Krylov Complexity in Quantum Dynamics

• Krylov complexity is defined as a quantitative measure of the spread of quantum states or operators via a parameter-free Lanczos recursion method.
• It discriminates between integrable and chaotic regimes by revealing distinct dynamical features such as low plateaus or linear growth in time evolution.
• Its robust mathematical foundations connect operator growth, quantum chaos, and holographic dualities, establishing it as a versatile tool in theoretical physics.

Krylov complexity is a quantitative, dynamical measure of the spread of a quantum state or operator under time evolution, formalized through the recursive structure of the Lanczos algorithm. Originally emerging in the context of quantum chaos and operator growth in many-body and field-theoretic systems, it has developed into a precise and technically robust tool for probing the nature of dynamical complexity in quantum physics and for connecting to holographic dualities. Krylov complexity is distinctive in that it is constructed without arbitrary control parameters—its definition is fixed once a seed operator or state and a physically motivated inner product are specified—and it directly quantifies the number of orthogonal directions explored under dynamical evolution, with a phenomenology that sharply discriminates between integrable and chaotic regimes and meaningful connections to gravitational observables.

1. Construction and Mathematical Foundations

Krylov complexity is based on the expansion of an operator (or state) in a dynamically adapted, orthonormal Krylov basis generated via successive actions of the Hamiltonian (for states) or the Liouvillian superoperator (for operators). The construction proceeds as follows:

• Starting from a seed operator 𝒪 or vector |ψ⟩, Krylov vectors are recursively generated:

$$|O_0) = \frac{O}{\|O\|},\quad |O_{n+1}) = \frac{\mathcal{L}|O_n) - b_n|O_{n-1})}{b_{n+1}},\quad b_{n+1} = \|\mathcal{L}|O_n) - b_n|O_{n-1})\|$$

where $\mathcal{L}$ is the Liouvillian, $\mathcal{L}(O) = [H, O]$.

• The time-evolved operator is expressed as:

$|O(t)) = \sum_n \phi_n(t) |O_n)$

• The Krylov complexity is defined as:

$K(t) = \sum_n n |\phi_n(t)|^2$

It represents the mean position of the operator or state on the Krylov chain, quantifying how many basis directions are needed to describe the spread induced by time evolution.

This tridiagonal, chain-like structure links directly to the mathematics of orthogonal polynomials via the recurrence relations satisfied by the polynomials associated with the Krylov basis (Mück et al., 2022).

2. Krylov Complexity Across Physical Contexts

Krylov complexity applies in a broad range of physical settings, from simple quantum mechanical models to quantum field theory (QFT) and models with holographic duals:

• Free Scalar QFT: Krylov complexity for a free scalar field (on a spatial lattice) reduces to a sum over oscillator modes, with analytic expressions allowing exploration of volume scaling and mode-resolved dynamics (Adhikari et al., 2022).
• Gaussian States: In the case where the Krylov basis coincides with the Fock basis, Krylov complexity is equal to the average particle number. For a two-mode squeezed vacuum,

$K(t) = \overline{n} = \sinh^2(r_k)$

This links the complexity of operator evolution directly to physically measurable occupation numbers, providing a concrete operational meaning (Adhikari et al., 2022, Adhikari et al., 2023).

• Integrable Models: Exactly solvable models (e.g., those with Krawtchouk or Hermite polynomial structure) often exhibit truncated Krylov spaces with vanishing Lanczos coefficients beyond a certain order, resulting in non-growing or oscillatory complexity ("non-complexity") (Sasaki, 11 Mar 2024).
• PT-Symmetric and Dissipative Systems: Krylov complexity captures phase transitions between equilibrium oscillations and non-equilibrium exponential growth, revealing both well-behaved and "ill-defined" vacua (e.g., the Bateman oscillator) (Beetar et al., 2023).
• Random Matrix Theory (RMT): In the large-N limit, Krylov complexity grows linearly in time once Lanczos coefficients reach a plateau, with the RMT linear growth speed acting as an upper bound for generic chaotic systems (Tang, 2023).

3. Krylov Complexity and Quantum Chaos

A principal application of Krylov complexity is as a probe of operator growth and quantum chaos:

• Integrable vs. Chaotic Dynamics: Krylov complexity exhibits lower late-time plateaus in integrable systems, reflecting localization of the operator wavefunction in Krylov space, while in chaotic systems the complexity saturates at high values, matching and sometimes being bounded by RMT predictions (Rabinovici et al., 2022).
• Anderson-like Localization: The dynamics on the Krylov chain can be mapped to an off-diagonal Anderson hopping model, where disorder in the Lanczos/Arnoldi coefficients leads to operator localization and lower complexity saturation (Rabinovici et al., 2022, Kannan et al., 30 Mar 2025).
• Spectral Features: The linear growth and eventual plateau of Lanczos coefficients encode information about scrambling and the spectral properties of the system, with exponential complexity growth at early times bounded by the chaos exponent (e.g., Lyapunov exponent $\lambda_L \leq 2\alpha$, where $\alpha$ is the Lanczos slope) (Sánchez-Garrido, 4 Jul 2024, Avdoshkin et al., 2022).
• Sensitivity to Initial Conditions: The complexity dynamics and saturation values depend monotonically on the inverse participation ratio (IPR) of the initial operator or state in the energy eigenbasis, limiting the universality of K-complexity as a chaos quantifier compared to OTOCs; this holds even under averaging (PG et al., 5 Mar 2025).

