Krylov Vector Ergodicity

• Krylov vector ergodicity is a quantification of how quantum dynamics generate nearly orthogonal, delocalized vectors in Krylov space from an initial state.
• It uses methods like Arnoldi and Lanczos iterations to translate unitary or Liouvillian actions into measures of complexity and state-space exploration.
• The framework distinguishes between ergodic spreading, fractality, and localization, providing insights into thermalization, operator dynamics, and numerical precision challenges.

Krylov vector ergodicity is a Krylov-space characterization of whether repeated unitary or Liouvillian action on a seed state or operator generates vectors that are broadly explored, nearly orthogonal, and dynamically delocalized, or instead remain confined to a small structured sector. In the recent literature the phrase has both a narrow and a broad usage. In a narrow sense, it refers to the extent to which dynamics generates a set of nearly orthonormal vectors over time from a single initial vector, quantified through the closeness of a map $R_2$ from an orthonormal basis to a Krylov set to a unitary transformation (Vikram, 10 Jul 2025). In a broader sense, several works use Krylov-space dynamics to diagnose ergodicity, nonthermalization, constrained dynamics, fractality, or weak ergodicity breaking through the structure of Lanczos or Arnoldi coefficients, the spreading of amplitudes over Krylov basis vectors, or the persistence of low-dimensional symmetry sectors (Scialchi et al., 2024, Malik et al., 23 Dec 2025, Loizeau et al., 11 Feb 2026).

1. Krylov constructions and core observables

Krylov vector ergodicity is formulated in both state and operator settings. For discrete or stroboscopic unitary evolution, one starts from an initial state $|\psi_0\rangle$ and a unitary $U$, defines $|\psi_t\rangle = U^t |\psi_0\rangle$, and considers the Krylov space

$$\mathcal K = \mathrm{span}\{ |\psi_0\rangle, U|\psi_0\rangle, U^2|\psi_0\rangle,\dots \}.$$

An orthonormal Krylov basis $\{|K_n\rangle\}$ is then generated by Arnoldi iteration,

$$b_n |K_n\rangle = U|K_{n-1}\rangle - \sum_{l=0}^{n-1} \langle K_l|U|K_{n-1}\rangle\, |K_l\rangle,$$

which yields an upper Hessenberg representation $(U_\mathcal K)_{mn}=\langle K_m|U|K_n\rangle$ (Scialchi et al., 2024).

For operator dynamics, one works in operator Hilbert space. A Hamiltonian $\hat H$ generates Heisenberg evolution

$$\hat O(t)=e^{i\hat H t}\hat O e^{-i\hat H t}, \qquad \mathcal L(\hat O)=[\hat H,\hat O],$$

so the operator Krylov space is

$|\psi_0\rangle$0

With the Hilbert–Schmidt inner product

$|\psi_0\rangle$1

or its finite-temperature variants, Lanczos or Arnoldi orthogonalization yields a Krylov basis and a tridiagonal or Hessenberg effective chain (Malik et al., 23 Dec 2025, Tan et al., 2024).

The central observables are likewise shared across settings. The Krylov amplitudes $|\psi_0\rangle$2, $|\psi_0\rangle$3, or $|\psi_0\rangle$4 define a wavefunction on the Krylov basis. Krylov complexity is the first moment of the basis index,

$|\psi_0\rangle$5

while the Lanczos or Arnoldi coefficients $|\psi_0\rangle$6 act as effective hopping amplitudes. In this language, ergodicity is associated with broad propagation of the Krylov wavefunction, whereas confinement, early closure, or persistent recurrence indicate constrained or nonergodic dynamics (Malik et al., 23 Dec 2025, Menzler et al., 2024).

2. Unitary evolution, Arnoldi structure, and the maximally ergodic regime

A distinct formulation of Krylov vector ergodicity was developed for unitary dynamics by adapting Krylov methods from Lanczos to Arnoldi form. In this framework, the unitary matrix elements in the Krylov basis are parametrized by

$|\psi_0\rangle$7

and asymptotic ergodicity is identified with the limit

$|\psi_0\rangle$8

The reduced unitary then approaches the pure lower-shift form

$|\psi_0\rangle$9

equivalently,

$U$0

This is the “maximally ergodic” regime in the Arnoldi approach (Scialchi et al., 2024).

