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  • Open Access

Quantum Flow Matching

Zidong Cui1,2, Pan Zhang3,4, and Ying Tang1,2,5,6,*

  • *Contact author: jamestang23@gmail.com
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PRX Intelligence 1, 013009 – Published 6 August, 2026

DOI: https://doi.org/10.1103/s6zj-vzdp

Abstract

The flow matching has rapidly become a dominant paradigm in classical generative modeling, offering an efficient way to interpolate between two complex distributions. We extend this idea to the quantum realm and introduce the quantum flow matching (QFM), a quantum-circuit realization that offers efficient interpolation between two density matrices. QFM offers systematic preparation of density matrices and generation of samples for accurately estimating observables, and can be realized on quantum computers without the need for costly circuit redesigns. We validate its versatility on a set of applications: (1) generating target states with prescribed magnetization and entanglement entropy, (2) estimating nonequilibrium free-energy differences to test the quantum Jarzynski equality, and (3) expediting the study on superdiffusion. These results position QFM as a unifying and promising framework for generative modeling across quantum systems.

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References (72)

  1. A. Polkovnikov, K. Sengupta, A. Silva, and M. Vengalattore, Colloquium: Nonequilibrium dynamics of closed interacting quantum systems, Rev. Mod. Phys. 83, 863 (2011).
  2. L. D’Alessio, Y. Kafri, A. Polkovnikov, and M. Rigol, From quantum chaos and eigenstate thermalization to statistical mechanics and thermodynamics, Adv. Phys. 65, 239 (2016).
  3. M. C. Banuls, R. Blatt, J. Catani, A. Celi, J. I. Cirac, M. Dalmonte, L. Fallani, K. Jansen, M. Lewenstein, and S. Montangero, Simulating lattice gauge theories within quantum technologies, Eur. Phys. J. D 74, 165 (2020).
  4. G. Carleo, I. Cirac, K. Cranmer, L. Daudet, M. Schuld, N. Tishby, L. Vogt-Maranto, and L. Zdeborová, Machine learning and the physical sciences, Rev. Mod. Phys. 91, 045002 (2019).
  5. K. Choo, G. Carleo, N. Regnault, and T. Neupert, Symmetries and many-body excitations with neural-network quantum states, Phys. Rev. Lett. 121, 167204 (2018).
  6. G. Torlai, G. Mazzola, J. Carrasquilla, M. Troyer, R. Melko, and G. Carleo, Neural-network quantum state tomography, Nat. Phys. 14, 447 (2018).
  7. M. Schmitt and M. Heyl, Simulating dynamics of correlated matter with neural quantum states, arXiv:2506.03124.
  8. F. Verstraete, J. J. García-Ripoll, and J. I. Cirac, Matrix product density operators: Simulation of finite-temperature and dissipative systems, Phys. Rev. Lett. 93, 207204 (2004).
  9. J.-G. Liu, L. Mao, P. Zhang, and L. Wang, Solving quantum statistical mechanics with variational autoregressive networks and quantum circuits, Mach. Learn.: Sci. Technol. 2, 025011 (2021).
  10. E. Stoudenmire and S. R. White, Minimally entangled typical thermal state algorithms, New J. Phys. 12, 055026 (2010).
  11. S. R. White, Minimally entangled typical quantum states at finite temperature, Phys. Rev. Lett. 102, 190601 (2009).
  12. L. Bassman Oftelie, K. Klymko, D. Liu, N. M. Tubman, and W. A. de Jong, Computing free energies with fluctuation relations on quantum computers, Phys. Rev. Lett. 129, 130603 (2022).
