Krylov/Spread Complexity in Quantum Systems

• Krylov/Spread Complexity is a nonperturbative, basis-independent measure quantifying quantum state and operator spread using the Lanczos recursion.
• It maps the evolution onto an effective tight-binding chain, where the behavior of hopping amplitudes distinguishes chaotic, integrable, and localized dynamics.
• The framework applies broadly from condensed matter and quantum chaos to black hole physics, offering both numerical diagnostics and experimental relevance.

Krylov/Spread Complexity is a nonperturbative, basis-independent dynamical measure that quantifies the spread of a quantum state or operator under time evolution, utilizing the unique structure of the "Krylov basis" constructed via the Lanczos recursion. Emerging from quantum chaos and operator growth studies, Krylov complexity has become a canonical probe for distinguishing chaotic, integrable, and localized dynamics in many-body systems, with far-reaching applications from condensed matter to black hole physics. Its mathematical foundation lies in mapping the generator of dynamics onto an effective tight-binding chain, where complexity growth, localization phenomena, and late-time saturation encode deep dynamical information.

1. Construction and Formal Definition

Given a quantum system (either a state or operator evolution), one first selects an initial vector $\lvert \psi_0 \rangle$ or operator $|\mathcal O)$. Time evolution is governed by either the Hamiltonian $H$ (Schrödinger picture) or the Liouvillian $\mathcal L = [H, \cdot]$ (Heisenberg picture for operators). The "Krylov subspace" is generated by repeated action: $$\text{span}\left\{ |\psi_0\rangle, H|\psi_0\rangle, H^2|\psi_0\rangle, \ldots \right\}$$ or

$$\text{span}\left\{ |\mathcal O), \mathcal L|\mathcal O), \mathcal L^2|\mathcal O), \ldots \right\}.$$

This sequence is orthonormalized via the Lanczos tridiagonalization, producing basis vectors $\{ |K_n\rangle \}$ with recursion: $$H|K_n\rangle = a_n |K_n\rangle + b_{n+1} |K_{n+1}\rangle + b_n |K_{n-1}\rangle, \quad (b_0 = 0, |K_{-1}\rangle = 0)$$ with $a_n = \langle K_n|H|K_n\rangle$ and $$b_{n+1} = \|H|K_n\rangle - a_n|K_n\rangle - b_n|K_{n-1}\rangle\|$$. For operators, the analogous recursion holds for the Liouvillian. The time-evolved state or operator is expanded as

$$|\psi(t)\rangle = \sum_{n=0}^{K-1} \psi_n(t)\, |K_n\rangle$$

with $\psi_n(t) = \langle K_n | \psi(t) \rangle$. Krylov (or spread) complexity is defined by

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

which is the expectation value of the "position operator" $\hat K = \sum_n n\,|K_n\rangle\langle K_n|$ on this emergent 1D chain (Rabinovici et al., 8 Jul 2025).

2. Anderson Localization Analogy and Growth Regimes

In the Krylov basis, the Hamiltonian (or Liouvillian) becomes a tridiagonal matrix, corresponding to a tight-binding chain with hopping amplitudes $b_n$ and on-site energies $a_n$. The particle's "spreading" along this chain is governed by fluctuations in $b_n$, which encode the system's underlying spectral statistics.

• Chaotic Systems: $b_n$ typically grows smoothly (nearly linearly) with $n$ up to a plateau, resulting in rapid delocalization and high late-time complexity ($\overline{C_K} \approx K/2$).
• Integrable/Localized Systems: $b_n$ exhibits strong fluctuations (off-diagonal disorder) due to Poissonian level statistics, leading to Anderson-like localization of the wavefunction, sharply suppressed spread complexity ($\overline{C_K} \ll K/2$), and a transition probability distribution $Q_{0n}$ skewed to small $n$ (Rabinovici et al., 2021).
• Dynamical Signatures: Early time: $C_K(t) \sim t^2$ (quadratic); intermediate: $C_K \sim e^{2\alpha t}$ for $b_n \sim \alpha n$ (exponential growth, Lyapunov regime); post-scrambling: linear ramp; long times: plateau or localization.