The same regime appears in the closely related unitary-superoperator framework of classical–quantum Krylov correspondence. There the maximal possible growth of Krylov complexity is linear,

$U$1

and it is attained when

$U$2

This is again called a maximally ergodic regime. The associated interpretation is that each time step requires a completely new Krylov state, so the wavepacket moves ballistically along the Krylov chain (Scialchi et al., 11 Mar 2026).

The Arnoldi formulation also gives a direct ergodicity measure. The distance between the actual reduced unitary $U$3 and the shift matrix $U$4 is quantified by

$U$5

In the random-matrix crossover model and in spin chains, this Krylov-based ergodicity measure tracks standard chaos indicators such as level-spacing ratios and eigenvector statistics, provided the unitary spectrum is sufficiently spread on the circle (Scialchi et al., 2024).

This regime is not a universal consequence of short-time linear growth alone. In the harmonic-oscillator example of quantum-to-classical correspondence, linear growth and ballistic propagation of the Krylov wavefunction occur within each period despite completely regular dynamics. The long-time persistence of the shift-like regime, rather than short-time ballistic propagation by itself, is therefore the relevant indicator (Scialchi et al., 11 Mar 2026).

3. Operator-space ergodicity, thermalization, and constrained dynamics

In operator Krylov space, ergodicity is typically discussed as operator delocalization or its suppression. A prominent example is the nonintegrable transverse-field Ising chain with a macroscopic spatial inhomogeneity,

$U$6

with inhomogeneity parameter

$U$7

There, increasing $U$8 suppresses ergodicity. The main Krylov signatures are the dispersion pattern of the Lanczos or Arnoldi coefficients, the slowing of initial Krylov-complexity growth, and the reduction of the fraction of Krylov basis states significantly occupied by a time-evolved operator. The participation-ratio-like quantity

$U$9

drops strongly as inhomogeneity increases, supporting the interpretation of Krylov-space confinement (Malik et al., 23 Dec 2025).

A complementary operator-space picture appears in finite-temperature chaos diagnostics. For the Gaussian orthogonal ensemble, the Lanczos coefficients computed with the Wightman inner product exhibit a linear regime

$|\psi_t\rangle = U^t |\psi_0\rangle$0

while the spectral form factor defines an ergodicity indicator

$|\psi_t\rangle = U^t |\psi_0\rangle$1

After suitable size scaling, larger Krylov growth rate $|\psi_t\rangle = U^t |\psi_0\rangle$2 or $|\psi_t\rangle = U^t |\psi_0\rangle$3 correlates with larger spectral ergodicity indicator. This establishes a direct relation between faster spreading across Krylov basis vectors and stronger spectral ergodicity (Tan et al., 2024).

The closed disordered Heisenberg spin chain studied in “Krylov space perturbation theory for quantum synchronization in closed systems” gives a further refinement of the operator-space viewpoint. Its Hamiltonian is

$|\psi_t\rangle = U^t |\psi_0\rangle$4

and in the clean limit it has the dynamical symmetry

$|\psi_t\rangle = U^t |\psi_0\rangle$5

The associated Krylov orbit closes on a two-site Krylov sector. Weak disorder only opens a narrow channel out of that sector through a single Lanczos coefficient $|\psi_t\rangle = U^t |\psi_0\rangle$6, so the dressed mode frequency shifts only at second order,

$|\psi_t\rangle = U^t |\psi_0\rangle$7

This produces a long-lived, nearly nonergodic mode. At stronger disorder, the global mode fragments into local synchronized patches, which the authors interpret as fragmentation of the global dynamical symmetry into local dynamical symmetries with distinct frequencies (Loizeau et al., 11 Feb 2026).

