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  • Letter
  • Open Access

Krylov complexity has it all

Wolfgang Mück

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Phys. Rev. D 114, L061702 – Published 28 September, 2026

DOI: https://doi.org/10.1103/3g6c-jx8t

Abstract

This paper establishes that Krylov complexity contains the entire information about the dynamics of a quantum operator, extending the list of equivalent quantities that can serve this purpose, such as the Lanczos coefficients, the return amplitude, and the spectral density. To demonstrate this equivalence, an explicit recursive algorithm is constructed to calculate Lanczos coefficients from the Taylor expansion of the Krylov complexity around t=0. Furthermore, the paper discusses the distinction between Krylov and spread complexity, clarifying why a similar recursive algorithm cannot exist for the latter without additional dynamical input. These results provide a “proof of principle” for using Krylov complexity as a complete characterization of operator evolution in quantum systems.

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References (34)

  1. D. E. Parker, X. Cao, A. Avdoshkin, T. Scaffidi, and E. Altman, A universal operator growth hypothesis, Phys. Rev. X 9, 041017 (2019).
  2. J. L. F. Barbón, E. Rabinovici, R. Shir, and R. Sinha, On the evolution of operator complexity beyond scrambling, J. High Energy Phys. 10 (2019) 264.
  3. A. Avdoshkin and A. Dymarsky, Euclidean operator growth and quantum chaos, Phys. Rev. Res. 2, 043234 (2020).
  4. P. Caputa, J. M. Magan, and D. Patramanis, Geometry of Krylov complexity, Phys. Rev. Res. 4, 013041 (2022).
  5. E. Rabinovici, A. Sánchez-Garrido, R. Shir, and J. Sonner, Operator complexity: A journey to the edge of Krylov space, J. High Energy Phys. 06 (2021) 062.
  6. V. Viswanath and G. Müller, The Recursion Method, Lecture Notes in Physics Vol. 23 (Springer-Verlag, Belin, 1994).
  7. C. Lanczos, An iteration method for the solution of the eigenvalue problem of linear differential and integral operators, J. Res. Nation al Bureau of Standards 45, 255 (1950).
  8. V. Balasubramanian, P. Caputa, J. M. Magan, and Q. Wu, Quantum chaos and the complexity of spread of states, Phys. Rev. D 106, 046007 (2022).
  9. M. Alishahiha and S. Banerjee, A universal approach to Krylov state and operator complexities, SciPost Phys. 15, 080 (2023).
  10. J. Erdmenger, S.-K. Jian, and Z.-Y. Xian, Universal chaotic dynamics from Krylov space, J. High Energy Phys. 08 (2023) 176.
  11. P. Nandy, A. S. Matsoukas-Roubeas, P. Martínez-Azcona, A. Dymarsky, and A. del Campo, Quantum dynamics in Krylov space: Methods and applications, Phys. Rep. 1125–1128, 1 (2025).
  12. E. Rabinovici, A. Sánchez-Garrido, R. Shir, and J. Sonner, Krylov complexity, arXiv:2507.06286.
  13. S. Baiguera, V. Balasubramanian, P. Caputa, S. Chapman, J. Haferkamp, M. P. Heller, and N. Y. Halpern, Quantum complexity in gravity, quantum field theory, and quantum information science, Phys. Rep. 1159, 1 (2026).
  14. W. Mück and Y. Yang, Krylov complexity and orthogonal polynomials, Nucl. Phys. B984, 115948 (2022).
  15. P. Caputa, B. Chen, R. W. McDonald, J. Simón, and B. Strittmatter, Spread complexity rate as proper momentum, Phys. Rev. D 113, L041901 (2026).
  16. Z.-Y. Fan, Momentum-Krylov complexity correspondence, arXiv:2411.04492.
  17. E. Rabinovici, A. Sánchez-Garrido, R. Shir, and J. Sonner, A bulk manifestation of Krylov complexity, J. High Energy Phys. 08 (2023) 213.
  18. P.-Z. He, Revisit the relationship between spread complexity rate and radial momentum, arXiv:2411.19172.
  19. J. Xu, On chord dynamics and complexity growth in double-scaled SYK, J. High Energy Phys. 06 (2025) 259.
  20. M. Ambrosini, E. Rabinovici, A. Sánchez-Garrido, R. Shir, and J. Sonner, Operator K-complexity in DSSYK: Krylov complexity equals bulk length, J. High Energy Phys. 08 (2025) 059.
  21. Z. Li and J. Tian, The holography of spread complexity: A story of observers, J. High Energy Phys. 07 (2026) 187.
  22. S. E. Aguilar-Gutierrez and J. Xu, Geometry of chord intertwiner, multiple shocks and switchback in double-scaled SYK, J. High Energy Phys. 02 (2026) 246.
  23. M. Ambrosini, E. Rabinovici, and J. Sonner, Holography of K-complexity: Switchbacks and shockwaves, J. High Energy Phys. 06 (2026) 060.
  24. H. Nastase, C. Nunez, and D. Roychowdhury, Holographic Krylov complexity for charged, composite and extended probes, arXiv:2604.07432.
  25. D. Zoakos, Holographic Krylov complexity in the Coulomb branch of N=4 SYM, J. High Energy Phys. 06 (2026) 066.
  26. Y. Fu, H.-S. Jeong, K.-Y. Kim, and J. F. Pedraza, Toward Krylov-based holography in double-scaled SYK, J. High Energy Phys. 05 (2026) 056.
  27. H.-S. Jeong, Krylov subspace dynamics as near-horizon AdS2 holography, arXiv:2602.11627.
  28. A. Fatemiabhari, H. Nastase, and D. Roychowdhury, Holographic Krylov complexity in N=4 SYM, Phys. Rev. D 113, 106033 (2026).
  29. A. Fatemiabhari, H. Nastase, C. Nunez, and D. Roychowdhury, Holographic Krylov complexity for conformal quiver gauge theories, Nucl. Phys. B1025, 117402 (2026).
  30. A. Fatemiabhari, H. Nastase, C. Nunez, and D. Roychowdhury, Holographic Krylov complexity in confining gauge theories, arXiv:2511.22717.
  31. A. Fatemiabhari and C. Nunez, Krylov complexity, confinement and universality, J. High Energy Phys. 07 (2026) 035.
  32. A. Fatemiabhari, C. Nunez, and R. T. Santamaria, Complexity and operator growth in holographic 6d SCFTs, Nucl. Phys. B1028, 117496 (2026).
  33. E. Alfinito and M. Beccaria, Krylov correlators in sl(2,R) models: Exact results and holographic complexity, J. High Energy Phys. 08 (2026) 144.
  34. D. Chatzis, M. Hammond, C. Nunez, A. V. Ramallo, and R. T. Santamaria, Holographic spread complexity from branes and strings, arXiv:2607.00074.

Outline

Information

Phys. Rev. D 114, L061702– Published 28 September, 2026
  • Received 4 June 2026
  • Accepted 9 September 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. Funded by SCOAP3.

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