{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,7,2]],"date-time":"2026-07-02T03:02:07Z","timestamp":1782961327391,"version":"3.54.5"},"reference-count":53,"publisher":"Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften","license":[{"start":{"date-parts":[[2025,11,20]],"date-time":"2025-11-20T00:00:00Z","timestamp":1763596800000},"content-version":"unspecified","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100012166","name":"National Key R&D Program of China","doi-asserted-by":"crossref","award":["2024YFA1016000"],"award-info":[{"award-number":["2024YFA1016000"]}],"id":[{"id":"10.13039\/501100012166","id-type":"DOI","asserted-by":"crossref"}]},{"name":"National Science Foundation","award":["DMS-2309378"],"award-info":[{"award-number":["DMS-2309378"]}]},{"name":"National Science Foundation","award":["DMS-2208163"],"award-info":[{"award-number":["DMS-2208163"]}]},{"name":"the U.S. Department of Energy, Office of Science, Accelerated Research in Quantum Computing Centers, Quantum Utility through Advanced Computational Quantum Algorithms","award":["DESC002557"],"award-info":[{"award-number":["DESC002557"]}]}],"content-domain":{"domain":["quantum-journal.org"],"crossmark-restriction":false},"short-container-title":["Quantum"],"abstract":"\n We study a qDRIFT-type randomized method to simulate Lindblad dynamics by decomposing its generator into an ensemble of Lindbladians,\n \n \n L<\/mml:mi>\n <\/mml:mrow>\n =<\/mml:mo>\n \n &#x2211;<\/mml:mo>\n \n a<\/mml:mi>\n &#x2208;<\/mml:mo>\n \n A<\/mml:mi>\n <\/mml:mrow>\n <\/mml:mrow>\n <\/mml:munder>\n \n \n L<\/mml:mi>\n <\/mml:mrow>\n a<\/mml:mi>\n <\/mml:msub>\n <\/mml:math>\n , where each\n \n \n \n L<\/mml:mi>\n <\/mml:mrow>\n a<\/mml:mi>\n <\/mml:msub>\n <\/mml:math>\n comprises a simple Hamiltonian and a single jump operator. Assuming an efficient quantum simulation is available for the Lindblad evolution\n \n \n e<\/mml:mi>\n \n t<\/mml:mi>\n \n \n L<\/mml:mi>\n <\/mml:mrow>\n a<\/mml:mi>\n <\/mml:msub>\n <\/mml:mrow>\n <\/mml:msup>\n <\/mml:math>\n , we implement\n \n \n e<\/mml:mi>\n \n t<\/mml:mi>\n \n \n L<\/mml:mi>\n <\/mml:mrow>\n a<\/mml:mi>\n <\/mml:msub>\n <\/mml:mrow>\n <\/mml:msup>\n <\/mml:math>\n for a randomly sampled\n \n \n \n L<\/mml:mi>\n <\/mml:mrow>\n a<\/mml:mi>\n <\/mml:msub>\n <\/mml:math>\n at each time step according to a probability distribution\n \n &#x03BC;<\/mml:mi>\n <\/mml:math>\n over the ensemble\n \n {<\/mml:mo>\n \n \n L<\/mml:mi>\n <\/mml:mrow>\n a<\/mml:mi>\n <\/mml:msub>\n \n }<\/mml:mo>\n \n a<\/mml:mi>\n &#x2208;<\/mml:mo>\n \n A<\/mml:mi>\n <\/mml:mrow>\n <\/mml:mrow>\n <\/mml:msub>\n <\/mml:math>\n . 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.\n <\/jats:p>","DOI":"10.22331\/q-2025-11-20-1917","type":"journal-article","created":{"date-parts":[[2025,11,20]],"date-time":"2025-11-20T15:58:18Z","timestamp":1763654298000},"page":"1917","update-policy":"https:\/\/doi.org\/10.22331\/q-crossmark-policy-page","source":"Crossref","is-referenced-by-count":8,"title":["A Randomized Method for Simulating Lindblad Equations and Thermal State Preparation"],"prefix":"10.22331","volume":"9","author":[{"given":"Hongrui","family":"Chen","sequence":"first","affiliation":[{"name":"Department of Mathematics, Stanford University, Stanford, CA 