Taken together, these results define a broad operator-Krylov notion of ergodicity: regular $|\psi_t\rangle = U^t |\psi_0\rangle$8, rapid early growth of Krylov complexity, and broad late-time support indicate ergodic operator spreading, whereas restricted exploration of the Krylov basis, persistence of small symmetry sectors, or fragmentation into local Krylov structures indicate constrained dynamics or nonthermalization (Malik et al., 23 Dec 2025, Loizeau et al., 11 Feb 2026).

4. Localization, fractality, and weak ergodicity breaking in Krylov space

A major line of work treats ergodicity breaking as localization or fractality in Krylov space itself. In “Krylov Delocalization/Localization across Ergodicity Breaking,” the Liouvillian is mapped to a semi-infinite tight-binding Krylov chain with hoppings $|\psi_t\rangle = U^t |\psi_0\rangle$9, and localization is diagnosed by the variance

$$\mathcal K = \mathrm{span}\{ |\psi_0\rangle, U|\psi_0\rangle, U^2|\psi_0\rangle,\dots \}.$$0

The paper interprets $$\mathcal K = \mathrm{span}\{ |\psi_0\rangle, U|\psi_0\rangle, U^2|\psi_0\rangle,\dots \}.$$1 as an inverse localization-length measure through

$$\mathcal K = \mathrm{span}\{ |\psi_0\rangle, U|\psi_0\rangle, U^2|\psi_0\rangle,\dots \}.$$2

Its proposal is that the ergodic regime corresponds to delocalization on the Krylov chain, while weakly ergodicity-broken regimes correspond to localization on that chain (Menzler et al., 2024).

The Rosenzweig–Porter model provides a static version of the same idea. There the edge Krylov vectors are analyzed in the original computational basis through the Krylov inverse participation ratio

$$\mathcal K = \mathrm{span}\{ |\psi_0\rangle, U|\psi_0\rangle, U^2|\psi_0\rangle,\dots \}.$$3

For $$\mathcal K = \mathrm{span}\{ |\psi_0\rangle, U|\psi_0\rangle, U^2|\psi_0\rangle,\dots \}.$$4, the edge vector $$\mathcal K = \mathrm{span}\{ |\psi_0\rangle, U|\psi_0\rangle, U^2|\psi_0\rangle,\dots \}.$$5 yields the Krylov fractal dimension

$$\mathcal K = \mathrm{span}\{ |\psi_0\rangle, U|\psi_0\rangle, U^2|\psi_0\rangle,\dots \}.$$6

Thus the Krylov vectors are ergodic for $$\mathcal K = \mathrm{span}\{ |\psi_0\rangle, U|\psi_0\rangle, U^2|\psi_0\rangle,\dots \}.$$7, fractal or nonergodic extended for $$\mathcal K = \mathrm{span}\{ |\psi_0\rangle, U|\psi_0\rangle, U^2|\psi_0\rangle,\dots \}.$$8, and localized for $$\mathcal K = \mathrm{span}\{ |\psi_0\rangle, U|\psi_0\rangle, U^2|\psi_0\rangle,\dots \}.$$9. The associated Lanczos profile is interpolated by a $\{|K_n\rangle\}$0-logarithmic form,

$\{|K_n\rangle\}$1

which reduces to the GOE-like $\{|K_n\rangle\}$2 form in the ergodic regime and to the logarithmic form in the localized regime (Bhattacharjee et al., 2024).

Non-Hermitian many-body dynamics yields a related but distinct phase structure. In the disordered non-Hermitian XY chain, the Krylov amplitudes obey

$\{|K_n\rangle\}$3

with nonreciprocal hoppings $\{|K_n\rangle\}$4. Two transitions are identified as the non-Hermitian disorder $\{|K_n\rangle\}$5 increases. A Krylov localization transition occurs at

$\{|K_n\rangle\}$6

diagnosed by the generalized Krylov variance built from

$\{|K_n\rangle\}$7

while a distinct chaotic-to-non-chaotic transition occurs at

$\{|K_n\rangle\}$8

diagnosed by the reciprocity measure

$\{|K_n\rangle\}$9

The important point is that weak ergodicity breaking and non-chaoticity do not coincide: there is an intermediate regime in which the Krylov wavefunction is localized in a finite region of Krylov space while the system remains chaotic by level statistics (Zhou et al., 27 Jan 2025).

