---
title: Krylov Vector Ergodicity
url: https://www.emergentmind.com/topics/krylov-vector-ergodicity
type: topic
---

# Krylov Vector Ergodicity

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 [2507.08067]. 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 [2407.06428, 2512.20285, 2602.11431].

## 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 \( (U_\mathcal K)_{mn}=\langle K_m|U|K_n\rangle \) [2407.06428].

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
\[
\mathrm{Span}\{\mathcal L^0(\hat O),\mathcal L^1(\hat O),\dots\}.
\]
With the Hilbert–Schmidt inner product
\[
\langle A|B\rangle=\frac{1}{D}\mathrm{Tr}(A^\dagger B),
\]
or its finite-temperature variants, Lanczos or Arnoldi orthogonalization yields a Krylov basis and a tridiagonal or Hessenberg effective chain [2512.20285, 2401.10499].

The central observables are likewise shared across settings. The Krylov amplitudes \(\beta_n(t)\), \(\varphi_n(t)\), or \(\phi_n(t)\) define a wavefunction on the Krylov basis. Krylov complexity is the first moment of the basis index,
\[
\mathcal K_C(t)=\sum_n n |\beta_n(t)|^2,
\qquad
C_K(t)=\sum_n n |\varphi_n(t)|^2,
\]
while the Lanczos or Arnoldi coefficients \(b_n\) 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 [2512.20285, 2403.14384].

## 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
\[
b_n = \langle K_n|U|K_{n-1}\rangle,\qquad
a_n=\langle K_n|U|K_n\rangle,\qquad
c_n=\langle K_0|U|K_n\rangle,
\]
and asymptotic ergodicity is identified with the limit
\[
a_n\to 0,\qquad b_n\to 1,\qquad c_n\to 0.
\]
The reduced unitary then approaches the pure lower-shift form
\[
U^{\mathrm{erg}}_{n,m}=\delta_{n,m+1},
\]
equivalently,
\[
U|K_n\rangle \approx |K_{n+1}\rangle.
\]
This is the “maximally ergodic” regime in the Arnoldi approach [2407.06428].

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,
\[
C_K(t)=t,
\]
and it is attained when
\[
U|K_n\rangle = |K_{n+1}\rangle.
\]
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 [2603.11034].

The Arnoldi formulation also gives a direct ergodicity measure. The distance between the actual reduced unitary \(U_\mathcal K\) and the shift matrix \(U^{\mathrm{erg}}\) is quantified by
\[
\Erg(U)^{-1} =
\frac{1}{\sqrt{2\,\mathrm{Tr}(\mathbb I)}}\,
\left\| U^{\mathrm{erg}}-U_\mathcal K \right\|_F.
\]
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 [2407.06428].

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 [2603.11034].

## 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,
\[
\hat H = -J_1 \sum_{i = 1}^{(N-1)/2} \hat{\sigma}^z_{i}\hat{\sigma}^z_{i+1}
-J_2\sum_{i = (N+1)/2}^{N - 1} \hat{\sigma}^z_{i}\hat{\sigma}^z_{i+1}
-h_x\sum_{i=1}^{N }\hat{\sigma}^x_{i}
-h_z\sum_{i=1}^{N}\hat{\sigma}^z_{i},
\]
with inhomogeneity parameter
\[
\mathcal J_r=\frac{J_2}{J_1}.
\]
There, increasing \(\mathcal J_r\) 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
\[
S_{\mathcal K}=\frac{1}{K}\bigg[\sum_i |\langle \mathcal W_i|\hat O(t_{sat})\rangle|^4\bigg]^{-1}
\]
drops strongly as inhomogeneity increases, supporting the interpretation of Krylov-space confinement [2512.20285].