94305"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Bowen","family":"Li","sequence":"additional","affiliation":[{"name":"Department of Mathematics, City University of Hong Kong, Kowloon Tong, Hong Kong SAR"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Jianfeng","family":"Lu","sequence":"additional","affiliation":[{"name":"Departments of Mathematics, Physics, and Chemistry, Duke University, Durham, NC 27708"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Lexing","family":"Ying","sequence":"additional","affiliation":[{"name":"Department of Mathematics and ICME, Stanford University, Stanford, CA 94305"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"9598","published-online":{"date-parts":[[2025,11,20]]},"reference":[{"key":"0","doi-asserted-by":"publisher","unstructured":"Heinz-Peter Breuer and Francesco Petruccione. ``The theory of open quantum systems''. Oxford University Press. (2007).","DOI":"10.1093\/acprof:oso\/9780199213900.001.0001"},{"key":"1","doi-asserted-by":"publisher","unstructured":"G. Lindblad. ``On the generators of quantum dynamical semigroups''. Communications in Mathematical Physics 48, 119 \u2013 130 (1976).","DOI":"10.1007\/BF01608499"},{"key":"2","doi-asserted-by":"publisher","unstructured":"E. B. Davies. ``Markovian master equations''. Communications in Mathematical Physics 39, 91 \u2013 110 (1974). 10.1007\/BF01608389.","DOI":"10.1007\/BF01608389"},{"key":"3","doi-asserted-by":"publisher","unstructured":"Upendra Harbola, Massimiliano Esposito, and Shaul Mukamel. ``Quantum master equation for electron transport through quantum dots and single molecules''. Physical Review B 74 (2006).","DOI":"10.1103\/PhysRevB.74.235309"},{"key":"4","doi-asserted-by":"publisher","unstructured":"Mario Bertolotti. ``An introduction to quantum optics: an open system approach''. Contemporary Physics 62, 57\u201358 (2021).","DOI":"10.1080\/00107514.2021.1924281"},{"key":"5","doi-asserted-by":"publisher","unstructured":"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\u20133791 (1995). 10.1103\/PhysRevLett.75.3788.","DOI":"10.1103\/PhysRevLett.75.3788"},{"key":"6","doi-asserted-by":"publisher","unstructured":"Chi-Fang Chen, Michael J. Kastoryano, Fernando G. S. L. Brand\u00e3o, and Andr\u00e1s Gily\u00e9n. ``Efficient quantum thermal simulation''. Nature (2025).","DOI":"10.1038\/s41586-025-09583-x"},{"key":"7","doi-asserted-by":"publisher","unstructured":"Zhiyan Ding, Chi-Fang Chen, and Lin Lin. ``Single-ancilla ground state preparation via Lindbladians''. Physical Review Research 6, 033147 (2024).","DOI":"10.1103\/PhysRevResearch.6.033147"},{"key":"8","unstructured":"Chi-Fang Chen, Michael J. Kastoryano, and Andr\u00e1s Gily\u00e9n. ``An efficient and exact noncommutative quantum Gibbs sampler'' (2023). arXiv:2311.09207."},{"key":"9","doi-asserted-by":"publisher","unstructured":"Zhiyan Ding, Bowen Li, and Lin Lin. ``Efficient quantum gibbs samplers with Kubo\u2013Martin\u2013Schwinger detailed balance condition''. Communications in Mathematical Physics 406, 67 (2025).","DOI":"10.1007\/s00220-025-05235-3"},{"key":"10","unstructured":"Andr\u00e1s Gily\u00e9n, Chi-Fang Chen, Joao F. Doriguello, and Michael J. Kastoryano. ``Quantum generalizations of glauber and metropolis dynamics'' (2024). arXiv:2405.20322."},{"key":"11","doi-asserted-by":"publisher","unstructured":"Yu Cao and Jianfeng Lu. ``Structure-preserving numerical schemes for lindblad equations''. J. Sci. Comput. 102 (2024).","DOI":"10.1007\/s10915-024-02707-x"},{"key":"12","doi-asserted-by":"publisher","unstructured":"Andrew M Childs and Tongyang Li. ``Efficient simulation of sparse Markovian quantum dynamics''. Quantum Information and Computation (2017).","DOI":"10.26421\/qic17.11-12"},{"key":"13","unstructured":"Richard Cleve and Chunhao Wang. ``Efficient quantum algorithms for simulating Lindblad evolution'' (2016)."},{"key":"14","unstructured":"Xiantao Li and Chunhao Wang. ``Simulating markovian open quantum systems using higher-order series expansion'' (2023). arXiv:2212.02051."},{"key":"15","doi-asserted-by":"publisher","unstructured":"Earl Campbell. ``Random compiler for fast Hamiltonian simulation''. Physical Review Letters 123 (2019).","DOI":"10.1103\/physrevlett.123.070503"},{"key":"16","doi-asserted-by":"publisher","unstructured":"Chi-Fang Chen, Hsin-Yuan Huang, Richard Kueng, and Joel A. Tropp. ``Concentration for random product formulas''. PRX Quantum 2 (2021).","DOI":"10.1103\/prxquantum.2.040305"},{"key":"17","doi-asserted-by":"publisher","unstructured":"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).","DOI":"10.1063\/1.3511335"},{"key":"18","unstructured":"De Huang, Jonathan Niles-Weed, Joel A. Tropp, and Rachel Ward. ``Matrix concentration for products'' (2020). arXiv:2003.05437."},{"key":"19","doi-asserted-by":"publisher","unstructured":"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).","DOI":"10.1103\/physreva.80.012304"},{"key":"20","unstructured":"Alexei Kitaev. ``A simple model of quantum holography'' (2015). Talks at KITP, April 7 and May 27, 2015."},{"key":"21","doi-asserted-by":"publisher","unstructured":"Renjie Feng, Gang Tian, and Dongyi Wei. ``Spectrum of SYK model''. Peking Mathematical Journal 2, 41\u201370 (2019).","DOI":"10.1007\/s42543-018-0007-1"},{"key":"22","doi-asserted-by":"publisher","unstructured":"O. Bohigas, M. J. Giannoni, and C. Schmit. ``Characterization of chaotic quantum spectra and universality of level fluctuation laws''. Phys. Rev. Lett. 52, 1\u20134 (1984).","DOI":"10.1103\/PhysRevLett.52.1"},{"key":"23","doi-asserted-by":"crossref","unstructured":"Jan Kochanowski, Alvaro M. Alhambra, Angela Capel, and Cambyse Rouz\u00e9. ``Rapid thermalization of dissipative many-body dynamics of commuting Hamiltonians'' (2024). arXiv:2404.16780.","DOI":"10.1007\/s00220-025-05353-y"},{"key":"24","doi-asserted-by":"crossref","unstructured":"Cambyse Rouz\u00e9, Daniel Stilck Franca, and Alvaro M. Alhambra. ``Efficient thermalization and universal quantum computing with quantum gibbs samplers'' (2024). arXiv:2403.12691.","DOI":"10.1145\/3717823.3718268"},{"key":"25","doi-asserted-by":"publisher","unstructured":"Thiago Bergamaschi, Chi-Fang Chen, and Yunchao Liu. ``Quantum computational advantage with constant-temperature gibbs sampling'' (2024). arXiv:2404.14639.","DOI":"10.1109\/FOCS61266.2024.00071"},{"key":"26","unstructured":"Chi-Fang Chen and Fernando G. S. L. Brand\u00e3o. ``Fast thermalization from the eigenstate thermalization hypothesis'' (2023). arXiv:2112.07646."