These works support a now-common interpretation: delocalized motion across Krylov basis vectors corresponds to ergodic dynamics, fractal or localized support corresponds to nonergodic or weakly ergodicity-broken dynamics, and the effective geometry of the Krylov chain can resolve transitions that conventional short-range spectral statistics may miss (Menzler et al., 2024, Bhattacharjee et al., 2024, Zhou et al., 27 Jan 2025).

5. Entanglement generation, scars, and algebraically closed Krylov sectors

The narrowest formal definition of Krylov vector ergodicity appears in “Enhanced entanglement from quantum ergodicity.” The setting uses a non-demolition Hamiltonian

$$b_n |K_n\rangle = U|K_{n-1}\rangle - \sum_{l=0}^{n-1} \langle K_l|U|K_{n-1}\rangle\, |K_l\rangle,$$0

with evolved state

$$b_n |K_n\rangle = U|K_{n-1}\rangle - \sum_{l=0}^{n-1} \langle K_l|U|K_{n-1}\rangle\, |K_l\rangle,$$1

The relevant Krylov set is

$$b_n |K_n\rangle = U|K_{n-1}\rangle - \sum_{l=0}^{n-1} \langle K_l|U|K_{n-1}\rangle\, |K_l\rangle,$$2

Writing these Bob-side vectors as

$$b_n |K_n\rangle = U|K_{n-1}\rangle - \sum_{l=0}^{n-1} \langle K_l|U|K_{n-1}\rangle\, |K_l\rangle,$$3

the paper defines Krylov vector ergodicity by the closeness of $$b_n |K_n\rangle = U|K_{n-1}\rangle - \sum_{l=0}^{n-1} \langle K_l|U|K_{n-1}\rangle\, |K_l\rangle,$$4 to a unitary map, quantified by

$$b_n |K_n\rangle = U|K_{n-1}\rangle - \sum_{l=0}^{n-1} \langle K_l|U|K_{n-1}\rangle\, |K_l\rangle,$$5

Perfect Krylov vector ergodicity means $$b_n |K_n\rangle = U|K_{n-1}\rangle - \sum_{l=0}^{n-1} \langle K_l|U|K_{n-1}\rangle\, |K_l\rangle,$$6, and the entanglement purity is exactly

$$b_n |K_n\rangle = U|K_{n-1}\rangle - \sum_{l=0}^{n-1} \langle K_l|U|K_{n-1}\rangle\, |K_l\rangle,$$7

Equivalently, in terms of the return probability

$$b_n |K_n\rangle = U|K_{n-1}\rangle - \sum_{l=0}^{n-1} \langle K_l|U|K_{n-1}\rangle\, |K_l\rangle,$$8

the purity obeys

$$b_n |K_n\rangle = U|K_{n-1}\rangle - \sum_{l=0}^{n-1} \langle K_l|U|K_{n-1}\rangle\, |K_l\rangle,$$9

This turns the generation of nearly orthonormal Krylov vectors into a direct operational resource for EPR-state preparation and operator transfer (Vikram, 10 Jul 2025).

