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arXiv:2605.28681 · Physical review. D/Physical review. D.

Krylov complexity has it all

Wolfgang Mück

hep-thquant-phSubmitted May 27, 2026• Announced May 28, 2026• 0 citations• Revised Jun 3, 2026 (v2)
v1 · May 27, 2026v2 · Jun 3, 2026v3 · Jul 21, 2026v4 · Sep 10, 2026
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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.

Journal reference
PRD 114, L061702 (2026)
DOI
10.1103/3g6c-jx8t
Comments
11 pages, 1 table, no figures, v.2: references added

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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=0t=0t=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.
Krylov complexity is a powerful diagnostic of quantum dynamics, with clear connections to other measures of quantum chaos and operator growth. One such measure is the Loschmidt amplitude, defined as the overlap of initially identical states evolved under two slightly different Hamiltonians. Its decay in certain systems is controlled by the classical Lyapunov exponent. Using the algebraic properties of the Krylov complexity operator, we express Krylov complexity as the derivative of a Loschmidt amplitude whose perturbation is parameterized by an angular variable φφφ. This formulation allows us to define a spectral propagator that encodes the entire complexity distribution, which we characterize for specific types of systems. We study the two-dimensional quantum geometry spanned by time and φφφ where the original and deformed trajectories reside, demonstrating that Krylov complexity is upper-bounded by its volume. We also express the time derivative of Krylov complexity in terms of a distinct Loschmidt amplitude. Depending on the growth of the Lanczos coefficients, the perturbation term in this amplitude can be truncated. We propose that the strength of this perturbation provides a classification scheme for Krylov complexity dynamics and relate it to the φφφ-derivative of the spectral propagator. Using this analytical framework, we derive general relations between the time-dependence of the survival amplitude and Krylov space measures.
A bstract We investigate the Krylov complexity of Schrödinger field theories, focusing on both bosonic and fermionic systems within the grand canonical ensemble which includes a chemical potential. Krylov complexity measures operator growth in quantum systems by analyzing how operators spread within the Krylov space, a subspace of the Hilbert space spanned by successive applications of the superoperator [ H , ·] on an initial operator. Using the Lanczos algorithm, we construct an orthonormal Krylov basis and derive the Lanczos coefficients, which govern the operator connectivity and thus characterize the complexity. Our study reveals that the Lanczos coefficients { b n } are almost independent of the chemical potential, while { a n } are dependent on the chemical potential. Both { a n } and { b n } show linear relationships with respect to n . For both bosonic and fermionic systems, the Krylov complexities behave similarly over time, especially at late times, due to the analogous profiles of the squared absolute values of their autocorrelation functions | φ 0 ( t )| 2 . The Krylov complexity grows exponentially with time, but its asymptotic scaling factor λ K is significantly smaller than the twice of the slope of the { b n } coefficients, contrasting to the relativistic field theories where the scaling aligns more closely with the twice of the slope of { b n }.
A bstract Quantifying complexity in quantum systems has witnessed a surge of interest in recent years, with Krylov-based measures such as Krylov complexity ( C K ) and Spread complexity ( C S ) gaining prominence. In this study, we investigate their interplay by considering the complexity of states represented by density matrix operators . After setting up the problem, we analyze a handful of analytical and numerical examples spanning generic two-dimensional Hilbert spaces, qubit states, quantum harmonic oscillators, and random matrix theories, uncovering insightful relationships. For generic pure states, our analysis reveals two key findings: (I) a correspondence between moment-generating functions (of Lanczos coefficients) and survival amplitudes, and (II) an early-time equivalence between C K and 2 C S . Furthermore, for maximally entangled pure states, we find that the moment-generating function of C K becomes the Spectral Form Factor and, at late-times, C K is simply related to NC S for N ≥ 2 within the N -dimensional Hilbert space. Notably, we confirm that C K = 2 C S holds across all times when N = 2. Through the lens of random matrix theories, we also discuss deviations between complexities at intermediate times and highlight subtleties in the averaging approach at the level of the survival amplitude.
In this work we study the relationship between quantum random walks on graphs and Krylov/spread complexity. We show that the latter's definition naturally emerges through a canonical method of reducing a graph to a chain, on which we can identify the usual Krylov structure. We use this identification to construct families of graphs corresponding to special classes of systems with known complexity features and conversely, to compute Krylov complexity for graphs of physical interest. The two main outcomes are the analytic computation of the Lanczos coefficients for the SYK model for an arbitrary number qqq of interacting fermions and the complete characterization of Krylov complexity for the hypercube graph in any number of dimensions. The latter serves as the starting point for an in-depth comparison between Krylov and circuit complexities as they purportedly arise in the context of black holes. We find that while under certain conditions Krylov complexity follows the growth and saturation pattern ascribed to such systems, the timescale at which saturation happens can generally be shorter than what is predicted by random unitary circuits, due to the effects of quantum speed-ups commonly occurring when comparing quantum and classical random walks.
Abstract Krylov complexity is considered to provide a measure of the growth of operators evolving under Hamiltonian dynamics. The main strategy is the analysis of the structure of Krylov subspace KM(H,η)\mathcal {K}_M(\mathcal {H},\eta )KM​(H,η) spanned by the multiple applications of the Liouville operator L\mathcal {L}L defined by the commutator in terms of a Hamiltonian H\mathcal {H}H, L:=[H,⋅]\mathcal {L}:=[\mathcal {H},\cdot ]L:=[H,⋅] acting on an operator η, KM(H,η)=span{η,Lη,…,LM−1η}\mathcal {K}_M(\mathcal {H},\eta )=\text{span}\lbrace \eta ,\mathcal {L}\eta ,\ldots ,\mathcal {L}^{M-1}\eta \rbraceKM​(H,η)=span{η,Lη,…,LM−1η}. For a given inner product ( ·, ·) of the operators, the orthonormal basis {On}\lbrace \mathcal {O}_n\rbrace{On​} is constructed from O0=η/(η,η)\mathcal {O}_0=\eta /\sqrt{(\eta ,\eta )}O0​=η/(η,η)​ by Lanczos algorithm. The moments μm=(O0,LmO0)\mu _m=(\mathcal {O}_0,\mathcal {L}^m\mathcal {O}_0)μm​=(O0​,LmO0​) are closely related to the important data {bn} called Lanczos coefficients. I present the exact and explicit expressions of the moments {μm} for 16 quantum mechanical systems which are exactly solvable both in the Schrödinger and Heisenberg pictures. The operator η is the variable of the eigenpolynomials. Among them six systems show a clear sign of ‘non-complexity’ as vanishing higher Lanczos coefficients bm = 0, m ≥ 3.
Krylov complexity provides a powerful framework for characterizing the dynamical evolution of quantum systems through the spreading of states in Krylov space. The motivation for this is rooted in the optimality of the Krylov basis for the analyzed cost function. In this work, we reinterpret the motivation for the Krylov basis from a dynamical perspective and show that it corresponds to a first-order approximation of the time-evolution operator. We extend this framework to higher-order generators and analytically disprove the optimality assumption by showing that an infinite-order generator can be constructed to exhibit smaller spread for arbitrary times. We propose a natural time scale for the construction of these higher-order generators and discuss results for matrices sampled from Gaussian Unitary Ensembles, demonstrating smaller Krylov complexity at all higher orders. These results extend the framework of Krylov complexity beyond the conventional Krylov basis by disproving the widely held assumption of optimality, extending the construction to higher-order generators, and introducing a physically motivated method for their construction. Our findings therefore suggest that previous statements and results concerning Krylov complexity may need to be reconsidered.