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Explicit connections between Krylov and Nielsen complexity

Ben Craps1, Gabriele Pascuzzi1, Juan F. Pedraza2, Le-Chen Qu2,3, and Shan-Ming Ruan4,5

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Phys. Rev. D 114, 046024 – Published 25 August, 2026

DOI: https://doi.org/10.1103/spd2-l836

Abstract

We establish a direct correspondence between Krylov and Nielsen complexity by choosing the Krylov basis to be part of the elementary gate set of Nielsen geometry and selecting a Nielsen complexity metric compatible with the Krylov metric. Up to normalization, the Krylov complexity of a Hermitian operator then equals the length squared of a straight-line trajectory on the manifold of unitaries that connects the identity operator with a precursor operator. The corresponding length provides an upper bound on Nielsen complexity that saturates whenever the straight line is a minimal geodesic. While for general systems we can only establish saturation in the limit of small precursors, we provide evidence that in broad classes of models and for suitable initial operators there is a precise correspondence between Krylov complexity and (the square of) Nielsen complexity for a finite range of precursors.

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References (79)

  1. M. A. Nielsen, A geometric approach to quantum circuit lower bounds, Quantum Inf. Comput. 6, 213 (2006).
  2. M. A. Nielsen, M. R. Dowling, M. Gu, and A. C. Doherty, Quantum computation as geometry, Science 311, 1133 (2006).
  3. M. R. Dowling and M. A. Nielsen, The geometry of quantum computation, Quantum Inf. Comput. 8, 0861 (2008).
  4. R. Jefferson and R. C. Myers, Circuit complexity in quantum field theory, J. High Energy Phys. 10 (2017) 107.
  5. R. Khan, C. Krishnan, and S. Sharma, Circuit complexity in fermionic field theory, Phys. Rev. D 98, 126001 (2018).
  6. L. Hackl and R. C. Myers, Circuit complexity for free fermions, J. High Energy Phys. 07 (2018) 139.
  7. S. Chapman, J. Eisert, L. Hackl, M. P. Heller, R. Jefferson, H. Marrochio, and R. C. Myers, Complexity and entanglement for thermofield double states, SciPost Phys. 6, 034 (2019).
  8. A. Bhattacharyya, A. Shekar, and A. Sinha, Circuit complexity in interacting QFTs and RG flows, J. High Energy Phys. 10 (2018) 140.
  9. M. Guo, J. Hernandez, R. C. Myers, and S.-M. Ruan, Circuit complexity for coherent states, J. High Energy Phys. 10 (2018) 011.
  10. A. Bernamonti, F. Galli, J. Hernandez, R. C. Myers, S.-M. Ruan, and J. Simón, First law of holographic complexity, Phys. Rev. Lett. 123, 081601 (2019).
  11. A. Bernamonti, F. Galli, J. Hernandez, R. C. Myers, S.-M. Ruan, and J. Simón, Aspects of the first law of complexity, J. Phys. A 53, 29 (2020).
  12. E. Caceres, S. Chapman, J. D. Couch, J. P. Hernandez, R. C. Myers, and S.-M. Ruan, Complexity of mixed states in QFT and holography, J. High Energy Phys. 03 (2020) 012.
  13. S.-M. Ruan, Purification complexity without purifications, J. High Energy Phys. 01 (2021) 092.
  14. S. S. Haque, G. Jafari, and B. Underwood, Universal early-time growth in quantum circuit complexity, J. High Energy Phys. 10 (2024) 101.
  15. D. E. Parker, X. Cao, A. Avdoshkin, T. Scaffidi, and E. Altman, A universal operator growth hypothesis, Phys. Rev. X 9, 041017 (2019).
  16. 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.
  17. A. Avdoshkin and A. Dymarsky, Euclidean operator growth and quantum chaos, Phys. Rev. Res. 2, 043234 (2020).
  18. 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.
  19. S.-K. Jian, B. Swingle, and Z.-Y. Xian, Complexity growth of operators in the SYK model and in JT gravity, J. High Energy Phys. 03 (2021) 014.
  20. A. Dymarsky and M. Smolkin, Krylov complexity in conformal field theory, Phys. Rev. D 104, L081702 (2021).
  21. N. Hörnedal, N. Carabba, A. S. Matsoukas-Roubeas, and A. del Campo, Ultimate speed limits to the growth of operator complexity, Commun. Phys. 5, 207 (2022).
  22. 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).
  23. J. Erdmenger, S.-K. Jian, and Z.-Y. Xian, Universal chaotic dynamics from Krylov space, J. High Energy Phys. 08 (2023) 176.
  24. P. Caputa, H.-S. Jeong, S. Liu, J. F. Pedraza, and L.-C. Qu, Krylov complexity of density matrix operators, J. High Energy Phys. 05 (2024) 337.
  25. M. Baggioli, K.-B. Huh, H.-S. Jeong, K.-Y. Kim, and J. F. Pedraza, Krylov complexity as an order parameter for quantum chaotic-integrable transitions, Phys. Rev. Res. 7, 023028 (2025).