A different route to nonergodic Krylov behavior is provided by quantum many-body scars in spin-1 models. In the spin-1 XY model, the nematic Néel state generates a Krylov chain that closes after only $(U_\mathcal K)_{mn}=\langle K_m|U|K_n\rangle$0 states, with Lanczos coefficients

$(U_\mathcal K)_{mn}=\langle K_m|U|K_n\rangle$1

described in the paper as an elliptical pattern. The exact Krylov complexity is

$(U_\mathcal K)_{mn}=\langle K_m|U|K_n\rangle$2

and the fidelity distribution is

$(U_\mathcal K)_{mn}=\langle K_m|U|K_n\rangle$3

In the spin-1 XXZ model, the spin helix state yields a $(U_\mathcal K)_{mn}=\langle K_m|U|K_n\rangle$4-dimensional Krylov sector with

$(U_\mathcal K)_{mn}=\langle K_m|U|K_n\rangle$5

and exact complexity

$(U_\mathcal K)_{mn}=\langle K_m|U|K_n\rangle$6

In both cases the finite-support square-root form of $(U_\mathcal K)_{mn}=\langle K_m|U|K_n\rangle$7 reflects a hidden SU(2) algebra, periodic fidelity revivals, and recurrent rather than ergodic spreading in Krylov space (Hu et al., 31 Mar 2025).

These results illustrate a broader principle. A small algebraically closed Krylov sector can produce exact or nearly exact revivals, bounded oscillatory complexity, and low entanglement despite a nonintegrable ambient Hamiltonian. This suggests that Krylov vector ergodicity is not merely a measure of spreading, but also a measure of whether the Krylov orbit closes on a simple representation-theoretic sector or explores a much larger generic one (Vikram, 10 Jul 2025, Hu et al., 31 Mar 2025).

6. Semiclassical interpretation, numerical caveats, and distinct usages

The semiclassical literature gives a controlled setting in which Krylov vectors acquire a geometric meaning. In the unitary-map framework with the correct quantum density-matrix inner product and classical $(U_\mathcal K)_{mn}=\langle K_m|U|K_n\rangle$8 inner product, the paper “Quantum-to-classical correspondence in Krylov complexity” proves that the classical Krylov space is obtained as the asymptotic $(U_\mathcal K)_{mn}=\langle K_m|U|K_n\rangle$9 expansion of the quantum Krylov space up to Ehrenfest time. In this representation, each new Krylov vector captures the part of the current evolved distribution that is genuinely new relative to earlier times, and more sustained production of new phase-space structure leads to continued ballistic packets in Krylov space. At the same time, the paper is explicit that it does not prove a rigorous ergodicity criterion, and that regular systems can also show transient ballistic growth (Scialchi et al., 11 Mar 2026).

A major numerical caveat is that long Lanczos sequences can cease to represent the exact Krylov subspace in finite precision. “Escaping the Krylov space during finite precision Lanczos” shows that even with reorthogonalization, computed Lanczos vectors can leave the true vector space spanned by the exact Lanczos vectors. The diagnostic overlap with the exact Krylov space,

$\hat H$0

can drop below $\hat H$1 well before the exact termination point, and nonzero coefficients may persist beyond the exact Krylov dimension. The implication is that apparent long-iteration delocalization or statistical “ergodic” behavior of computed Krylov vectors can be a numerical artifact unless confinement to the exact Krylov space is verified (Eckseler et al., 5 May 2025).

There is also a distinct matrix-theoretic usage of “vector ergodicity.” For constant row-sum matrices, the vector-norm-based ergodicity coefficients

$\hat H$2

bound nontrivial eigenvalues and asymptotically recover the largest nontrivial eigenvalue modulus through $\hat H$3. This framework is related to invariant subspaces and can be suggestive for Krylov-type computations, but it is not the same concept as the quantum-dynamical notion of Krylov vector ergodicity developed in the recent many-body literature (Marsli et al., 2019).

A plausible synthesis is that the contemporary term denotes a family of closely related diagnostics rather than a single universal invariant. In one branch it is a formal measure of near-orthonormal state generation and entanglement utility (Vikram, 10 Jul 2025). In another it is a shift-structure criterion for Arnoldi dynamics of unitary maps (Scialchi et al., 2024). In a broader many-body setting it is an interpretation of how uniformly and extensively a state or operator explores its Krylov basis, with localization, fractality, symmetry-induced closure, and finite-precision leakage providing the main mechanisms by which that exploration can fail (Menzler et al., 2024, Bhattacharjee et al., 2024, Eckseler et al., 5 May 2025).