Universal Krylov Complexity Growth Laws

• Universal Krylov Complexity Growth Laws is a framework quantifying operator spreading in many-body quantum systems by projecting Heisenberg-evolved operators onto a Krylov basis derived from the Lanczos algorithm.
• It reveals distinct temporal regimes—exponential, linear, and power-law growth—that reflect the system’s chaotic, integrable, or critical nature through the asymptotic behavior of Lanczos coefficients.
• The universal laws link mathematical structure with physical phenomena by connecting operator entropy, symmetry effects, and holographic dualities to the evolution of complexity.

Universal Krylov Complexity Growth Laws

Krylov complexity quantifies the operator growth in quantum many-body dynamics by projecting Heisenberg-evolved operators onto a specially constructed Krylov basis derived via the Lanczos algorithm. Its evolution—captured by the interplay of the Liouvillian superoperator, the underlying system Hamiltonian, and the associated sequence of Lanczos coefficients—displays a set of universal growth laws. These laws encode essential distinctions between integrable, chaotic, and open quantum systems, as well as structural features such as symmetry, temperature, and quantum criticality.

1. Foundational Framework: Krylov Basis and Operator Growth Hypothesis

The construction of Krylov complexity begins with the Krylov (Lanczos) basis: For an initial operator $O$, the operator evolution is governed by the Liouvillian $\mathcal{L}[\,\cdot\,] = [H,\,\cdot\,]$, generating the Krylov basis $\{|O_n)\}$ recursively via

$b_{n+1}|O_{n+1}) = \mathcal{L}|O_n) - b_n|O_{n-1})$

with $b_0=0$ and $|O_0)=O/\|O\|$. In this basis, $\mathcal{L}$ becomes tridiagonal, with off-diagonal elements $b_n$. Krylov complexity is then defined as

$K(t) = \sum_{n=0}^{K-1} n\,|\varphi_n(t)|^2,$

where $|\varphi_n(t)|^2$ is the probability amplitude of the operator wavepacket on the $n$th Krylov site.

The Universal Operator-Growth Hypothesis (UOGH) posits that in generic many-body quantum systems, the off-diagonal Lanczos coefficients $b_n$ grow at most linearly with $n$ at large index: $b_n \lesssim \alpha n \quad (n \to \infty)$ with the "Lanczos slope" $\alpha$ achieving its maximal value in fully chaotic or nonintegrable (e.g., SYK-type) systems and growing sub-linearly in integrable or free systems (Bhattacharjee et al., 2022, Gamayun et al., 4 Apr 2025, Rabinovici et al., 8 Jul 2025).

2. Universal Growth Laws: Temporal Regimes

Krylov complexity exhibits a sequence of universal regimes connected with the dynamical structure of the $b_n$ sequence. The archetypal behavior, valid for a wide class of isolated, closed quantum systems, consists of the following stages (Bhattacharjee et al., 2022, Rabinovici et al., 8 Jul 2025, Tang, 2023):

Regime	Lanczos Asymptotics	Krylov Complexity $K(t)$	Typical Models
Early time	$b_n \approx \alpha n$	$K(t) \sim e^{2\alpha t}$	Chaotic/fast scrambling systems
Crossover	$b_n$ enters plateau $b_\infty$	$K(t) \sim b_\infty t$	RMT, post-scrambling in local chaos
Saturation	$b_n \to 0$ (finite chain edge)	$K(t) \to K_\mathrm{max}$	Finite Hilbert space, Heisenberg time

For polynomial $b_n \sim n^\delta$ with $0 \leq \delta < 1$ (integrable or critical scaling), one finds

$K(t) \sim t^{2\delta/(1-\delta)}$

Sublinear $b_n$ leads to power-law or bounded growth only.

Notably, exact analytical solutions for Krylov complexity exist for numerous solvable instances (e.g., Meixner–Pollaczek, continuous Hahn, alternating odd-even tail families), confirming exponential growth rates $2\alpha$ for any $b_n \sim \alpha n + \gamma + O(1/n)$ at large $n$ (Gamayun et al., 4 Apr 2025).

