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This research establishes Krylov complexity as a comprehensive descriptor for quantum operator dynamics, offering an explicit recursive method to derive Lanczos coefficients directly from its Taylor expansion. The study clarifies the theoretical boundaries between different complexity measures and provides a proof of principle for characterizing evolution in isolated quantum systems without auxiliary dynamical inputs.

Surse care acoperăarXiv, The Quantum Insider

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Krylov complexity has it all

arXiv22 iulie 20262 min de cititArticolul și perspectiva sunt în engleză.

arXiv:2605.28681v3 Announce Type: replace-cross 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.

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.

11 pages, 1 table, no figures, v.2: references added, v.3: extended caveat of the algorithm, one more reference

arXiv:2605.28681 [hep-th]

(or arXiv:2605.28681v3 [hep-th] for this version)

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