- Physical Review Journals
All JournalsPhysics Magazine

Physical Review Research

Reuse & Permissions

It is not necessary to obtain permission to reuse this article or its components as it is available under the terms of the Creative Commons Attribution 4.0 International license. This license permits unrestricted use, distribution, and reproduction in any medium, provided attribution to the author(s) and the published article's title, journal citation, and DOI are maintained. Please note that some figures may have been included with permission from other third parties. It is your responsibility to obtain the proper permission from the rights holder directly for these figures.

Export citation

Export citation

Choose format for download:

Download Citation
  • Open Access

Quantum-ready formulation for uncompromised data assimilation with high-rank covariances

Takemasa Miyoshi*

  • *Contact author: takemasa.miyoshi@riken.jp
PDF

Phys. Rev. Research 8, 033287 – Published 8 September, 2026

DOI: https://doi.org/10.1103/x91t-v327

Abstract

Data assimilation (DA) optimally fuses observational data with mathematical models, serving as a core computational framework for forecasting complex dynamical systems like the weather. Because manipulating large-scale high-rank error covariances is classically intractable, conventional DA is forced into severe structural compromises such as artificial localization or static parametrizations that destroy genuine cross-variable correlations. Future fault-tolerant quantum computing (FTQC) could offer transformative advantages to overcome these classical computational limits. While quantum linear system solvers theoretically provide exponential speedups for matrix inversion, the practical mapping of large-scale, uncompromised DA onto quantum architectures remains largely unexplored. Our framework builds upon two emerging technologies: FTQC hardware and efficient block-encoded oracles of the error covariance and observation operators, which we envision being supplied by future artificial intelligence surrogate models. Under this premise, we identify a potential bottleneck in the quantum implementation. We show that a direct mapping of the DA equations leads to a quadratic increase in the subnormalization factor (α) associated with block encoding when typical nonlocal observations are utilized. To mitigate this algorithmic overhead, we propose a quantum-ready, combined state-observation space formulation. By reconfiguring the DA problem into an expanded saddle-point block-matrix system, our approach avoids the explicit multiplication of nonlocal observation operators. This maintains a linear scaling of α and confines the conditioning of the quantum solver to that of the observation error covariance, decoupled from the severely ill-conditioned prior. Numerical verification using the chaotic Lorenz-96 model demonstrates that our formulation achieves perfect fidelity with the uncompromised DA solution. Although reading out the full analysis state reduces the exponential speedup to a polynomial one, the framework retains a substantial advantage over the classical matrix inversion. This work bridges computational geosciences and quantum information science, providing the essential algorithmic foundation for large-scale DA to become a compelling application for FTQC.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (34)