This Anderson-type mapping offers a precise framework—via the variance and profile of the Lanczos sequence—for distinguishing quantum chaos, integrability, and localization on spectral and dynamical grounds (Rabinovici et al., 2021, Rabinovici et al., 8 Jul 2025).

3. Operator Growth, Integrability Diagnostics, and Robustness

Krylov complexity measures dynamical operator growth in operator space, distinguished from combinatorial "operator size" measures. In practice:

• Chaotic (ETH) systems exhibit broad, essentially uniform spreading on the Krylov chain, with late-time complexity reflecting true operator scrambling (Craps et al., 2024).
• Integrable systems localize or "stall" operator growth; the long-time plateau is systematically lower and robustly so under changes to system size or specific operator seeds (Rabinovici et al., 2021).

Numerical verification in models such as the XXZ spin chain and SYK model shows that the late-time saturation level $\overline{C_K}$ and the variance $\sigma^2$ of logarithmic $b_n$ ratios are effective, basis-independent integrability diagnostics (Rabinovici et al., 2021). Multiseed (block) Krylov complexity, which initiates evolution from a set of local/few-body operators (block-Lanczos), further improves discriminatory power, removing seed dependence and yielding a reliable integrability–chaos separator across model classes (Craps et al., 2024).

4. Phenomenological and Toy Models

Toy models reproducing the salient features are constructed using model Lanczos sequences:

• "Clean" linear/ascent–descent $b_n$ profiles plus i.i.d. Gaussian noise (with disorder strength $W$).
• Tuning $W$ increases localization, suppresses complexity, and skews occupation probability toward the Krylov chain origin.
• Even flat off-diagonal disordered chains show that the emergence of localization in Krylov space is sufficient to suppress operator growth.

These phenonenological models both quantitatively and qualitatively reproduce integrable behavior seen in XXZ chains and match the effects of disorder on complexity saturation (Rabinovici et al., 2021).

5. Connections to Quantum Chaos, Localization, and Experimental Relevance

Krylov complexity complements, but is largely independent of, traditional chaos quantifiers (level statistics, OTOC exponents, operator size) and localization diagnostics:

• Chaos–integrability contrast: Early-time dynamics is similar, but late-time spread complexity differentiates by plateau value and participation bias (delocalized vs. localized wavepacket).
• Localization: Strong disorder in $b_n$ produces sharply localized Krylov chains, directly encoding Anderson- or many-body–like localization phenomena (Rabinovici et al., 2021).
• Experimental perspective: Krylov/Spread complexity provides a computationally and conceptually minimal diagnostic—requiring only access to return amplitudes or moments—open to direct dynamical or spectroscopic measurement through temporal snapshots or basis tomography.

Furthermore, the mapping to an effective 1D Anderson model facilitates the extraction of universal bounds, scaling laws, and scaling collapse in the presence of disorder, making Krylov complexity a unifying language for complex dynamics across physical platforms.

6. Extensions, Limitations, and Open Problems

While Krylov complexity is highly sensitive to the structure of operator spreading and an effective probe of integrability and chaos, it has important limitations:

• Its absolute value can depend on the choice of the initial operator or seed; this seed-dependence can be remedied by multi-seed averaging or block constructions (Craps et al., 2024).
• In systems where saddle-dominated scrambling or specific coherent structures dominate, Krylov complexity alone may not unambiguously diagnose genuine quantum chaos; additional spectral diagnostics are typically necessary (Huh et al., 2023).
• There are open questions regarding full nonperturbative scaling in higher dimensions, implications for bulk dualities (e.g., beyond JT gravity), and the influence of finite-size, disorder realizations, or non-unitary effects.

Key outstanding directions include formalizing spectral- and dynamical–correlator connections, extending block Krylov diagnostics to operator ensembles, and developing experimental protocols for direct complexity observation.

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Key References:

• Krylov Localization and Suppression of Complexity (Rabinovici et al., 2021)
• Krylov Complexity (review) (Rabinovici et al., 8 Jul 2025)
• Multiseed Krylov Complexity (Craps et al., 2024)