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arXiv:2602.06150 · Physical review. D/Physical review. D.

Krylov Distribution

Mohsen Alishahiha, Mohammad Javad Vasli

hep-thcond-mat.stat-mechquant-phSubmitted Feb 5, 2026• Announced Feb 9, 2026• 0 citations• Revised Feb 15, 2026 (v2)
v1 · Feb 5, 2026v2 · Feb 15, 2026
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Abstract

We introduce the Krylov distribution $\mathcal{D}(ξ)$, a static Krylov-space diagnostic that characterizes how inverse-energy response is organized in Hilbert space. The central object is the resolvent-dressed state $(H-ξ)^{-1}|ψ_0\rangle$, whose decomposition in the Krylov basis generated from a reference state defines a normalized distribution over Krylov levels. Unlike conventional spectral functions, which resolve response solely along the energy axis, the Krylov distribution captures how the resolvent explores the dynamically accessible subspace as the spectral parameter $ξ$ is varied. Using asymptotic analysis, exact results in solvable models, and numerical studies of an interacting spin chain, we identify three universal regimes: saturation outside the spectral support, extensive growth within continuous spectra, and sublinear or logarithmic scaling near spectral edges and quantum critical points. We further show that fidelity susceptibility and the quantum geometric tensor admit natural decompositions in terms of Krylov-resolved resolvent amplitudes.

Comments
22 pages, 9 figures. V2:Comments added, Refs. added

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We introduce the Krylov distribution D(ξ)\mathcal{D}(ξ)D(ξ), a static Krylov-space diagnostic that characterizes how inverse-energy response is organized in Hilbert space. The central object is the resolvent-dressed state (H−ξ)−1∣ψ0⟩(H-ξ)^{-1}|ψ_0\rangle(H−ξ)−1∣ψ0​⟩, whose decomposition in the Krylov basis generated from a reference state defines a normalized distribution over Krylov levels. Unlike conventional spectral functions, which resolve response solely along the energy axis, the Krylov distribution captures how the resolvent explores the dynamically accessible subspace as the spectral parameter ξξξ is varied. Using asymptotic analysis, exact results in solvable models, and numerical studies of an interacting spin chain, we identify three universal regimes: saturation outside the spectral support, extensive growth within continuous spectra, and sublinear or logarithmic scaling near spectral edges and quantum critical points. We further show that fidelity susceptibility and the quantum geometric tensor admit natural decompositions in terms of Krylov-resolved resolvent amplitudes.
We establish a direct correspondence between the Krylov hierarchy of quantum Fisher information (QFI) and a Krylov-resolved decomposition of the symmetric logarithmic derivative (SLD). Building on the existing Krylov formulation of QFI, we show that the metrological information resolved at each Krylov level is exactly encoded in the incremental gain of the Lanczos hierarchy, giving the Krylov distribution a direct information-theoretic interpretation. A dual spectral formulation distinguishes the Hilbert--Schmidt geometry used to construct the computational Lanczos hierarchy from the state-dependent geometry in which the SLD and QFI are naturally defined, and relates their associated Liouville measures. This distinction clarifies the origin of different convergence regimes: we recover exponential convergence for gapped spectra and derive an exact algebraic hard-edge convergence law for the Jacobi class, including the exact finite-order error and the marginal case separating finite from divergent QFI.
Krylov state complexity, or spread complexity, has emerged as a sharp and versatile diagnostic of quantum chaos, information spreading, and many-body dynamics. Built from the Lanczos algorithm and grounded in the optimal-basis theorem, Krylov complexity thereby provides a robust spectroscopic window into quantum dynamics. A central theme is the characteristic overshoot observed in chaotic systems: a complexity peak in which chaotic evolution drives the state deeper into the Krylov chain than in integrable systems before relaxing to equilibrium. This behavior, tied to random-matrix universality classes of spectral statistics, is illustrated across a broad range of models, including quantum billiards, quantum spin chains, and variants of the SYK model. We also discuss proposed holographic descriptions of Krylov complexity in Einstein gravity, and conclude by outlining future directions and open problems, including time-dependent systems and quantum-field-theoretic formulations. A Mathematica notebook is provided for numerical exploration of Krylov complexity and spectral statistics across models.
A central challenge in diagnosing quantum chaos is to distinguish genuine many-body scrambling from kinematic effects of the spectrum. Krylov state complexity, or spread complexity, has emerged as a powerful diagnostic, with its characteristic growth, peak, and relaxation often taken as signatures of chaos. However, previous work has shown that this criterion can give false positives: saddle-dominated integrable systems may display prominent peaks even without random-matrix level correlations. We argue, based on complementary numerical and analytical evidence, that this ambiguity can be resolved by unfolding the spectrum prior to constructing the ensuing Krylov dynamics. By removing the non-universal smooth density of states while retaining microscopic spectral correlations, unfolding suppresses spurious peaks in integrable systems while preserving the universal spectral signatures of chaotic systems. Analytically, the formulation of the Lanczos iteration in terms of orthogonal polynomials yields an exact complexity kernel with a robust near-diagonal structure whose fine-grained features reflect the underlying spectral correlations. Moreover, for the logarithmic model, unfolding can be performed exactly, mapping the spectrum to a uniform lattice and yielding an analytic spread complexity that removes the false-positive peak. These findings establish unfolded Krylov complexity as a more reliable probe of genuine many-body scrambling.
Abstract Krylov complexity measures the spread of the wavefunction in the Krylov basis, which is constructed using the Hamiltonian and an initial state. We investigate the evolution of the maximally entangled state in the Krylov basis for both chaotic and non-chaotic systems. For this purpose, we derive an Ehrenfest theorem for the Krylov complexity, which reveals its close relation to the spectrum. Our findings suggest that neither the linear growth nor the saturation of Krylov complexity is necessarily associated with chaos. However, for chaotic systems, we observe a universal rise-slope-ramp-plateau behavior in the transition probability from the initial state to one of the Krylov basis states. Moreover, a long ramp in the transition probability is a signal for spectral rigidity, characterizing quantum chaos. Also, this ramp is directly responsible for the late-time peak of Krylov complexity observed in the literature. On the other hand, for non-chaotic systems, this long ramp is absent. Therefore, our results help to clarify which features of the wave function time evolution in Krylov space characterize chaos. We exemplify this by considering the Sachdev-Ye-Kitaev model with two-body or four-body interactions.
Quantum Krylov subspace methods can extract ground and excited states by diagonalizing the Hamiltonian in a compact variational space. In practice, these spaces are almost always generated by real or imaginary time evolution, forcing a timestep trade-off between dynamical accuracy and basis collapse and often producing ill-conditioned overlap matrices that stall convergence. Here we introduce Quantum Krylov using Unitary Decomposition (QKUD), a time-evolution-free construction that maps Hamiltonian powers to implementable unitaries via the Hermitian transform sin⁡(εH)/ε\sin(εH)/εsin(εH)/ε. QKUD reduces to the exact Hamiltonian-power Krylov recursion as ε→0ε\rightarrow0ε→0, while finite εεε provides a controllable deformation that tunes subspace geometry and improves conditioning. Across molecular active-space benchmarks and a frustrated 2D J1-J2 Heisenberg model, QKUD reproduces exact-Krylov convergence in well-conditioned regimes and systematically restores variational improvement when both exact Krylov and time-evolution Krylov stagnate. These results identify overlap conditioning, instead of time-evolution fidelity, is the key resource for robust quantum Krylov simulation and provide a resilient way forward for accurate quantum simulation of challenging quantum many-body problems.
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.