  13. J. C. Getelina, N. Gomes, T. Iadecola, P. P. Orth, and Y.-X. Yao, Adaptive variational quantum minimally entangled typical thermal states for finite temperature simulations, SciPost Phys. 15, 102 (2023).
  14. B. Zhang, P. Xu, X. Chen, and Q. Zhuang, Generative quantum machine learning via denoising diffusion probabilistic models, Phys. Rev. Lett. 132, 100602 (2024).
  15. F. G. Brandao, A. W. Harrow, and M. Horodecki, Local random quantum circuits are approximate polynomial-designs, Commun. Math. Phys. 346, 397 (2016).
  16. P. Ćwikliński, M. Horodecki, M. Mozrzymas, Ł. Pankowski, and M. Studziński, Local random quantum circuits are approximate polynomial-designs: Numerical results, J. Phys. A: Math. Theor. 46, 305301 (2013).
  17. S. Lloyd, Universal quantum simulators, Science 273, 1073 (1996).
  18. A. Schuckert, O. Katz, L. Feng, E. Crane, A. De, M. Hafezi, A. V. Gorshkov, and C. Monroe, Observation of a finite-energy phase transition in a one-dimensional quantum simulator, Nat. Phys. 21, 374 (2025).
  19. K. Kumaran, M. Sajjan, B. Pokharel, J. Gibbs, J. Cohn, B. Jones, S. Mostame, S. Kais, and A. Banerjee, Superdiffusion resilience in Heisenberg chains with 2D interactions on a quantum processor, Phys. Rev. B 113, 174408 (2026).
  20. N. Ma, M. Goldstein, M. S. Albergo, N. M. Boffi, E. Vanden-Eijnden, and S. Xie, SiT: Exploring flow and diffusion-based generative models with scalable interpolant transformers, arXiv:2401.08740.
  21. Y. Lipman, R. T. Chen, H. Ben-Hamu, M. Nickel, and M. Le, Flow matching for generative modeling, arXiv:2210.02747.
  22. D. Rezende and S. Mohamed, Variational inference with normalizing flows, in International Conference on Machine Learning (JMLR.org, Lille, France, 2015), p. 1530.
  23. X. Liu, J. Zhuang, W. Hou, and Y.-Z. You, Measurement-based quantum diffusion models, arXiv:2508.08799.
  24. F. Hu, G. Liu, Y. Zhang, and X. Gao, Local diffusion models and phases of data distributions, arXiv:2508.06614.
  25. X. Wang, B. Qi, Y. Wang, and D. Dong, Entanglement-variational hardware-efficient ansatz for eigensolvers, Phys. Rev. Appl. 21, 034059 (2024).
  26. M. Motta, C. Sun, A. T. Tan, M. J. O’Rourke, E. Ye, A. J. Minnich, F. G. Brandao, and G. K.-L. Chan, Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution, Nat. Phys. 16, 205 (2020).
  27. S. McArdle, T. Jones, S. Endo, Y. Li, S. C. Benjamin, and X. Yuan, Variational ansatz-based quantum simulation of imaginary time evolution, npj Quantum Inf. 5, 75 (2019).
  28. E. Grant, L. Wossnig, M. Ostaszewski, and M. Benedetti, An initialization strategy for addressing barren plateaus in parametrized quantum circuits, Quantum 3, 214 (2019).
  29. J. Carrasquilla and R. G. Melko, Machine learning phases of matter, Nat. Phys. 13, 431 (2017).
  30. W. Roga, D. Spehner, and F. Illuminati, Geometric measures of quantum correlations: Characterization, quantification, and comparison by distances and operations, J. Phys. A: Math. Theor. 49, 235301 (2016).
  31. C. Jarzynski, Nonequilibrium equality for free energy differences, Phys. Rev. Lett. 78, 2690 (1997).
  32. J. Kurchan, A quantum fluctuation theorem, arXiv:cond-mat/0007360.
  33. H. Tasaki, Jarzynski relations for quantum systems and some applications, arXiv:cond-mat/0009244.
  34. S. Mukamel, Quantum extension of the Jarzynski relation: Analogy with stochastic dephasing, Phys. Rev. Lett. 90, 170604 (2003).
