Krylov Complexity Growth Rate

Updated 28 October 2025

Krylov complexity growth rate is a quantitative measure that tracks how a quantum operator spreads in its Krylov basis through time evolution.
It reveals universal dynamics with early quadratic behavior and late-time exponential growth, providing insights into operator spreading and chaos.
The growth rate connects boundary quantum dynamics with holographic bulk momentum, offering a bridge between quantum information theory and gravity duals.

Krylov complexity growth rate quantitatively characterizes the rate at which a quantum operator explores increasingly complex directions in its Krylov (or Lanczos) basis under time evolution. As a function of time, this growth rate provides insights into universal quantum dynamical behaviors, quantum chaos, operator spreading, and their duality relations in holographic settings.

1. Definition and Fundamental Relation

Krylov complexity (K-complexity) for an operator in a quantum system is defined as the mean position of its evolving “wavefunction” in the Krylov (Lanczos) chain, i.e.,

$K(t) = \sum_{n=0}^\infty n\, |\varphi_n(t)|^2$

where $\varphi_n(t)$ are the time-dependent amplitudes of the Heisenberg-evolved operator projected onto the orthogonally generated Krylov basis. The core dynamical quantity dictating operator growth in this space is the sequence of Lanczos coefficients $b_n$ , the “hopping amplitudes” in the tridiagonal representation of the Liouvillian.

The growth rate of Krylov complexity is tightly tied to the behavior of the Lanczos coefficients:

If $b_n \sim \alpha n$ for large $n$ , then the complexity grows exponentially: $K(t) \sim \exp(2\alpha t)$ .
For more general asymptotics $b_n \sim n^\delta$ , the growth law is $K(t) \sim t^{2/(2-\delta)}$ .
The late-time exponential growth rate is thus $\lambda_K = 2\alpha$ when linear growth dominates.

2. Holographic Momentum-Krylov Complexity Correspondence

In holographic quantum systems, recent work establishes a direct semiclassical correspondence between the growth rate of boundary Krylov complexity and the radial momentum $P$ of a bulk infalling particle in AdS black hole backgrounds (Fan, 7 Nov 2024). The central relation is: $\frac{dK}{dt} = P$ where $K$ is boundary Krylov complexity and $P$ is the proper radial momentum as measured in the AdS bulk. For massless particles in static AdS black hole metrics: $P(r) = \frac{E}{\sqrt{h(r)}}$ and Krylov complexity integrates this along the infalling trajectory: $K = \int_{r}^{\infty} \frac{E\, dr}{\sqrt{f(r)} h(r)}$ This geometric prescription recovers both universal initial (quadratic) growth and late-time exponential growth, with exponent set by near-horizon Rindler physics: $K(t) \sim e^{2\alpha t}, \quad \alpha = \frac{\pi}{\beta}$ where $\beta$ is the inverse black hole temperature. This universality underpins "fast scrambling" in holographic CFTs.

Notably, for the three-dimensional BTZ black hole, the holographic calculation matches exactly the boundary CFT $_2$ Krylov complexity at finite temperature. This represents the first precise all-times correspondence between holographic and field-theoretic Krylov complexity.

3. Universal Early- and Late-Time Scaling Laws

The time evolution of Krylov complexity displays two robust universal regimes:

Early-time (short $t$ ): Quadratic growth

$K(t) = b_1^2 t^2 + \mathcal{O}(t^4)$

with $b_1$ the first Lanczos coefficient, directly related to the initial commutator structure, e.g.

$K(t) = \frac{E t^2}{2 \ell_{AdS}^2}$

in vacuum AdS.

Late-time: Exponential growth for chaotic or fast scrambling systems

$K(t) \sim e^{2\alpha t}$

with $\alpha = \frac{\pi}{\beta}$ in holographic CFTs.

This dichotomy reflects initial ballistic operator spreading, then fast scrambling governed by the Lyapunov exponent or its upper bound. The rate of exponential growth coincides with the "maximal chaos" bound set by thermal quantum field theory.

Centrally, Krylov complexity is always even in time: $K(-t) = K(t)$ , expressing time-reversal symmetry inherent in the symmetrized inner product.