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
\[
b_n \approx \alpha(T)n,
\qquad
\alpha(T)=\pi k_B T,
\]
while the spectral form factor defines an ergodicity indicator
\[
g=\log_{10}\!\left(\frac{t_H}{t_{\rm Th}}\right).
\]
After suitable size scaling, larger Krylov growth rate \(\alpha(T)\) or \(A(T)\) correlates with larger spectral ergodicity indicator. This establishes a direct relation between faster spreading across Krylov basis vectors and stronger spectral ergodicity [2401.10499].

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
\[
H=\sum_i \left(X_iX_{i+1}+Y_iY_{i+1}+Z_iZ_{i+1}\right)+\sum_i (1+w h_i)X_i,
\]
and in the clean limit it has the dynamical symmetry
\[
[H,S^+]=-2S^+,
\qquad
S^+=\sum_j (Z_j+iY_j).
\]
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 \(b_2\), so the dressed mode frequency shifts only at second order,
\[
E\approx 2+w^2 E^{(2)}.
\]
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 [2602.11431].

Taken together, these results define a broad operator-Krylov notion of ergodicity: regular \(b_n\), 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 [2512.20285, 2602.11431].

## 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 \(b_n\), and localization is diagnosed by the variance
\[
\sigma^2=\mathrm{Var}(x_j),
\qquad
x_j=\ln\!\left(\frac{b_{2j-1}}{b_{2j}}\right).
\]
The paper interprets \(\sigma\) as an inverse localization-length measure through
\[
l_{\mathrm{loc}} \propto \frac{\sqrt{L}}{\sigma}.
\]
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 [2403.14384].

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
\[
\mathrm{IPR}^{\ell}_{\rm K}(\varphi_k)=\sum_{n=1}^N |\langle n|\varphi_k\rangle|^{2\ell}.
\]
For \(\ell=2\), the edge vector \(\varphi_{N-1}\) yields the Krylov fractal dimension
\[
\mathrm{IPR}^{2}_{\rm K}(\varphi_{N-1})\sim N^{-\mathcal D_2},
\qquad
\mathcal D_2=
\begin{cases}
1, & \gamma\le 1,\\
2-\gamma, & 1<\gamma\le 2,\\
0, & \gamma>2.
\end{cases}
\]
Thus the Krylov vectors are ergodic for \(\gamma\le 1\), fractal or nonergodic extended for \(1<\gamma\le 2\), and localized for \(\gamma>2\). The associated Lanczos profile is interpolated by a \(q\)-logarithmic form,
\[
\mathsf b_\gamma(x)^2
=
p\left(\frac{1-x^{1-q}}{1-q}\right)
=
-p\ln_q x,
\]
which reduces to the GOE-like \(\sqrt{1-x}\) form in the ergodic regime and to the logarithmic form in the localized regime [2407.07399].

Non-Hermitian many-body dynamics yields a related but distinct phase structure. In the disordered non-Hermitian XY chain, the Krylov amplitudes obey
\[
i \dot\varphi_n = b_n \varphi_{n-1}+a_n\varphi_n+c_{n+1}\varphi_{n+1},
\]
with nonreciprocal hoppings \(b_n\neq c_n^*\). Two transitions are identified as the non-Hermitian disorder \(W_\gamma\) increases. A Krylov localization transition occurs at
\[
W_L\approx 0.0351,
\]
diagnosed by the generalized Krylov variance built from
\[
j_n=|b_n c_n|^{1/2},
\]
while a distinct chaotic-to-non-chaotic transition occurs at
\[
W_C\approx 1.647,
\]
diagnosed by the reciprocity measure
\[
R_K^{(d)}=\frac{1}{d}\sum_{n=1}^{d}\cos\theta_n,
\qquad
\theta_n=\arg(b_n c_n).
\]
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 [2501.15982].

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 [2403.14384, 2407.07399, 2501.15982].