},{"key":"27","doi-asserted-by":"publisher","unstructured":"Vittorio Gorini, Andrzej Kossakowski, and Ennackal Chandy George Sudarshan. ``Completely positive dynamical semigroups of n-level systems''. J. Math. Phys. 17, 821\u2013825 (1976).","DOI":"10.1063\/1.522979"},{"key":"28","doi-asserted-by":"publisher","unstructured":"Goran Lindblad. ``On the generators of quantum dynamical semigroups''. Commun. Math. Phys. 48, 119\u2013130 (1976).","DOI":"10.1007\/BF01608499"},{"key":"29","doi-asserted-by":"publisher","unstructured":"Alexander M\u00fcller-Hermes and Daniel Stilck Franca. ``Sandwiched R\u00e9nyi convergence for quantum evolutions''. Quantum 2, 55 (2018).","DOI":"10.22331\/q-2018-02-27-55"},{"key":"30","doi-asserted-by":"publisher","unstructured":"D\u00e9nes Petz and Csaba Sud\u00e1r. ``Geometries of quantum states''. Journal of Mathematical Physics 37, 2662\u20132673 (1996).","DOI":"10.1017\/CBO9780511535048"},{"key":"31","doi-asserted-by":"publisher","unstructured":"Andrew Lesniewski and Mary Beth Ruskai. ``Monotone Riemannian metrics and relative entropy on noncommutative probability spaces''. Journal of Mathematical Physics 40, 5702\u20135724 (1999).","DOI":"10.1063\/1.533053"},{"key":"32","doi-asserted-by":"publisher","unstructured":"Eric A. Carlen and Jan Maas. ``Non-commutative calculus, optimal transport and functional inequalities in dissipative quantum systems''. Journal of Statistical Physics 178, 319\u2013378 (2019).","DOI":"10.1007\/s10955-019-02434-w"},{"key":"33","doi-asserted-by":"publisher","unstructured":"Zhiyan Ding, Xiantao Li, and Lin Lin. ``Simulating open quantum systems using hamiltonian simulations''. PRX Quantum 5, 020332 (2024).","DOI":"10.1103\/PRXQuantum.5.020332"},{"key":"34","doi-asserted-by":"publisher","unstructured":"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).","DOI":"10.1103\/physrevx.11.011020"},{"key":"35","doi-asserted-by":"publisher","unstructured":"M. Kliesch, T. Barthel, C. Gogolin, M. Kastoryano, and J. Eisert. ``Dissipative quantum church-turing theorem''. Physical Review Letters 107 (2011).","DOI":"10.1103\/physrevlett.107.120501"},{"key":"36","doi-asserted-by":"publisher","unstructured":"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).","DOI":"10.1103\/PhysRevLett.116.237201"},{"key":"37","doi-asserted-by":"publisher","unstructured":"Evan Borras and Milad Marvian. ``Quantum algorithm to simulate Lindblad master equations''. Physical Review Research 7, 023076 (2025).","DOI":"10.1103\/PhysRevResearch.7.023076"},{"key":"38","doi-asserted-by":"publisher","unstructured":"Sirui Peng, Xiaoming Sun, Qi Zhao, and Hongyi Zhou. ``Quantum-trajectory-inspired lindbladian simulation''. PRX Quantum 6, 030358 (2025).","DOI":"10.1103\/ssrs-8x32"},{"key":"39","unstructured":"I. J. David, I. Sinayskiy, and F. Petruccione. ``Faster quantum simulation of markovian open quantum systems via randomisation'' (2024). arXiv:2408.11683."},{"key":"40","doi-asserted-by":"publisher","unstructured":"Andrew M. Childs, Aaron Ostrander, and Yuan Su. ``Faster quantum simulation by randomization''. Quantum 3, 182 (2019).","DOI":"10.22331\/q-2019-09-02-182"},{"key":"41","doi-asserted-by":"publisher","unstructured":"Michel Ledoux and Michel Talagrand. ``Probability in Banach spaces: isoperimetry and processes''. Springer Science & Business Media. (2013).","DOI":"10.1007\/978-3-642-20212-4"},{"key":"42","unstructured":"Sinho Chewi. ``Log-concave sampling''. Book draft available at https:\/\/chewisinho.github.io (2023)."