3. Growth in Distinct Physical Contexts

3.1 Integrable and Saddle-Dominated Scrambling

Even in integrable systems, exponential Krylov complexity growth can emerge via saddle-dominated scrambling, where unstable saddles in phase space induce effective linear $b_n$ scaling: $$b_n \approx \alpha n + \gamma \implies K(t) \sim \exp(2\alpha t).$$ For the integrable Lipkin–Meshkov–Glick model, semiclassical analysis yields $\alpha = \frac{1}{2}\sqrt{2J - 1}$ (Bhattacharjee et al., 2022). This demonstrates that exponential operator growth is not a necessary signature of chaos.

3.2 Random Matrix Theory and Quantum Chaos Benchmarking

In the GUE (random matrix theory) setting, at infinite temperature, $b_n$ rapidly approaches a plateau, and the maximal linear Krylov complexity growth $K(t)\sim b_\mathrm{RMT} t$ provides a benchmark: the linear growth speed in arbitrary local chaotic systems is conjectured not to exceed the RMT plateau value, imposing a sharp universal bound (consistent up to $O(1)$ fluctuations) (Tang, 2023). At low temperatures, $b_n$ grows linearly with $n$ up to a plateau, with slope saturating the chaos bound $\lambda_L = 2\pi/\beta$; Krylov complexity then exhibits $K(t) \sim e^{2\pi t/\beta}$ until scrambling time $t_* \sim \beta\log\beta$, before entering linear growth.

3.3 Quantum Field Theory, UV Cutoff, and Holography

Across quantum field theoretic models, the universality of linear $b_n$ (i.e., $b_n \sim \frac{\pi}{\beta} n + O(1)$ in continuum theories) holds irrespective of chaos, tying the exponential growth rate to temperature scale: $K(t)\sim e^{2\pi t/\beta}$ This exponent is ubiquitous in both free and strongly coupled theories and is realized in holographic (AdS/CFT) models as a precise match—the growth of boundary Krylov complexity is dual to the radial momentum of an infalling probe in AdS, yielding $K(t)\sim e^{2\pi t/\beta}$ in the "Lyapunov" regime (Avdoshkin et al., 2022, Fan, 2024). However, in lattice realizations with finite ultraviolet cutoff, $b_n$ saturates to a constant at large $n$, causing a transition to linear $K(t)$ growth (or even bounded/oscillatory behavior in finite volume or integrable models).

4. Fundamental Bounds and Generalizations

4.1 Dispersion/Uncertainty Bound

For isolated quantum systems, Krylov complexity growth is universally bounded by a speed-limit derived from a Robertson-type uncertainty relation: $|\dot K(t)| \leq 2b_1\, \Delta K(t)$ where $\Delta K$ is the position variance on the Krylov chain; saturation occurs for coherent-state-like dynamics (e.g., linear $b_n$ or SU(2)/Heisenberg-Weyl algebras) (Hörnedal et al., 2022).

In open (Markovian) quantum systems, the generalized non-Hermitian uncertainty relation constrains the normalized Krylov complexity $\widetilde C(t)$ so that exponential early-time growth induced by linear $b_n$ is always attenuated at late times by decoherence, causing complexity to decay (Bhattacharya et al., 2024).

4.2 Lyapunov Bound

Chaotic operator growth, as measured by the out-of-time-ordered correlator (OTOC) exponent $\lambda_\mathrm{OTOC}$, is universally bounded above by twice the Lanczos slope: $$\lambda_\mathrm{OTOC} \leq \lambda_K = 2\alpha \leq 2\pi/\beta$$ the final inequality saturating in holographic models and random matrix theory (Avdoshkin et al., 2022, Tang, 2023, Guo, 2022). Slow scramblers and integrable systems exhibit $\alpha < \pi/\beta$.

