{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,23]],"date-time":"2026-01-23T15:20:00Z","timestamp":1769181600102,"version":"3.49.0"},"reference-count":42,"publisher":"MDPI AG","issue":"2","license":[{"start":{"date-parts":[[2026,1,23]],"date-time":"2026-01-23T00:00:00Z","timestamp":1769126400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Natural Science Foundation of Shandong Province, China","award":["ZR2024MA046"],"award-info":[{"award-number":["ZR2024MA046"]}]},{"name":"Natural Science Foundation of Shandong Province, China","award":["ZR2021LLZ004"],"award-info":[{"award-number":["ZR2021LLZ004"]}]},{"name":"Fundamental Research Funds for the Central Universities, China","award":["202364008"],"award-info":[{"award-number":["202364008"]}]},{"name":"Basque Country Government","award":["IT1470-22"],"award-info":[{"award-number":["IT1470-22"]}]},{"name":"Basque Country Government","award":["PGC2018-101355-B-I00"],"award-info":[{"award-number":["PGC2018-101355-B-I00"]}]},{"name":"QUANTUM ENIA project"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>The dynamics of open quantum systems play a crucial role in quantum information science. However, obtaining numerically exact solutions for the Lindblad master equation is often computationally expensive. Recently, machine learning techniques have gained considerable attention for simulating open quantum system dynamics. In this paper, we propose a deep learning model based on time series prediction (TSP) to forecast the dynamical evolution of open quantum systems. We employ the positive operator-valued measure (POVM) approach to convert the density matrix of the system into a probability distribution and construct a TSP model based on Transformer neural networks. This model effectively captures the historical evolution patterns of the system and accurately predicts its future behavior. Our results show that the model achieves high-fidelity predictions of the system\u2019s evolution trajectory in both short- and long-term scenarios, and exhibits robust generalization under varying initial states and coupling strengths. Moreover, we successfully predicted the steady-state behavior of the system, further proving the practicality and scalability of the method.<\/jats:p>","DOI":"10.3390\/e28020133","type":"journal-article","created":{"date-parts":[[2026,1,23]],"date-time":"2026-01-23T13:55:34Z","timestamp":1769176534000},"page":"133","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Time Series Prediction of Open Quantum System Dynamics by Transformer Neural Networks"],"prefix":"10.3390","volume":"28","author":[{"given":"Zhao-Wei","family":"Wang","sequence":"first","affiliation":[{"name":"College of Physics and Optoelectronic Engineering, Ocean University of China, Qingdao 266100, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Lian-Ao","family":"Wu","sequence":"additional","affiliation":[{"name":"Department of Physics, University of the Basque Country UPV\/EHU, 48080 Bilbao, Spain"},{"name":"IKERBASQUE, Basque Foundation for Science, 48013 Bilbao, Spain"},{"name":"EHU Quantum Center, University of the Basque Country UPV\/EHU, 48940 Leioa, Spain"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-8976-9515","authenticated-orcid":false,"given":"Zhao-Ming","family":"Wang","sequence":"additional","affiliation":[{"name":"College of Physics and Optoelectronic Engineering, Ocean University of China, Qingdao 266100, China"},{"name":"Engineering Research Center of Advanced Marine Physical Instruments and Equipment of Ministry of Education, Ocean University of China, Qingdao 266100, China"},{"name":"Qingdao Key Laboratory of Optics and Optoelectronics, College of Physics and Optoelectronic Engineering, Ocean University of China, Qingdao 266100, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2026,1,23]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Yang, N., and Yu, T. (2025). Quantum Synchronization via Active\u2013Passive Decomposition Configuration: An Open Quantum-System Study. Entropy, 27.","DOI":"10.3390\/e27040432"},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Cariolaro, G. (2015). Quantum Communications, Springer.","DOI":"10.1007\/978-3-319-15600-2"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"633","DOI":"10.1038\/nphys1342","article-title":"Quantum computation and quantum-state engineering driven by dissipation","volume":"5","author":"Verstraete","year":"2009","journal-title":"Nat. Phys."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"248","DOI":"10.1016\/j.cjph.2021.05.001","article-title":"Quantum computation: Algorithms and Applications","volume":"72","author":"Cho","year":"2021","journal-title":"Chin. J. Phys."},{"key":"ref_5","doi-asserted-by":"crossref","unstructured":"Mui, K., Couvertier, A., and Yu, T. (2025). Enhanced quantum state swapping via environmental memory. APL Quantum, 2.","DOI":"10.1063\/5.0253875"},{"key":"ref_6","unstructured":"Nielsen, M.A., and Chuang, I.L. (2010). Quantum Computation and Quantum Information, Cambridge University Press. [10th anniversary ed.]