},{"key":"43","doi-asserted-by":"publisher","unstructured":"James D. Whitfield, Jacob Biamonte, and Al\u00e1n Aspuru-Guzik. ``Simulation of electronic structure Hamiltonians using quantum computers''. Molecular Physics 109, 735\u2013750 (2011).","DOI":"10.1080\/00268976.2011.552441"},{"key":"44","doi-asserted-by":"publisher","unstructured":"Persi Diaconis and Daniel Stroock. ``Geometric Bounds for Eigenvalues of Markov Chains''. The Annals of Applied Probability 1, 36 \u2013 61 (1991).","DOI":"10.1214\/aoap\/1177005980"},{"key":"45","doi-asserted-by":"publisher","unstructured":"Luca D\u2019Alessio, Yariv Kafri, Anatoli Polkovnikov, and Marcos Rigol. ``From quantum chaos and eigenstate thermalization to statistical mechanics and thermodynamics''. Advances in Physics 65, 239\u2013362 (2016).","DOI":"10.1080\/00018732.2016.1198134"},{"key":"46","doi-asserted-by":"publisher","unstructured":"Chi-Fang Chen, Alexander M. Dalzell, Mario Berta, Fernando G. S. L. Brand\u00e3o, and Joel A. Tropp. ``Sparse random hamiltonians are quantumly easy''. Phys. Rev. X 14, 011014 (2024).","DOI":"10.1103\/PhysRevX.14.011014"},{"key":"47","unstructured":"Akshar Ramkumar and Mehdi Soleimanifar. ``Mixing time of quantum gibbs sampling for random sparse hamiltonians'' (2024). arXiv:2411.04454."},{"key":"48","doi-asserted-by":"publisher","unstructured":"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).","DOI":"10.22331\/q-2025-08-29-1843"},{"key":"49","doi-asserted-by":"publisher","unstructured":"Antonio M Garc\u00eda-Garc\u00eda and Jacobus JM Verbaarschot. ``Spectral and thermodynamic properties of the Sachdev-Ye-Kitaev model''. Physical Review D 94, 126010 (2016).","DOI":"10.1103\/PhysRevD.94.126010"},{"key":"50","doi-asserted-by":"publisher","unstructured":"Antonio M Garc\u00eda-Garc\u00eda and Jacobus JM Verbaarschot. ``Analytical spectral density of the Sachdev-Ye-Kitaev model at finite N''. Physical Review D 96, 066012 (2017).","DOI":"10.1103\/PhysRevD.96.066012"},{"key":"51","unstructured":"James D. Watson. ``Randomly compiled quantum simulation with exponentially reduced circuit depths'' (2025). arXiv:2411.04240."},{"key":"52","doi-asserted-by":"publisher","unstructured":"Kouhei Nakaji, Mohsen Bagherimehrab, and Alan Aspuru-Guzik. ``qswift: High-order randomized compiler for Hamiltonian simulation'' (2023). arXiv:2302.14811. newblock.","DOI":"10.1103\/PRXQuantum.5.020330"}],"container-title":["Quantum"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/quantum-journal.org\/papers\/q-2025-11-20-1917\/pdf\/","content-type":"unspecified","content-version":"vor","intended-application":"text-mining"}],"deposited":{"date-parts":[[2025,11,20]],"date-time":"2025-11-20T15:58:22Z","timestamp":1763654302000},"score":1,"resource":{"primary":{"URL":"https:\/\/quantum-journal.org\/papers\/q-2025-11-20-1917\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,11,20]]},"references-count":53,"URL":"https:\/\/doi.org\/10.22331\/q-2025-11-20-1917","archive":["CLOCKSS"],"relation":{},"ISSN":["2521-327X"],"issn-type":[{"value":"2521-327X","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,11,20]]},"article-number":"1917"}}