  1. Y. Sasaki, Some basic formalisms in numerical variational analysis, Mon. Weather Rev. 98, 875 (1970).
  2. P. Bauer, A. Thorpe, and G. Brunet, The quiet revolution of numerical weather prediction, Nature (London) 525, 47 (2015).
  3. A. C. Lorenc, Analysis methods for numerical weather prediction, Q. J. R. Meteorol. Soc. 112, 1177 (1986).
  4. P. Bauer et al., The digital revolution of Earth-system science, Nat. Comput. Sci. 1, 104 (2021).
  5. G. Evensen, Sequential data assimilation with a nonlinear quasi-geostrophic model using Monte Carlo methods to forecast error statistics, J. Geophys. Res. 99, 10143 (1994).
  6. P. L. Houtekamer and F. Zhang, Review of the ensemble Kalman filter for atmospheric data assimilation, Mon. Weather Rev. 144, 4489 (2016).
  7. D. J. Patil et al., Local low dimensionality of atmospheric dynamics, Phys. Rev. Lett. 86, 5878 (2001).
  8. T. Miyoshi, K. Kondo, and T. Imamura, The 10 240-member ensemble Kalman filtering with an intermediate AGCM, Geophys. Res. Lett. 41, 5264 (2014).
  9. T. Miyoshi, K. Kondo, and K. Terasaki, Big ensemble data assimilation in numerical weather prediction, Computer 48, 15 (2015).
  10. K. Kondo and T. Miyoshi, Impact of removing covariance localization in an ensemble Kalman filter: Experiments with 10 240 members using an intermediate AGCM, Mon. Weather Rev. 144, 4849 (2016).
  11. A. Chattopadhyay, E. Nabizadeh, E. Bach, and P. Hassanzadeh, Deep learning-enhanced ensemble-based data assimilation for high-dimensional nonlinear dynamical systems, J. Comput. Phys. 477, 111918 (2023).
  12. M. A. Sacco, J. J. Ruiz, M. Pulido, and P. Tandeo, Evaluation of machine learning techniques for forecast uncertainty quantification, Q. J. R. Meteorol. Soc. 148, 3470 (2022).
  13. S. Kotsuki, F. Kawasaki, and M. Ohashi, Quantum data assimilation: A new approach to solving data assimilation on quantum annealers, Nonlinear Processes Geophys. 31, 237 (2024).
  14. A. W. Harrow, A. Hassidim, and S. Lloyd, Quantum algorithm for linear systems of equations, Phys. Rev. Lett. 103, 150502 (2009).
  15. J.-P. Liu, H. Ø. Kolden, H. K. Krovi, N. F. Loureiro, K. Trivisa, and A. M. Childs, Efficient quantum algorithm for dissipative nonlinear differential equations, Proc. Natl. Acad. Sci. USA 118, e2026805118 (2021).
  16. J.-P. Liu, D. An, D. Fang, J. Wang, G. H. Low, and S. Jordan, Efficient quantum algorithm for nonlinear reaction-diffusion equations and energy estimation, Commun. Math. Phys. 404, 963 (2023).
  17. H. Shi, G. Zhang, and M. Zhang, A quantum algorithm for the Kalman filter using block encoding, arXiv:2404.04554.
  18. N. Binandeh Dehaghani, R. Wisniewski, and A. P. Aguiar, Quantum solution framework for finite-horizon LQG control via block encodings and QSVT, in Proceedings of 2025 IEEE International Conference on Quantum Computing and Engineering (QCE) (IEEE, Piscataway, NJ, 2025), p. 282.
  19. A. Gilyén, Y. Su, G. H. Low, and N. 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 (Association for Computing Machinery, New York, NY, 2019), p. 193.
  20. E. N. Lorenz, Predictability: A problem partly solved, Proc. Seminar on Predictability 1 (ECMWF, Reading, UK, 1996).
  21. E. N. Lorenz and K. A. Emanuel, Optimal sites for supplementary weather observations: Simulation with a small model, J. Atmos. Sci. 55, 399 (1998).
  22. P. Courtier, Dual formulation of four-dimensional variational assimilation, Q. J. R. Meteorol. Soc. 123, 2449 (1997).
  23. P. Courtier, J.-N. Thépaut, and A. Hollingsworth, A strategy for operational implementation of 4D-Var, using an incremental approach, Q. J. R. Meteorol. Soc. 120, 1367 (1994).
  24. B. R. Hunt, E. J. Kostelich, and I. Szunyogh, Efficient data assimilation for spatiotemporal chaos: A local ensemble transform Kalman filter, Physica D 230, 112 (2007).
  25. M. K. Tippett, J. L. Anderson, C. H. Bishop, T. M. Hamill, and J. S. Whitaker, Ensemble square root filters, Mon. Weather Rev. 131, 1485 (2003).
  26. P. L. Houtekamer and H. L. Mitchell, A sequential ensemble Kalman filter for atmospheric data assimilation, Mon. Weather Rev. 129, 123 (2001).
  27. A. M. Childs, R. Kothari, and R. D. Somma, Quantum algorithm for systems of linear equations with exponentially improved dependence on precision, SIAM J. Comput. 46, 1920 (2017).
  28. M. Benzi, G. H. Golub, and J. Liesen, Numerical solution of saddle point problems, Acta Numer. 14, 1 (2005).
  29. Å. Björck, Numerical Methods for Least Squares Problems (SIAM, Philadelphia, 1996).
  30. K. Terasaki and T. Miyoshi, Including the horizontal observation error correlation in the ensemble Kalman filter: Idealized experiments with NICAM-LETKF, Mon. Weather Rev. 152, 277 (2024).
  31. P. P. Weston, W. Bell, and J. R. Eyre, Accounting for correlated error in the assimilation of high-resolution sounder data, Q. J. R. Meteorol. Soc. 140, 2420 (2014).
  32. M. Benedetti, E. Lloyd, S. Sack, and M. Fiorentini, Parameterized quantum circuits as machine learning models, Quantum Sci. Technol. 4, 043001 (2019).
  33. C. Zoufal, A. Lucchi, and S. Woerner, Quantum generative adversarial networks for learning and loading random distributions, npj Quantum Inf. 5, 103 (2019).
  34. T. Miyoshi, takemasa-miyoshi/quantum-da-lorenz96: v1.0.0: Code for PRR publication, Zenodo, 2026, https://doi.org/10.5281/zenodo.21972189; development version available at https://github.com/takemasa-miyoshi/quantum-da-lorenz96.

Outline

Information

Phys. Rev. Research 8, 033287– Published 8 September, 2026
  • Received 11 May 2026
  • Accepted 11 August 2026
Crossmark - Check for updates button
Creative Commons Logo

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.

Published by the American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation