---
title: Krylov Spread Complexity Explained
url: https://www.emergentmind.com/topics/krylov-spread-complexity-80a50441-92e3-4e00-9695-bcf4bc01c67e
type: topic
---

# Krylov Spread Complexity Explained

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Krylov spread complexity is a state-complexity measure defined from the spread of a time-evolved state over the Krylov basis generated by repeated action of the dynamical generator on a chosen reference state. If
\[
|\psi(t)\rangle=\sum_n \phi_n(t)\,|K_n\rangle,
\]
the standard definition is
\[
C_K(t)=\sum_n n\,|\phi_n(t)|^2,
\]
so the complexity is the first moment of a probability distribution on the Krylov chain. In the literature this quantity also appears under notations such as \(S_K(t)\), \(\mathcal C(t)\), or \(K(t)\), and it is the Schrödinger-picture analogue of operator Krylov complexity [2312.12593, 2507.06286].

## 1. Formal definition and basis-optimized interpretation

The basic construction starts from a reference state \(|\psi_0\rangle\) and the orbit generated by repeated action of the Hamiltonian,
\[
|\psi_0\rangle,\quad H|\psi_0\rangle,\quad H^2|\psi_0\rangle,\quad \dots
\]
After orthonormalization by the Lanczos procedure, one obtains the Krylov basis \(\{|K_n\rangle\}\). In this basis the Hamiltonian becomes tridiagonal, and the amplitudes obey a nearest-neighbor Schrödinger equation on an effective one-dimensional chain,
\[
i\,\partial_t \psi_n(t)=a_n \psi_n(t)+b_{n+1}\psi_{n+1}(t)+b_n\psi_{n-1}(t),
\]
with \(\psi_n(0)=\delta_{n0}\). The spread complexity is then the average Krylov-site index occupied by the state [2312.12593].

A central structural result is that spread complexity is basis-optimized. For an ordered orthonormal basis \(\{\ket{\mathcal V_n}\}\), one may define
\[
C_{\mathcal V}=\sum_n \mu_n\, |\braket{\psi_t|\mathcal V_n}|^2,
\]
with \(\mu_n\) a positive increasing sequence; the conventional choice is \(\mu_n=n\). The literature then identifies the Krylov basis generated from the initial state by the Hamiltonian as the basis minimizing this cost. In that sense, Krylov spread complexity is not merely a basis-dependent spread observable, but a basis-dependent observable evaluated in the dynamically preferred basis [2603.25724].

The same framework admits a statistical formulation. The spreading operator is
\[
K=\sum_n n\,|K_n\rangle\langle K_n|,
\]
and a projective measurement of \(K\) at time \(t\) yields the value \(n\) with probability \(P_n(t)=|\phi_n(t)|^2\). The characteristic function
\[
\chi_K(u,t)=\sum_n e^{-iun}|\phi_n(t)|^2
\]
generates the full hierarchy of generalized spread complexities
\[
\mathcal C_m(t)=\sum_n n^m |\phi_n(t)|^2,
\]
of which the ordinary spread complexity is the first moment, \(\mathcal C_1(t)\) [2411.09390].

## 2. State, operator, and multiseed formulations

Although the phrase “spread complexity” is usually reserved for states, the formalism is parallel to operator Krylov complexity. In the operator setting one works in operator Hilbert space, uses the Liouvillian
\[
\mathcal L=[H,\cdot],
\]
constructs the operator Krylov basis from
\[
|\mathcal O),\ \mathcal L|\mathcal O),\ \mathcal L^2|\mathcal O),\dots,
\]
and defines the operator complexity as the first moment of the probability distribution over Krylov sites. The review literature explicitly treats the state version once called spread complexity as the state analogue of the same general Krylov construction [2507.06286].

Seed dependence is a persistent issue in both formulations. A recent response is the multiseed or block-Lanczos construction, in which one starts not from a single operator but from an entire orthonormal set of simple operators
\[
\Omega_0=\{|\mathcal O_{0,0}\rangle,\dots,|\mathcal O_{0,m-1}\rangle\}
\]
and generates block Krylov subspaces \(\Omega_0,\Omega_1,\dots\). The resulting multiseed complexity averages the block-level spreading of all simple seeds and is designed to reduce the severe seed dependence of ordinary single-seed Krylov complexity [2409.15666].

This suggests a useful distinction. Ordinary Krylov spread complexity measures spreading relative to a particular reference state or operator, whereas multiseed constructions attempt to characterize the spreading of an entire simple sector. A plausible implication is that the former is better suited to state-specific dynamical questions, while the latter is better suited to system-level discrimination between integrable and chaotic dynamics.

## 3. Dynamical behavior in closed many-body systems

At short times, spread complexity has a universal quadratic onset. For the state amplitudes in Krylov space one finds
\[
C(0)=0,\qquad \dot C(0)=0,\qquad \ddot C(0)=2b_1^2,
\]
hence
\[
C(t)=b_1^2 t^2+\mathcal O(t^3).
\]
This early-time law is independent of the higher Lanczos data and has been verified in several models [2312.12593].

