Universal Operator Growth Hypothesis

• Universal Operator Growth Hypothesis is a framework describing the linear asymptotic growth of Lanczos coefficients, capturing operator complexity evolution in nonintegrable quantum systems.
• It connects Krylov space dynamics with exponential operator spreading, establishing universal limits for chaos measures such as Lyapunov exponents in models like SYK and spin chains.
• The hypothesis underpins analytical approaches to high-frequency spectral decay and hydrodynamic transport, providing a robust basis for understanding thermalization in many-body dynamics.

The Universal Operator Growth Hypothesis (OGH) posits that in generic, nonintegrable many-body quantum systems, the growth of operator complexity under time evolution is controlled in a universal fashion by the asymptotic linear growth of Lanczos coefficients. This linear growth encapsulates the most rapid operator spreading compatible with locality in the system, setting fundamental limits for the dynamics of information scrambling, chaos, and transport.

1. Foundational Statement and Mathematical Framework

The OGH asserts that for a generic quantum many-body Hamiltonian with local interactions, if one starts with a simple local operator $O$ having zero overlap with conserved quantities, the successive Lanczos coefficients $\{b_n\}$ from the recursion (Krylov) method grow asymptotically linearly at large $n$: $b_n = \alpha n + \gamma + o(1) \qquad (d > 1)$ Here, $\alpha$ is a positive energy scale (the operator growth rate), $\gamma$ is a constant, and $o(1)$ denotes subleading corrections. In $d=1$, geometric constraints reduce the maximal growth rate, leading instead to a logarithmic correction: $b_n \sim \frac{n}{\ln n}$

The autocorrelation function $C(t) = \mathrm{Tr}[O(t)O]$ admits a continued-fraction (or Krylov) representation with these coefficients. The high-frequency spectral function,

$\Phi(\omega) = \int dt \, e^{i\omega t} C(t),$

decays exponentially,

$$\Phi(\omega) \sim e^{-|\omega|/\omega_0}, \quad \omega_0 = \frac{2}{\pi}\alpha,$$

with $\omega_0$ set by the operator growth rate $\alpha$. Thus, $\alpha$ simultaneously controls the exponential spreading of operator complexity and the large-frequency response.

2. Operator Complexity and Krylov Space Dynamics

In the Krylov framework, the Liouvillian $\mathcal{L} = [H, \cdot]$ is tridiagonalized on the Krylov basis $\{O_n\}$, such that the operator evolution is equivalent to a quantum walk on a semi-infinite one-dimensional chain: $|O(t)\rangle = \sum_n i^n \varphi_n(t)|O_n)$ The amplitude $|\varphi_n(t)|^2$ represents the probability that $O(t)$ has Krylov-complexity level $n$ at time $t$. When $b_n \sim \alpha n$, the operator's mean "position" in Krylov space grows exponentially: $\langle n \rangle_t \sim e^{2\alpha t}$ For generic complexity measures ("q-complexities") $Q(t)$, the hypothesis rigorously implies a universal bound: $Q(t) \leq C\,\langle n \rangle_t$ for some constant $C$. Thus, $\alpha$ not only governs Krylov complexity growth, but upper-bounds all reasonable operator complexity measures, including those related to out-of-time-order correlators (OTOCs).

3. Connection to Quantum Chaos and Lyapunov Exponents

The OGH yields a direct constraint on chaos as measured by Lyapunov exponents. In chaotic quantum systems, the OTOC (or any "q-complexity") may grow at most exponentially with rate $\lambda_L$ no greater than twice the operator growth rate: $\lambda_L \leq 2\alpha$ This bound is nearly saturated in maximally chaotic models such as the large-$q$ limit of the Sachdev-Ye-Kitaev (SYK) model, where explicit calculation yields $b_n \sim \mathcal{J} n$ and $\alpha = \mathcal{J}$ (the effective interaction strength). Compared to the Maldacena-Shenker-Stanford bound $\lambda_L \leq 2\pi T$, the OGH-based constraint is model-independent and can sharpen the universal chaos bound in certain regimes, e.g., low-$T$ SYK where $2\alpha_T = 2\pi T$.

