp. ``Sparse random hamiltonians are quantumly easy''. Phys. Rev. X 14, 011014 (2024).
https://doi.org/10.1103/PhysRevX.14.011014

[48] Akshar Ramkumar and Mehdi Soleimanifar. ``Mixing time of quantum gibbs sampling for random sparse hamiltonians'' (2024). arXiv:2411.04454.
arXiv:2411.04454

[49] 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 (2025).
https://doi.org/10.22331/q-2025-08-29-1843

[50] Antonio M García-García and Jacobus JM Verbaarschot. ``Spectral and thermodynamic properties of the Sachdev-Ye-Kitaev model''. Physical Review D 94, 126010 (2016).
https://doi.org/10.1103/PhysRevD.94.126010

[51] Antonio M García-García and Jacobus JM Verbaarschot. ``Analytical spectral density of the Sachdev-Ye-Kitaev model at finite N''. Physical Review D 96, 066012 (2017).
https://doi.org/10.1103/PhysRevD.96.066012

[52] James D. Watson. ``Randomly compiled quantum simulation with exponentially reduced circuit depths'' (2025). arXiv:2411.04240.
arXiv:2411.04240

[53] Kouhei Nakaji, Mohsen Bagherimehrab, and Alan Aspuru-Guzik. ``qswift: High-order randomized compiler for Hamiltonian simulation'' (2023). arXiv:2302.14811. newblock.
https://doi.org/10.1103/PRXQuantum.5.020330
arXiv:2302.14811

Cited by

[1] Evan Borras and Milad Marvian, "Quantum simulation algorithms based on quantum trajectories", Quantum 10, 2063 (2026).

[2] 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).

[3] Michele Minervini, Dhrumil Patel, and Mark M. Wilde, "Evolved quantum Boltzmann machines", Physical Review A 113 3, 032427 (2026).

[4] Joel Rajakumar and James D. Watson, "Gibbs Sampling gives Quantum Advantage at Constant Temperatures with O(1)-Local Hamiltonians", Quantum 10, 1981 (2026).

[5] Michele Minervini, Madison Chin, Jacob Kupperman, Nana Liu, Ivy Luo, Meghan Ly, Soorya Rethinasamy, Kathie Wang, and Mark M. Wilde, "Constrained free energy minimization for the design of thermal states and stabilizer thermodynamic systems", Physical Review A 113 4, 042407 (2026).

[6] Yu Cao, Mingfeng He, and Xiantao Li, "Dynamically optimal unraveling schemes for simulating Lindblad equations", Journal of Physics A: Mathematical and Theoretical 59 16, 165301 (2026).

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

[8] Limin Xu, Zhen Huang, and Zhennan Zhou, "Frozen Gaussian sampling algorithms for simulating Markovian open quantum systems in the semiclassical regime", (2026).

[9] Limin Xu, Zhen Huang, and Zhennan Zhou, "Gaussian sampling algorithms for simulating Markovian open quantum systems in the semiclassical regime", Journal of Computational Physics 565, 115186 (2026).

[10] Francesco Turro and Xiaojun Yao, "Emergent hydrodynamic mode on SU(2) plaquette chains and quantum simulation", Physical Review D 111 9, 094502 (2025).

[11] Dhrumil Patel and Mark M. Wilde, "Natural gradient and parameter estimation for quantum Boltzmann machines", Physical Review A 112 5, 052421 (2025).

[12] Dhrumil Patel, Daniel Koch, Saahil Patel, and Mark M. Wilde, "Quantum Boltzmann machine learning of ground-state energies", arXiv:2410.12935, (2024).

[13] Yu Tong and Yongtao Zhan, "Fast Mixing of Weakly Interacting Fermionic Systems at Any Temperature", PRX Quantum 6 3, 030301 (2025).

[14] Jumpei Kato, Kaito Wada, Kosuke Ito, and Naoki Yamamoto, "Exponentially accurate open quantum simulation via randomized dissipation with minimal ancilla", arXiv:2412.19453, (2024).

[15] 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).

[16] 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).

[17] Michele Minervini, Dhrumil Patel, and Mark M. Wilde, "Quantum natural gradient with thermal-state initialization", Physical Review A 112 2, 022424 (2025).

[18] Bowen Li and Jianfeng Lu, "Quantum space-time Poincaré inequality for Lindblad dynamics", arXiv:2406.09115, (2024).

[19] Ruchira V Bhat, Rahul Bhowmick, Avinash Singh, and Krishna Kumar Sabapathy, "Meta-learning of Gibbs states for many-body Hamiltonians with applications to Quantum Boltzmann Machines", arXiv:2507.16373, (2025).

[20] Tianren Wang, Zongkang Zhang, Bing-Nan Lu, Mauro Cirio, and Ying Li, "Super-bath Quantum Eigensolver", arXiv:2412.19599, (2024).

[21] Pegah Mohammadipour and Xiantao Li, "Reducing circuit depth in Lindblad simulations via step-size extrapolation", Physical Review A 112 6, 062206 (2025).

[22] I. J. David, I. Sinayskiy, and F. Petruccione, "Faster Quantum Simulation Of Markovian Open Quantum Systems Via Randomisation", arXiv:2408.11683, (2024).

[23] Jia-Cheng Huang, Hao-En Li, Yi-Cheng Wang, Guang-Ze Zhang, Jun Li, and Han-Shi Hu, "Towards Robust Variational Quantum Simulation of Lindblad Dynamics via Stochastic Magnus Expansion", PRX Quantum 6 4, 040312 (2025).

[24] Joao Basso, Shirshendu Ganguly, Alistair Sinclair, Nikhil Srivastava, Zachary Stier, and Thuy-Duong Vuong, "On quantum to classical comparison for Davies generators", arXiv:2510.07267, (2025).

[25] I. J. David, I. Sinayskiy, and F. Petruccione, "Tighter Error Bounds for the qDRIFT Algorithm", arXiv:2506.17199, (2025).

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

[27] Di Fang, Jianfeng Lu, Yu Tong, and Chu Zhao, "Quantum Gibbs sampling through the detectability lemma", arXiv:2604.07214, (2026).

The above citations are from Crossref's cited-by service (last updated successfully 2026-08-07 17:14:34) and SAO/NASA ADS (last updated successfully 2026-08-07 17:14:35). 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.