Beyond short times, the phenomenology is more model-dependent. In integrable systems with unstable saddles, such as the Lipkin-Meshkov-Glick model and the inverted harmonic oscillator in a bounded quartic potential, the spread complexity of the thermofield double state displays the pattern
\[
\text{ramp} \rightarrow \text{peak} \rightarrow \text{slope} \rightarrow \text{plateau},
\]
even though these systems are not chaotic in the spectral-statistical sense. This directly shows that spread complexity alone is not a definitive diagnostic of true quantum chaos [2312.12593].

A further limitation is strong initial-state dependence. For both states and operators, the complexity varies monotonically with an inverse participation ratio defined from the initial condition in the eigenbasis of the generator of dynamics. In the state case, greater eigenbasis delocalization means faster growth and larger saturation; in the operator case, smaller diagonal overlap with the energy basis means larger complexity. The same dependence persists even in fully chaotic regimes, and averaging over many initial conditions still does not recover a robust chaos indicator [2503.03400].

In disordered spin chains, however, Krylov spread complexity becomes a sharp dynamical diagnostic of ergodicity versus many-body localization when the question is phrased appropriately. For a disordered interacting spin chain, the infinite-time spread complexity scales as
\[
\langle S_{K,\infty}\rangle \sim H^\alpha,
\]
with \(\alpha=1\) in the ergodic phase and \(\alpha<1\) in the MBL phase, where \(H\) denotes the Hilbert-space dimension. The infinite-time profile on the Krylov chain is approximately flat in the ergodic phase but stretched exponential in the MBL phase, reflecting a broad distribution of eigenstate-dependent decay lengths [2603.25724].

A complementary study of the disordered Heisenberg spin-\(\tfrac12\) chain shows that the relevant diagnostic depends strongly on the initial state. For the infinite-temperature thermofield double, the ergodic-to-MBL transition is identified by the disappearance of a pre-saturation peak in spread complexity. For the Néel state and other sparse computational-basis states, the late-time saturation values of spread complexity and Krylov inverse participation ratio distinguish the ergodic phase from the integrable phases, while Haar-random states can distinguish the disorder-free integrable regime from the MBL regime [2409.02186].

## 4. Floquet, graph, non-Hermitian, and measurement-induced extensions

For periodically driven systems, the relevant dynamical generator is the Floquet operator rather than a Hermitian Hamiltonian, and the natural iterative procedure is Arnoldi rather than Lanczos. Starting from
\[
\{|\psi_0\rangle,\ U_F|\psi_0\rangle,\ U_F^2|\psi_0\rangle,\dots\},
\]
Arnoldi orthogonalization produces the Floquet Krylov basis, and spread complexity is defined from the resulting amplitudes exactly as in the continuous-time case. In periodically kicked Ising spin chains with non-integrable deformations, chaotic dynamics is associated with suppressed fluctuations in the Arnoldi coefficients and with larger saturation values of spread complexity than regular dynamics [2405.16182].

On finite graphs, especially for continuous-time quantum walks, the Krylov basis can acquire a direct graph interpretation. For path graphs seeded from an endpoint, the Krylov basis coincides with the vertex basis, and the long-time averaged Krylov complexity becomes the first moment of the walk’s limiting distribution. In this setting the paper derives an exact lower bound
\[
\bar{\mathcal C}_{\min}=2\frac{\mathcal D-1}{\mathcal D^2}
\]
for connected graph Hamiltonians in the class considered, attained by the complete graph, and proposes the empirical upper law
\[
\bar{\mathcal C}_{\max}\approx 0.66\,\mathcal D-1.31
\]
from graph optimization data [2401.00526].

Non-Hermitian dynamics requires additional modifications. For measurement-induced non-unitary evolution generated by repeated projections, the Schrödinger-picture spread-complexity framework is extended using bi-Lanczos for general non-Hermitian Hamiltonians and a reduced-memory complex-symmetric algorithm for complex symmetric cases. In the one-dimensional tight-binding problem with repeated measurements, spread complexity shows initial growth, an extended decay regime, and eventual saturation; decreasing the interval between measurements delays the onset of growth, and in the limit of vanishing interval the onset time tends to infinity, which is the Krylov-space manifestation of the quantum Zeno effect [2312.11635].

Two further non-Hermitian developments are notable. In a non-Hermitian SSH chain with an imaginary chemical potential, spread complexity detects both the usual \(\mathcal{PT}\)-transition from real to complex spectrum and a second transition inside the broken phase from complex to purely imaginary spectrum; in the purely imaginary regime a Krylov spread fidelity
\[
\mathcal F(t)=|\mathcal C(t)-\mathcal C_\Omega|
\]
reveals additional dynamical phase structure [2503.18936]. Separately, an SVD-based formulation replaces the non-Hermitian Hamiltonian by \(\sqrt{H^\dagger H}\), defines spread complexity from the resulting singular-value Krylov chain, and finds that chaotic non-Hermitian models exhibit a characteristic peak associated with singular-value repulsion, while integrable cases do not [2411.09309].