  26. B. Craps, O. Evnin, and G. Pascuzzi, Multiseed Krylov complexity, Phys. Rev. Lett. 134, 050402 (2025).
  27. K.-B. Huh, H.-S. Jeong, L. A. Pando Zayas, and J. F. Pedraza, Krylov complexity in mixed phase space, Phys. Rev. D 111, L121902 (2025).
  28. 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).
  29. 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, 1–77 (2026).
  30. E. Rabinovici, A. Sánchez-Garrido, R. Shir, and J. Sonner, Krylov complexity, arXiv:2507.06286.
  31. L. Susskind, Computational complexity and black hole horizons, Fortschr. Phys. 64, 24 (2016); 64, 44(A) (2016).
  32. D. Stanford and L. Susskind, Complexity and shock wave geometries, Phys. Rev. D 90, 126007 (2014).
  33. A. R. Brown, D. A. Roberts, L. Susskind, B. Swingle, and Y. Zhao, Holographic complexity equals bulk action?, Phys. Rev. Lett. 116, 191301 (2016).
  34. A. R. Brown, D. A. Roberts, L. Susskind, B. Swingle, and Y. Zhao, Complexity, action, and black holes, Phys. Rev. D 93, 086006 (2016).
  35. R.-G. Cai, S.-M. Ruan, S.-J. Wang, R.-Q. Yang, and R.-H. Peng, Action growth for AdS black holes, J. High Energy Phys. 09 (2016) 161.
  36. A. Belin, R. C. Myers, S.-M. Ruan, G. Sárosi, and A. J. Speranza, Does complexity equal anything?, Phys. Rev. Lett. 128, 081602 (2022).
  37. A. Belin, R. C. Myers, S.-M. Ruan, G. Sárosi, and A. J. Speranza, Complexity equals anything II, J. High Energy Phys. 01 (2023) 154.
  38. R. Carrasco, J. F. Pedraza, A. Svesko, and Z. Weller-Davies, Gravitation from optimized computation: Einstein and beyond, J. High Energy Phys. 09 (2023) 167.
  39. E. Jørstad, R. C. Myers, and S.-M. Ruan, Complexity=anything: Singularity probes, J. High Energy Phys. 07 (2023) 223.
  40. H.-Y. Jiang, M.-T. Wang, and Y.-X. Liu, Holographic complexity and phase transition for AdS black holes, Phys. Rev. D 110, 046013 (2024).
  41. E. Caceres, R. Carrasco, and V. Patil, Lorentzian threads and generalized complexity, J. High Energy Phys. 04 (2024) 010.
  42. R. C. Myers and S.-M. Ruan, Complexity equals (almost) anything, in Proceedings of Gravity, Strings and Fields (Springer, Cham, 2025).
  43. D. Areán, H.-S. Jeong, J. F. Pedraza, and L.-C. Qu, Kasner interiors from analytic hairy black holes, J. High Energy Phys. 11 (2024) 138.
  44. H.-Y. Jiang and Y.-X. Liu, Complexity equals anything for multi-horizon black holes, J. High Energy Phys. 12 (2025) 072.
  45. M. Miyaji, S.-M. Ruan, S. Shibuya, and K. Yano, Non-perturbative overlaps in JT gravity: From spectral form factor to generating functions of complexity, J. High Energy Phys. 06 (2025) 251.
  46. M. Miyaji, S.-M. Ruan, S. Shibuya, and K. Yano, Universal time evolution of holographic and quantum complexity, Phys. Rev. Lett. 136, 151602 (2026).
  47. E. Cáceres, R. Carrasco, V. Patil, J. F. Pedraza, and A. Svesko, The landscape of complexity measures in 2D gravity, J. High Energy Phys. 10 (2025) 218.
  48. H. W. Lin, The bulk Hilbert space of double scaled SYK, J. High Energy Phys. 11 (2022) 060.
  49. E. Rabinovici, A. Sánchez-Garrido, R. Shir, and J. Sonner, A bulk manifestation of Krylov complexity, J. High Energy Phys. 08 (2023) 213.
  50. M. P. Heller, J. Papalini, and T. Schuhmann, Krylov spread complexity as holographic complexity beyond JT gravity, Phys. Rev. Lett. 135, 151602 (2025).
  51. V. Balasubramanian, J. M. Magan, P. Nandi, and Q. Wu, Spread complexity and the saturation of wormhole size, Phys. Rev. D 113, 046004 (2026).
  52. S. E. Aguilar-Gutierrez, Towards complexity in de Sitter space from the doubled-scaled Sachdev-Ye-Kitaev model, J. High Energy Phys. 10 (2024) 107.
  53. J. Xu, On chord dynamics and complexity growth in double-scaled SYK, J. High Energy Phys. 06 (2025) 259.
  54. M. P. Heller, F. Ori, J. Papalini, T. Schuhmann, and M.-T. Wang, De Sitter holographic complexity from Krylov complexity in DSSYK, arXiv:2510.13986.
  55. M. Ambrosini, E. Rabinovici, and J. Sonner, Holography of K-complexity: Switchbacks and shockwaves, J. High Energy Phys. 06 (2026) 060.
  56. 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.
  57. C. Lv, R. Zhang, and Q. Zhou, Building Krylov complexity from circuit complexity, Phys. Rev. Res. 6, L042001 (2024).
  58. P. Caputa, J. M. Magan, and D. Patramanis, Geometry of Krylov complexity, Phys. Rev. Res. 4, 013041 (2022).
  59. B. Craps, O. Evnin, and G. Pascuzzi, A relation between Krylov and Nielsen complexity, Phys. Rev. Lett. 132, 160402 (2024).
  60. S. E. Aguilar-Gutierrez and A. Rolph, Krylov complexity is not a measure of distance between states or operators, Phys. Rev. D 109, L081701 (2024).