."},{"key":"ref_7","doi-asserted-by":"crossref","unstructured":"Tanimura, Y. (2020). Numerically \u201cexact\u201d approach to open quantum dynamics: The hierarchical equations of motion (HEOM). J. Chem. Phys., 153.","DOI":"10.1063\/5.0011599"},{"key":"ref_8","doi-asserted-by":"crossref","unstructured":"Kast, D., and Ankerhold, J. (2013). Persistence of Coherent Quantum Dynamics at Strong Dissipation. Phys. Rev. Lett., 110.","DOI":"10.1103\/PhysRevLett.110.010402"},{"key":"ref_9","doi-asserted-by":"crossref","unstructured":"Kosionis, S.G., Biswas, S., Fouseki, C., Stefanatos, D., and Paspalakis, E. (2025). Efficient population transfer in a quantum dot exciton under phonon-induced decoherence via shortcuts to adiabaticity. Phys. Rev. B, 112.","DOI":"10.1103\/m1cc-l6ng"},{"key":"ref_10","doi-asserted-by":"crossref","unstructured":"Chen, P., Yang, N., Couvertier, A., Ding, Q., Chatterjee, R., and Yu, T. (2024). Chaos in Optomechanical Systems Coupled to a Non-Markovian Environment. Entropy, 26.","DOI":"10.3390\/e26090742"},{"key":"ref_11","doi-asserted-by":"crossref","unstructured":"Wang, Z.M., Ren, F.H., Luo, D.W., Yan, Z.Y., and Wu, L.A. (2021). Quantum state transmission through a spin chain in finite-temperature heat baths. J. Phys. A Math. Theor., 54.","DOI":"10.1088\/1751-8121\/abe751"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"569","DOI":"10.1016\/S0375-9601(97)00717-2","article-title":"The non-Markovian stochastic Schr\u00f6dinger equation for open systems","volume":"235","author":"Strunz","year":"1997","journal-title":"Phys. Lett. A"},{"key":"ref_13","doi-asserted-by":"crossref","unstructured":"Martyn, J.M., Najafi, K., and Luo, D. (2023). Variational Neural-Network Ansatz for Continuum Quantum Field Theory. Phys. Rev. Lett., 131.","DOI":"10.1103\/PhysRevLett.131.081601"},{"key":"ref_14","doi-asserted-by":"crossref","unstructured":"Nagy, A., and Savona, V. (2019). Variational quantum Monte Carlo method with a neural-network ansatz for open quantum systems. Phys. Rev. Lett., 122.","DOI":"10.1103\/PhysRevLett.122.250501"},{"key":"ref_15","doi-asserted-by":"crossref","unstructured":"Rodr\u00edguez, L.E.H., Ullah, A., Espinosa, K.J.R., Dral, P.O., and Kananenka, A.A. (2022). A comparative study of different machine learning methods for dissipative quantum dynamics. Mach. Learn. Sci. Technol., 3.","DOI":"10.1088\/2632-2153\/ac9a9d"},{"key":"ref_16","doi-asserted-by":"crossref","unstructured":"Reh, M., Schmitt, M., and G\u00e4rttner, M. (2021). Time-dependent variational principle for open quantum systems with artificial neural networks. Phys. Rev. Lett., 127.","DOI":"10.1103\/PhysRevLett.127.230501"},{"key":"ref_17","doi-asserted-by":"crossref","unstructured":"Luo, D., Chen, Z., Carrasquilla, J., and Clark, B.K. (2022). Autoregressive neural network for simulating open quantum systems via a probabilistic formulation. Phys. Rev. Lett., 128.","DOI":"10.1103\/PhysRevLett.128.090501"},{"key":"ref_18","doi-asserted-by":"crossref","unstructured":"Viteritti, L.L., Rende, R., and Becca, F. (2023). Transformer Variational Wave Functions for Frustrated Quantum Spin Systems. Phys. Rev. Lett., 130.","DOI":"10.1103\/PhysRevLett.130.236401"},{"key":"ref_19","doi-asserted-by":"crossref","unstructured":"Norambuena, A., Mattheakis, M., Gonz\u00e1lez, F.J., and Coto, R. (2024). Physics-Informed Neural Networks for Quantum Control. Phys. Rev. Lett., 132.","DOI":"10.1103\/PhysRevLett.132.010801"},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"1906","DOI":"10.1109\/TAI.2025.3531330","article-title":"Robust Control of Uncertain Quantum Systems Based on Physics-Informed Neural Networks and Sampling Learning","volume":"6","author":"Zhang","year":"2025","journal-title":"IEEE Trans. Artif. Intell."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"7095","DOI":"10.1109\/JLT.2022.3199782","article-title":"Physics-Informed Neural Network for Optical Fiber Parameter Estimation From the Nonlinear Schr\u00f6dinger Equation","volume":"40","author":"Jiang","year":"2022","journal-title":"J. Light. Technol."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"2052","DOI":"10.1039\/D4DD00153B","article-title":"Physics-informed neural networks and beyond: Enforcing physical constraints in quantum dissipative dynamics","volume":"3","author":"Ullah","year":"2024","journal-title":"Digit. Discov."},{"key":"ref_23","doi-asserted-by":"crossref","unstructured":"Chen, L., and Wu, Y. (2022). Learning quantum dissipation by the neural ordinary differential equation. Phys. Rev. A, 106.","DOI":"10.1103\/PhysRevA.106.022201"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"6037","DOI":"10.1021\/acs.jpclett.2c01242","article-title":"One-shot trajectory learning of open quantum systems dynamics","volume":"13","author":"Ullah","year":"2022","journal-title":"J. Phys. Chem. Lett."