## 5. Geometric, entanglement, and experimental perspectives

Krylov spread complexity has also been connected to standard quantum resources. For bipartite systems, the entanglement entropy of the evolved state is upper bounded by a weighted average of the entanglement entropies of the Krylov basis vectors plus a function of the spread complexity,
\[
f(K)=(1+K)\log(1+K)-K\log K.
\]
For qubits, the spread complexity is directly related to the \(\ell_1\)-coherence of the initial state in the energy eigenbasis:
\[
K(t)=C_{\ell_1}^2\sin^2\!\left(\frac{\omega t}{2}\right).
\]
For qutrits, the corresponding expression depends on pairwise coherences and energy gaps [2603.26619].

In two-band Hamiltonians, the complexity can be expressed purely geometrically in terms of Bloch-sphere data. If \(\hat n_{\rm ref}(k)\) is the Bloch vector of the reference state and \(\hat n_{\rm target}(k)\) that of the target state, then
\[
C_k=\frac{1-\hat n_{\rm ref}(k)\cdot \hat n_{\rm target}(k)}{2}.
\]
For momentum-independent reference states, the derivative of the averaged complexity is bounded by fidelity susceptibility,
\[
|\partial_\lambda C(\lambda)| \le 4\pi \sum_{i=1}^3 |\mathcal Q_i|\,\sqrt{\chi_F^i(\lambda)},
\]
which shows that the derivative of spread complexity is sensitive to any gap closing, topological or trivial. In the SSH model, generic reference states yield a logarithmically divergent derivative at the topological transition; in the massive Dirac model, an analogous divergence occurs at a trivial gap closing [2605.18594].

The framework has also been reformulated in experimentally accessible terms. A measurable alternative to the standard Krylov space is built from sampled time-evolved states,
\[
|g_i\rangle=e^{-iHt_i}|\Psi_0\rangle.
\]
The paper proves the exact finite-dimensional identity
\[
\mathrm{H}_n^n=\mathrm{K}_n
\]
for truncated Taylor-sample spaces and shows that the measurable sampled-state space \(\mathrm G_m\) approximates the conventional Krylov space. It furthermore states that the Krylov-space dimension equals the number of pairwise distinct eigenvalues, more precisely the number of distinct eigenvalues with nonzero support in the initial state. The corresponding measurable spread complexity reproduces the conventional spread dynamics with almost identical behavior when the sampled-state basis is sufficiently expressive [2404.13089].

## 6. Holographic interpretation, conceptual status, and open questions

The most explicit holographic realization of Krylov spread complexity appears in double-scaled SYK and its dual sine-dilaton gravity. In that setting the chord basis coincides with the Krylov basis,
\[
|K_n\rangle\equiv |n\rangle,
\]
the bulk length spectrum obeys
\[
L=2|\log q|\,n,
\]
and the disk-level gravitational observable satisfies the exact relation
\[
2|\log q|\,C_K(t)_\beta=\langle \hat L\rangle.
\]
Equivalently, the paper identifies the operators themselves,
\[
2|\log q|\,\hat K=\hat L,
\]
so the complexity = volume proposal is realized as an operator-level identity between the boundary Krylov position operator and the bulk renormalized length operator [2412.17785].

This relation is substantially more general than the earlier JT-gravity, infinite-temperature result. It is exact in the DSSYK nonlocality parameter \(q\) and inverse temperature \(\beta\) at disk level, and the finite-temperature state is not defined by merely evolving a thermalized reference state. Instead, the relevant state is
\[
|\psi(t)\rangle_\beta = Z_\beta^{-1/2}e^{-i\hat T(t-i\beta/2)}|0\rangle,
\]
so the Euclidean preparation segment contributes intrinsically to the complexity. The same analysis isolates the first quantum correction to complexity = volume and suggests interpreting it as a complexity of bulk quantum fields, although that interpretation is presented as conjectural rather than proven [2412.17785].

These developments sharpen the conceptual status of Krylov spread complexity. On one hand, it is not generically the same as circuit complexity, Nielsen geometry, operator size, or complexity = action, and several bodies of work show that it is not, by itself, a universal chaos diagnostic [2412.17785, 2507.06286]. On the other hand, in specific solvable settings it admits an exact operator-level bulk dual, and in many-body systems it can cleanly encode dynamical distinctions such as ergodic versus MBL spreading. This suggests that its most reliable interpretation is as a dynamically generated spread observable whose meaning is strongest when the choice of reference state, symmetry sector, and dynamical setting are tightly controlled.

Several open questions recur across the literature. Higher-topology and nonperturbative corrections remain outside the disk-level holographic analysis; the switchback effect has not yet been fully established in the finite-temperature DSSYK/sine-dilaton setting; the extension of the holographic paradigm beyond two-dimensional gravity and SYK-like models is unresolved; and, outside specially controlled settings, reference-state dependence remains a basic limitation rather than a secondary technicality [2412.17785, 2507.06286].

Source: https://www.emergentmind.com/topics/krylov-spread-complexity-80a50441-92e3-4e00-9695-bcf4bc01c67e