4. Support in Paradigmatic Models

The OGH is corroborated in a diversity of quantum dynamical systems:

• Nonintegrable Spin Chains: Numerical results for 1D spin-$1/2$ chains with integrability-breaking perturbations reveal that $b_n$ transitions from bounded or sublinear to linear in $n$ as interactions become strong. In higher dimensions, linear growth is universal. Integrable/free models, by contrast, display significantly slower (typically root-$n$) growth.
• SYK Model: Analytic computation at large-$N$ and large-$q$ yields $b_n \sim \mathcal{J}n$, with Krylov complexity and q-complexity bounds sharply realized.
• Classical Analogues: Investigation of classical spin models using the recursion method suggests a robust upper bound $\lambda_L \leq 2\alpha$ (sometimes rigorously $\lambda_L \leq 4\alpha$), indicating continuity between quantum and classical operator scrambling.

5. Implications for High-Frequency Response and Diffusion

The exponential decay in the spectral function at high frequencies, as rigorously established via the equivalence with linear $b_n$ growth, sets the radius of convergence for time-ordered moment expansions and underlies thermalization rates in strongly chaotic systems. Semi-analytic computation of hydrodynamic transport coefficients (e.g., diffusion constant $D$) is feasible using the continued fraction expansion for the Green's function: $$G(z) = \frac{1}{z - \dfrac{b_1^2}{z - \dfrac{b_2^2}{z - \cdots}}}$$ The universal asymptotic for $b_n$ enables an analytic "stitching" at large $n$ that, together with the early numerically computed $b_n$, allows computation of $D$ from the poles of $G(z)$, matching state-of-the-art numerics within a few percent.

6. Generalizations, Limitations, and Outlook

The OGH applies robustly to generic, locally interacting, nonintegrable many-body Hamiltonians in the thermodynamic limit. Important generalizations and caveats include:

• Dimensionality: In $d=1$ the maximal growth is $b_n \sim n/\ln n$, with the logarithmic slowdown reflecting boundary-limited operator growth.
• Complexity Measures: All "q-complexities" grow no faster than Krylov complexity, but the hypothesis does not necessarily control their exact scaling in integrable or weakly chaotic systems.
• Thermal Effects: At finite $T$, the rate $\alpha$ becomes $T$-dependent through the appropriate thermal inner product.
• Numerical Boundaries: In practical finite-size and finite-time numerics, true asymptotic linear scaling may be difficult to access, and subleading corrections (constant and $1/n$ pieces) can impact the observed regime.
• Classical Correspondence: In classical cases, a similar maximal operator growth principle leads to weaker bounds but the structural analogy holds.
• Open Systems and Extended Symmetries: Recent work shows that modifications are required in Lindbladian open quantum systems and in systems with higher-spin symmetries (which can violate the OGH bound by exhibiting, e.g., $b_n \sim n^2$ growth).

7. Summary Table of Universal Operator Growth Predictions

Property	Nonintegrable (Generic)	Integrable/Free	1D Correction	Universal Bound
$b_n$ scaling	$\sim n$	constant/$\sqrt{n}$	$n/\ln n$ (1D)	$b_n\leq \mathrm{(geom\,bound)}$
$\langle n\rangle_t$	$e^{2\alpha t}$	subexponential	$e^{2\alpha t/\ln t}$
$\Phi(\omega)$	$e^{-|{\omega}|/\omega_0}$	faster decay	$e^{-|{\omega}|/(\omega_0\ln|\omega|)}$
Lyapunov $\lambda_L$	$\leq 2\alpha$	N/A	$\leq 2\alpha_{1D}$	$\leq 2\pi T$ (MSS)
$D$ algorithm	Continued fraction with universal $b_n$	N/A	Crossover via stitching	Validated numerically

This framework unifies quantum chaos, thermalization, and operator spreading under the constraint of maximally efficient operator complexity growth. The OGH endows the growth rate $\alpha$—directly accessible in experiment and computation—with central importance as a bound for chaos, complexity, and hydrodynamics in quantum dynamics.