## 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
\[
H=N_A\otimes H_B,
\qquad
U_{t_0}=e^{-iH_B t_0},
\]
with evolved state
\[
|\Psi(t_0)\rangle=
\frac{1}{\sqrt{d_A}}
\sum_{n=0}^{d_A-1}
|n\rangle_A\otimes U_{t_0}^n|\phi\rangle_B.
\]
The relevant Krylov set is
\[
\mathcal K_{d_A}(U_{t_0},|\phi\rangle_B)
=
\{|\phi\rangle_B,\,
U_{t_0}|\phi\rangle_B,\,
\dots,\,
U_{t_0}^{d_A-1}|\phi\rangle_B\}.
\]
Writing these Bob-side vectors as
\[
|\mathcal K_j\rangle_2=R_2|j\rangle_2,
\]
the paper defines Krylov vector ergodicity by the closeness of \(R_2\) to a unitary map, quantified by
\[
\eta_2=\frac{1}{d_2}\mathrm{Tr}_2[(R_2R_2^\dagger)^2]-1.
\]
Perfect Krylov vector ergodicity means \(\eta_2=0\), and the entanglement purity is exactly
\[
\mathcal P(t_0)=\frac{1+\eta_2}{d_A}.
\]
Equivalently, in terms of the return probability
\[
p_\phi(t)=\left|\langle\phi|e^{-iH_B t}|\phi\rangle_B\right|^2,
\]
the purity obeys
\[
\mathcal P(t_0)
=
\frac{1}{d_A}
+
\frac{2}{d_A}
\sum_{\tau=1}^{d_A-1}
\left(1-\frac{\tau}{d_A}\right)p_\phi(\tau t_0).
\]
This turns the generation of nearly orthonormal Krylov vectors into a direct operational resource for EPR-state preparation and operator transfer [2507.08067].

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 \(L+1\) states, with Lanczos coefficients
\[
a_n=0,
\qquad
b_n=h\sqrt{n(L+1-n)},
\]
described in the paper as an elliptical pattern. The exact Krylov complexity is
\[
C_{\mathcal K}(t)=L\sin^2(ht),
\]
and the fidelity distribution is
\[
F_n(t)=\cos^{2L}(ht)\tan^{2n}(ht)\,\mathbb C_L^n.
\]
In the spin-1 XXZ model, the spin helix state yields a \((2L+1)\)-dimensional Krylov sector with
\[
b_n=\frac{1}{2}h\sin\alpha\sqrt{n(2L+1-n)},
\qquad
a_n=J\Delta L+h\cos\alpha(L-n),
\]
and exact complexity
\[
C(t)=2L\sin^2\alpha\,\sin^2\frac{ht}{2}.
\]
In both cases the finite-support square-root form of \(b_n\) reflects a hidden SU(2) algebra, periodic fidelity revivals, and recurrent rather than ergodic spreading in Krylov space [2503.24073].

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 [2507.08067, 2503.24073].

## 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 \(L^2\) inner product, the paper “Quantum-to-classical correspondence in Krylov complexity” proves that the classical Krylov space is obtained as the asymptotic \(\hbar\to 0\) 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 [2603.11034].

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,
\[
C_n=
\sqrt{\sum_{p=1}^{\dim(\mathcal K)}
|(O_n|\bar E_p^{\mathcal L})|^2},
\]
can drop below \(1\) 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 [2505.02670].

There is also a distinct matrix-theoretic usage of “vector ergodicity.” For constant row-sum matrices, the vector-norm-based ergodicity coefficients
\[
T_p(A)=
\max_{\substack{x\in\mathbb R^n\\x^Te=0,\ \|x\|_p=1}}
\|A^Tx\|_p
\]
bound nontrivial eigenvalues and asymptotically recover the largest nontrivial eigenvalue modulus through \(\bigl(T_p(A^k)\bigr)^{1/k}\). 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 [1911.06139].

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 [2507.08067]. In another it is a shift-structure criterion for Arnoldi dynamics of unitary maps [2407.06428]. 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 [2403.14384, 2407.07399, 2505.02670].

Source: https://www.emergentmind.com/topics/krylov-vector-ergodicity