5. Universality and Extensions

5.1 Symmetry-Resolved Krylov Complexity and Equipartition

Systems with global symmetries (e.g., U(1) charge) admit a block structure in the Krylov basis. The symmetry-resolved Krylov complexity $K_q(t)$ for each sector matches (to $O(t^2)$) the total complexity averaged over sector weights. Exact equipartition ($K(t) = K_q(t)$ for all $q$) occurs when the operator is energy-local or when sector Lanczos chains coincide; generically, sector mixing causes $K(t)\geq \sum_q p_q K_q(t)$, with deviations determined by inter-sector variance (Caputa et al., 2 Jul 2025).

5.2 Growth Laws at Quantum Phase Transitions

When a system is driven across a quantum critical point, Krylov complexity exhibits power-law growth controlled by Kibble–Zurek scaling. For a linear quench,

$C(t=0) \propto L^{d-D} \tau_Q^{-\nu/(1+z\nu)},$

where $\nu$ and $z$ are critical exponents, $d$ spatial dimension, $D$ defect dimensionality; higher cumulants of the Krylov probability distribution share the same scaling, converging to a Gaussian distribution in the adiabatic limit (Grabarits et al., 15 Oct 2025).

5.3 Entropic Growth and Irreversibility

At long times, whenever the operator wavefunction irreversibly spreads over an infinite Krylov chain (i.e., in systems exhibiting true scrambling), the operator entropy $S_K$ universally satisfies

$S_K(t) = \tilde\eta\, \log K(t) + \beta$

with $\tilde\eta=1$ for fully chaotic (linear $b_n$) systems, $\tilde\eta < 1$ for integrable or sublinear growth (Fan, 2022). This relation is a hallmark of quantum irreversibility and fails for finite or periodic systems.

5.4 Generalized Krylov Complexities and Master Inequalities

Extensions to generalized Krylov complexities (moments of order $\delta$, $K^{(\delta)}(t)$) reveal that (at long times) $K^{(\delta)}(t) \sim [K(t)]^\delta$ holds universally for "slow scramblers," with model-dependent prefactors in fast scramblers (e.g., SYK-type). The relative variances satisfy the master inequality

$$\frac{\Delta K}{K} \leq \frac{\Delta K^{(\delta)}}{\delta K^{(\delta)}}$$

with the direction of stringency depending on the scrambler class (Fan, 2023).

6. Key Model Realizations and Exact Solutions

A range of exactly solvable models provide analytic access to the universal regime:

• Linear $b_n$ (e.g., $b_n = n$): archetype for maximal exponential growth, $K(t) = \sinh^2 t$.
• Meixner–Pollaczek type ($b_n = \sqrt{n(n-1+\eta)}$): retains universal exponential rate with tunable subleading corrections.
• Alternating families, continuous-Hahn, or double-slope $b_n$: realize bounded/oscillatory $K(t)$ or two-slope behavior as seen in finite volume QFT or thermal AdS backgrounds (Gamayun et al., 4 Apr 2025, Avdoshkin et al., 2022).

These families confirm that subleading or staggered corrections in $b_n$ affect only transients, not the late-time universal law.

7. Synthesis: Universality Classes and Physical Implications

The universal growth laws for Krylov complexity offer a precise characterization of operator spreading and information scrambling in diverse quantum many-body settings. The growth rate and crossover structure of $K(t)$ are direct fingerprints of the asymptotics of the Lanczos coefficients, serving as both a diagnostic for chaos/integrability and a kinematic "speed limit" for quantum dynamics. They uniformly govern systems as diverse as integrable chains, random matrix models, quantum field theories, and black hole duals, with explicit dynamical signatures in the structure of $b_n$.

Universal Krylov complexity growth laws thus provide a robust organizing framework to quantify and classify complex operator dynamics in quantum systems, independent of model-specific control parameters, and establish sharp links between physical universality, mathematical structure, and information-theoretic limits (Bhattacharjee et al., 2022, Avdoshkin et al., 2022, Rabinovici et al., 8 Jul 2025, Gamayun et al., 4 Apr 2025, Caputa et al., 2 Jul 2025, Tang, 2023, Grabarits et al., 15 Oct 2025).