Regime	Krylov Complexity $K(t)$	Physics/Geometry
Early time	$K(t) \propto t^2$	AdS vacuum, pre-chaotic spreading
Late time	$K(t) \propto e^{2\alpha t}$ , $\alpha = \pi/\beta$	Near-horizon, fast scrambling

4. Operator Growth, Quantum Phase Transitions, and Universal Scaling

Krylov complexity can exhibit distinctive universal power-law scaling when a system is driven across a quantum phase transition. During a quench through a second-order quantum critical point, the complexity growth rate and cumulants scale with the same exponents as the Kibble-Zurek defect density: $K_q \sim 2C L^{d-D} \tau^{-\alpha(d-D)}$ where $L$ is the system size, $\tau$ is the quench (ramp) time, $d$ spatial dimension, $D$ defect dimensionality, and $\alpha = \nu/(z\nu+1)$ (with $\nu, z$ the correlation length and dynamical exponents respectively) (Grabarits et al., 15 Oct 2025). The full distribution becomes asymptotically Gaussian in the Kibble-Zurek regime. This demonstrates universal complexity growth laws in critical dynamics.

5. Krylov Complexity Growth in Integrable and Non-Chaotic Systems

Exponential growth of Krylov complexity, and its underlying linear Lanczos coefficients, can occur in both genuinely chaotic systems and in integrable systems with saddle-dominated dynamics or unbounded spectra. Examples include:

Inverted harmonic oscillator: linear $b_n$ ( $b_n = \lambda n$ ), exponential $K(t) \sim e^{2\lambda t}$ (Baek, 2022).
1-matrix quantum mechanics and certain integrable models: numerical and analytic evidence for linear $b_n$ , and exponential growth, even in non-chaotic settings (Vardian, 28 Jun 2024, Bhattacharjee et al., 2022).
In field theory and free CFTs, Krylov complexity generically grows exponentially, with universal exponent $2\pi/\beta$ , regardless of chaos (Dymarsky et al., 2021).

This establishes that exponential Krylov growth is not a definitive diagnostic of chaos; rather, it reflects the possibility of rapid operator spreading whenever the spectral measure (power spectrum) admits analytic behavior enabling linear recurrence.

6. Constraints, Generalizations, and Bounds

The exponential growth rate of Krylov complexity, where it exists, is bounded as: $\lambda_{OTOC} \leq \lambda_K \leq \frac{2\pi}{\beta}$ with $\lambda_{OTOC}$ the OTOC Lyapunov exponent (Avdoshkin et al., 2022). Exponential growth with the maximal rate saturates this chaos bound, realized in maximally chaotic (black-hole dual) CFTs. UV cutoffs, finite lattice effects, or compactification cause late-time saturation or slower growth, with the onset of a linear regime.

Generalized Krylov complexities ( $K_\delta = \sum_{n=0}^\infty n^\delta\, \varphi_n^2(t)$ ) show similar universal early-time and model-dependent late-time behaviors, with inequalities connecting their variances and bounding the growth rate, distinguishing fast from slow scrambling (Fan, 2023).

Additional structure arises when accounting for symmetry resolution: at early times, the full complexity equals the average over charge sectors, while at late times, inter-sector correlations enhance the complexity beyond sector averages (Caputa et al., 2 Jul 2025).

7. Implications for Holography and Quantum Dynamics

The direct relation between boundary Krylov complexity growth rate and bulk radial momentum provides a geometric and physical interpretation of operator growth in the holographic dual. Late-time exponential growth of K-complexity is governed by near-horizon (Rindler) geometry, determining the universal chaotic bound. For higher-dimensional AdS black holes, the matching holds approximately at early and late times, with possible subleading corrections at intermediate times. For AdS $_3$ (BTZ)/CFT $_2$ , exact matching is achieved at all times.

This correspondence places K-complexity as a quantitative boundary dual to coarse-grained bulk observables associated with quantum information scrambling, chaos, and black hole interior physics.

Summary Table of Universal Krylov Complexity Growth Laws

Context	Lanczos $b_n$ behavior	$K(t)$ growth	Exponent / Scaling	Diagnostic Value
Holographic CFT/Black Hole	$\sim \alpha n$	Exponential ( $e^{2\alpha t}$ )	$\alpha = \pi/\beta$	Universal (maximal chaos)
Integrable with unbounded spectrum	$\sim \alpha n$	Exponential	model-dependent	Not diagnostic
Lattice/Finite system	saturates	Linear/Plateau	—	Integrable/chaotic distinction emerges late
Across phase transitions	—	Power-law ( $\tau^{-\eta}$ )	universal scaling	Kibble-Zurek universality

The growth rate of Krylov complexity thus serves as a central quantity encoding the interplay of operator growth, chaos bounds, universality in critical phenomena, and holographic duality, with both theoretical and diagnostic implications across quantum statistical, field-theoretic, and gravity dual systems.