  35. P. Talkner, E. Lutz, and P. Hänggi, Fluctuation theorems: Work is not an observable, Phys. Rev. E 75, 050102(R) (2007).
  36. J. Richter and A. Pal, Simulating hydrodynamics on noisy intermediate-scale quantum devices with random circuits, Phys. Rev. Lett. 126, 230501 (2021).
  37. J. Benton, G. Deligiannidis, and A. Doucet, Error bounds for flow matching methods, Trans. Machine Learn. Res. (2024), https://openreview.net/forum?id=uqQPyWFDhY.
  38. M. C. Caro, H.-Y. Huang, N. Ezzell, J. Gibbs, A. T. Sornborger, L. Cincio, P. J. Coles, and Z. Holmes, Out-of-distribution generalization for learning quantum dynamics, Nat. Commun. 14, 3751 (2023).
  39. D. Luo, Z. Chen, J. Carrasquilla, and B. K. Clark, Autoregressive neural network for simulating open quantum systems via a probabilistic formulation, Phys. Rev. Lett. 128, 090501 (2022).
  40. Y. Tang, J. Liu, J. Zhang, and P. Zhang, Learning nonequilibrium statistical mechanics and dynamical phase transitions, Nat. Commun. 15, 1117 (2024).
  41. E. Zapusek, K. Kirova, W. Hahn, M. Marthaler, and F. Reiter, Variational quantum thermalizers based on weakly-symmetric nonunitary multi-qubit operations, Quantum Sci. Technol. 11, 025006 (2026).
  42. A. D. King, A. Nocera, M. M. Rams, J. Dziarmaga, R. Wiersema, W. Bernoudy, J. Raymond, N. Kaushal, N. Heinsdorf, and R. Harris, Beyond-classical computation in quantum simulation, Science 388, 199 (2025).
  43. X. Wu, C. Zhu, J. Wang, and X. Wang, BOSS: Blocking algorithm for optimizing shuttling scheduling in ion trap, 2025 IEEE International Symposium on High Performance Computer Architecture (HPCA) (IEEE Computer Society, Los Alamitos, CA, USA, 2025), pp. 290–303.
  44. D. Gao et al., Establishing a new benchmark in quantum computational advantage with 105-qubit Zuchongzhi 3.0 processor, Phys. Rev. Lett. 134, 090601 (2025).
  45. R. Acharya, D. A. Abanin, L. Aghababaie-Beni, I. Aleiner, T. I. Andersen, M. Ansmann, F. Arute, K. Arya, A. Asfaw, and N. Astrakhantsev, Quantum error correction below the surface code threshold, Nature (London) 638, 920 (2025).
  46. S. Bravyi, A. W. Cross, J. M. Gambetta, D. Maslov, P. Rall, and T. J. Yoder, High-threshold and low-overhead fault-tolerant quantum memory, Nature (London) 627, 778 (2024).
  47. H. Cao, F. Pan, D. Feng, Y. Wang, and P. Zhang, Generative decoding for quantum error-correcting codes, arXiv:2503.21374.
  48. P. Zhang, Correcting a noisy quantum computer using a quantum computer, arXiv:2506.08331.
  49. J. Ho, A. Jain, and P. Abbeel, Denoising diffusion probabilistic models, Advances in Neural Information Processing Systems (NeurIPS 2020), Vol. 33 (Curran Associates, Inc., 2020), pp. 6840–6851.
  50. Y. Song, J. Sohl-Dickstein, D. P. Kingma, A. Kumar, S. Ermon, and B. Poole, Score-based generative modeling through stochastic differential equations, in International Conference on Learning Representations (ICLR) (Curran Associates, Inc., New Orleans, LA, USA, 2021), pp. 37799–37812.
  51. J. Song, C. Meng, and S. Ermon, Denoising diffusion implicit models, International Conference on Learning Representations (ICLR, 2021).
  52. R. Rombach, A. Blattmann, D. Lorenz, P. Esser, and B. Ommer, High-resolution image synthesis with latent diffusion models, Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR) (IEEE, 2022), pp. 10684–10695.
  53. Y. Song, P. Dhariwal, M. Chen, and I. Sutskever, Consistency models, arXiv:2303.01469.