  61. Unlike some conventions in the literature, we absorb the factor in into the basis |On) so that each element is manifestly Hermitian.

  62. L. Chen, B. Mu, H. Wang, and P. Zhang, Dissecting quantum many-body chaos in the Krylov space, Phys. Rev. Lett. 134, 190403 (2025).

  63. Since f(0)=0, the matrix is only positive semidefinite;

    we set f(0)=δ and take δ→0 at the end. More generally, any monotonically increasing f(n) is admissible [22].

  64. L. Susskind and Y. Zhao, Switchbacks and the bridge to nowhere, arXiv:1408.2823.
  65. D. A. Roberts, D. Stanford, and L. Susskind, Localized shocks, J. High Energy Phys. 03 (2015) 051.
  66. See Supplemental Material at http://link.aps.org/supplemental/10.1103/spd2-l836 for the large-q analysis of Krylov wavefunctions and operator size in the SYK model, the analysis of conjugate points in single-qubit Nielsen geometry, and the small-precursor expansion of Nielsen complexity.
  67. L. Susskind, New concepts for old black holes, arXiv:1311.3335.
  68. T. Ali, A. Bhattacharyya, S. S. Haque, E. H. Kim, N. Moynihan, and J. Murugan, Chaos and complexity in quantum mechanics, Phys. Rev. D 101, 026021 (2020).
  69. L.-C. Qu, J. Chen, and Y.-X. Liu, Chaos and complexity for inverted harmonic oscillators, Phys. Rev. D 105, 126015 (2022).
  70. L.-C. Qu, H.-Y. Jiang, and Y.-X. Liu, Chaos and multifold complexity for an inverted harmonic oscillator, J. High Energy Phys. 12 (2022) 065.

  71. Higher-order corrections in this expansion were previously discussed in Ref. [14] and can be directly applied to the precursor considered here, as explained in part C of the Supplemental Material [66].

  72. The chosen trace pairing (TI,TJ)=δIJ naturally identifies the dual algebra g* with the Lie algebra g.

  73. Z. Zimborás, R. Zeier, M. Keyl, and T. Schulte-Herbrüggen, A dynamic systems approach to fermions and their relation to spins, Eur. Phys. J. Quantum Technol. 1, 11 (2014).
  74. T. Frankel, The Geometry of Physics: An Introduction (Cambridge University Press, Cambridge, England, 2004).
  75. V. Balasubramanian, M. Decross, A. Kar, and O. Parrikar, Quantum complexity of time evolution with chaotic Hamiltonians, J. High Energy Phys. 01 (2020) 134,

  76. The time dependence is encoded in the eigenstates as |n⟩≡|n(t)⟩=eiHt|n0⟩, where O(0)|n0⟩=ωn|n0⟩.

  77. S. Sachdev and J. Ye, Gapless spin-fluid ground state in a random quantum Heisenberg magnet, Phys. Rev. Lett. 70, 3339 (1993).
  78. A. Kitaev, A simple model of quantum holography, Talks at KITP (2015), available at: http://online.kitp.ucsb.edu/online/entangled15/kitaev/.
  79. J. Maldacena and D. Stanford, Remarks on the Sachdev-Ye-Kitaev model, Phys. Rev. D 94, 106002 (2016).

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Phys. Rev. D 114, 046024– Published 25 August, 2026
  • Received 6 December 2025
  • Revised 19 May 2026
  • Accepted 7 August 2026
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