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"7833","DOI":"10.1109\/JSEN.2019.2923982","article-title":"A review of deep learning models for time series prediction","volume":"21","author":"Han","year":"2019","journal-title":"IEEE Sens. J."},{"key":"ref_26","first-page":"11106","article-title":"Informer: Beyond efficient transformer for long sequence time-series forecasting","volume":"35","author":"Zhou","year":"2021","journal-title":"AAAI Conf. Artif. Intell."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"76","DOI":"10.1111\/joes.12429","article-title":"Machine learning advances for time series forecasting","volume":"37","author":"Masini","year":"2023","journal-title":"J. Econ. Surv."},{"key":"ref_28","doi-asserted-by":"crossref","unstructured":"Naicker, K., Sinayskiy, I., and Petruccione, F. (2022). Statistical and machine learning approaches for prediction of long-time excitation energy transfer dynamics. arXiv.","DOI":"10.1103\/PhysRevResearch.4.033175"},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"2476","DOI":"10.1021\/acs.jpclett.1c00079","article-title":"Convolutional neural networks for long time dissipative quantum dynamics","volume":"12","author":"Kananenka","year":"2021","journal-title":"J. Phys. Chem. Lett."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"10225","DOI":"10.1021\/acs.jpclett.1c02672","article-title":"Simulation of open quantum dynamics with bootstrap-based long short-term memory recurrent neural network","volume":"12","author":"Lin","year":"2021","journal-title":"J. Phys. Chem. Lett."},{"key":"ref_31","doi-asserted-by":"crossref","unstructured":"Wu, D., Hu, Z., Li, J., and Sun, X. (2021). Forecasting nonadiabatic dynamics using hybrid convolutional neural network\/long short-term memory network. J. Chem. Phys., 155.","DOI":"10.1063\/5.0073689"},{"key":"ref_32","first-page":"1","article-title":"Attention is all you need","volume":"30","author":"Vaswani","year":"2017","journal-title":"Adv. Neural Inf. Process. Syst."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"171101","DOI":"10.1063\/5.0232871","article-title":"A short trajectory is all you need: A transformer-based model for long-time dissipative quantum dynamics","volume":"161","author":"Kananenka","year":"2024","journal-title":"J. Chem. Phys."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"821","DOI":"10.1063\/1.522979","article-title":"Completely positive dynamical semigroups of N-level systems","volume":"17","author":"Gorini","year":"1976","journal-title":"J. Math. Phys."},{"key":"ref_35","doi-asserted-by":"crossref","unstructured":"Carrasquilla, J., Luo, D., P\u00e9rez, F., Milsted, A., Clark, B.K., Volkovs, M., and Aolita, L. (2021). Probabilistic simulation of quantum circuits using a deep-learning architecture. Phys. Rev. A, 104.","DOI":"10.1103\/PhysRevA.104.032610"},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"155","DOI":"10.1038\/s42256-019-0028-1","article-title":"Reconstructing quantum states with generative models","volume":"1","author":"Carrasquilla","year":"2019","journal-title":"Nat. Mach. Intell."},{"key":"ref_37","first-page":"1","article-title":"Pytorch: An imperative style, high-performance deep learning library","volume":"32","author":"Paszke","year":"2019","journal-title":"Adv. Neural Inf. Process. Syst."},{"key":"ref_38","unstructured":"Goodfellow, I., Bengio, Y., and Courville, A. (2016). Deep Learning, MIT Press."},{"key":"ref_39","unstructured":"Kingma, D.P., and Ba, J. (2014). Adam: A method for stochastic optimization. arXiv."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"260","DOI":"10.19101\/IJACR.PID77","article-title":"Determining the impact of window length on time series forecasting using deep learning","volume":"9","author":"Azlan","year":"2019","journal-title":"Int. J. Adv. Comput. Res."},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"1760","DOI":"10.1016\/j.cpc.2012.02.021","article-title":"QuTiP: An open-source Python framework for the dynamics of open quantum systems","volume":"183","author":"Johansson","year":"2012","journal-title":"Comput. Phys. Commun."},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"5837","DOI":"10.1021\/acs.jctc.2c00702","article-title":"Automatic evolution of machine-learning-based quantum dynamics with uncertainty analysis","volume":"18","author":"Lin","year":"2022","journal-title":"J. Chem. Theory Comput."}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/28\/2\/133\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,1,23]],"date-time":"2026-01-23T13:57:26Z","timestamp":1769176646000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/28\/2\/133"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,1,23]]},"references-count":42,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2026,2]]}},"alternative-id":["e28020133"],"URL":"https:\/\/doi.org\/10.3390\/e28020133","relation":{},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2026,1,23]]}}}