  54. P. Guo and A. G. Schwing, Variational rectified flow matching, Proceedings of the 42nd International Conference on Machine Learning (PMLR, 2025), Vol. 267, pp. 20921–20940.
  55. Z. Cui, The code for “Quantum flow matching”, GitHub (2026), https://github.com/Machine-learning-and-complex-systems/QFM.git.
  56. A. Kandala, A. Mezzacapo, K. Temme, M. Takita, M. Brink, J. M. Chow, and J. M. Gambetta, Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets, Nature (London) 549, 242 (2017).
  57. R. Wiersema, C. Zhou, Y. de Sereville, J. F. Carrasquilla, Y. B. Kim, and H. Yuen, Exploring entanglement and optimization within the Hamiltonian variational ansatz, PRX Quantum 1, 020319 (2020).
  58. C. Lyu, X. Xu, M.-H. Yung, and A. Bayat, Symmetry enhanced variational quantum spin eigensolver, Quantum 7, 899 (2023).
  59. J. Romero, R. Babbush, J. R. McClean, C. Hempel, P. J. Love, and A. Aspuru-Guzik, Strategies for quantum computing molecular energies using the unitary coupled cluster ansatz, Quantum Sci. Technol. 4, 014008 (2019).
  60. J. M. Arrazola, O. Di Matteo, N. Quesada, S. Jahangiri, A. Delgado, and N. Killoran, Universal quantum circuits for quantum chemistry, Quantum 6, 742 (2022).
  61. A. J. McRoberts and R. Moessner, Parametrically long lifetime of superdiffusion in nonintegrable spin chains, Phys. Rev. Lett. 133, 256301 (2024).
  62. K. Jacobs and D. A. Steck, A straightforward introduction to continuous quantum measurement, Contemp. Phys. 47, 279 (2006).
  63. M. Schuld and N. Killoran, Quantum machine learning in feature Hilbert spaces, Phys. Rev. Lett. 122, 040504 (2019).
  64. S. Lloyd, S. Garnerone, and P. Zanardi, Quantum algorithms for topological and geometric analysis of data, Nat. Commun. 7, 10138 (2016).
  65. H.-Y. Huang, M. Broughton, M. Mohseni, R. Babbush, S. Boixo, H. Neven, and J. R. McClean, Power of data in quantum machine learning, Nat. Commun. 12, 2631 (2021).
  66. J. Zhang, G. Pagano, P. W. Hess, A. Kyprianidis, P. Becker, H. Kaplan, A. V. Gorshkov, Z.-X. Gong, and C. Monroe, Observation of a many-body dynamical phase transition with a 53-qubit quantum simulator, Nature (London) 551, 601 (2017).
  67. F. Arute, K. Arya, R. Babbush, D. Bacon, J. C. Bardin, R. Barends, R. Biswas, S. Boixo, F. G. Brandao, and D. A. Buell, Quantum supremacy using a programmable superconducting processor, Nature (London) 574, 505 (2019).
  68. H. J. Kimble, The quantum internet, Nature (London) 453, 1023 (2008).
  69. S. Wehner, D. Elkouss, and R. Hanson, Quantum internet: A vision for the road ahead, Science 362, eaam9288 (2018).
  70. A. Nahum, J. Ruhman, S. Vijay, and J. Haah, Quantum entanglement growth under random unitary dynamics, Phys. Rev. X 7, 031016 (2017).
  71. M. Z. Hasan and C. L. Kane, Colloquium: Topological insulators, Rev. Mod. Phys. 82, 3045 (2010).
  72. Z.-Y. Wang, X.-C. Cheng, B.-Z. Wang, J.-Y. Zhang, Y.-H. Lu, C.-R. Yi, S. Niu, Y. Deng, X.-J. Liu, and S. Chen, Realization of an ideal Weyl semimetal band in a quantum gas with 3D spin-orbit coupling, Science 372, 271 (2021).

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PRX Intelligence 1, 013009– Published 6 August, 2026
  • Received 30 August 2025
  • Accepted 25 June 2026
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Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

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