4. Connections to Holography and Bulk Duals

Krylov complexity has deep links to holographic dualities, especially in models such as SYK that admit a gravity dual:

• Complexity = Volume Proposal: In settings where the Krylov basis matches the Fock basis, Krylov complexity tracks average particle number, and hence physical volume ("complexity equals volume") (Adhikari et al., 2022).
• Bulk-Krylov Dictionary: In double-scaled SYK, the Krylov basis elements correspond to fixed-chord-number states that map precisely to length eigenstates of the two-sided AdS$_2$ wormhole; the time-dependent Krylov complexity matches the renormalized wormhole length in JT gravity (Rabinovici et al., 2023, Sánchez-Garrido, 4 Jul 2024).
• Geometric Interpretation: In models with sufficient symmetry, Krylov complexity is related to geodesic motion in the appropriate manifold (sphere, hyperboloid), and can be given a geometric interpretation as the height or coordinate in a sigma-model picture (Lv et al., 2023, Sánchez-Garrido, 4 Jul 2024).
• Contrast with Holographic Complexity: Krylov complexity does not always mirror holographic complexity (e.g., in compact QFTs or low-temperature AdS backgrounds it can be bounded/oscillatory while holographic complexity may continue to grow) (Avdoshkin et al., 2022).

5. Methodological Properties and Universal Features

A distinctive technical advantage of Krylov complexity is its foundation in a parameter-free, intrinsic dynamical basis:

• Lanczos Algorithm: Krylov basis vectors are generated via the Lanczos recursion, ensuring orthogonality and optimal basis adaptation for operator evolution; this leads to a tridiagonal representation of the Liouvillian.
• Orthogonal Polynomials: The evolution is mathematically equivalent to the dynamics of a wavefunction on a 1D chain with hopping amplitudes set by the Lanczos coefficients, mapping to the recurrence relations of orthogonal polynomials (Mück et al., 2022).
• Generality: The Krylov complexity construction is intrinsic and independent of any arbitrary choices (e.g., tolerance, gate set), once the inner product is selected. This makes it robust across both finite-dimensional and field-theoretic settings.
• Optimality Theorems: For any reasonable class of generalized complexity measures (q-complexities, defined via a complexity operator with bounded step size), K-complexity is provably minimal and upper-bounds all comparably defined complexity measures (Rabinovici et al., 8 Jul 2025, Sánchez-Garrido, 4 Jul 2024).
• Algorithmic Considerations: For time-dependent (Floquet) systems, Krylov bases can be constructed by generalized Arnoldi iterations, extending the parameter-free nature to periodically-driven dynamics (Nizami et al., 2023).

6. Diagnostic Powers and Limitations

Krylov complexity is a sensitive probe of dynamical spreading and localization; its key features include:

Diagnostic	Integrable Systems	Chaotic Systems	Localization Regimes
Late-time plateau	Low, due to localization	Maximal/large, due to delocalization	Type and degree reflected in plateau value
Early-time growth	Oscillatory or bounded	Exponential, saturates then linear	Linear/oscillatory/power-law depending on type
Arnoldi/Lanczos coeffs	Strong fluctuations, saturate	Initial linear, then plateau	Mappings to nature of localization (e.g., anti-resonance, dynamical localization)

• Multiseed Krylov complexity, using a set of "simple" initial operators and the block-Lanczos algorithm, removes seed dependence and provides a robust diagnostic of chaos vs. integrability (Craps et al., 24 Sep 2024).
• In quantum kicked rotor and similar models, K-complexity and the scaling of Arnoldi coefficients can distinguish between dynamical localization (quantum interference) and classical-induced localization (Kannan et al., 30 Mar 2025).
• Krylov complexity, unlike Nielsen or circuit complexity, does not satisfy the triangle inequality and is not a distance on the space of states/operators; modifications do not resolve this, defining a strict conceptual difference (Aguilar-Gutierrez et al., 2023).

7. Open Problems and Research Directions

Krylov complexity is the focus of ongoing research, with several outstanding directions:

• Extension to Open and Time-dependent Systems: Generalization to non-unitary (open-system) and time-dependent dynamics (e.g., with Lindbladians) is under development, with implications for dissipative quantum information (e.g., in cosmology) (Zhai et al., 27 Nov 2024).
• Numerics and Large-scale Simulations: Efficient implementation of the Lanczos algorithm and Arnoldi iterations in large many-body systems, and controlling numerical instabilities, are active areas (Sánchez-Garrido, 4 Jul 2024).
• Comparison with Other Complexities: The geometric relation between Krylov and Nielsen complexity is still incomplete; universal or model-specific links between these measures are being clarified (Lv et al., 2023, Sánchez-Garrido, 4 Jul 2024).
• Bulk-to-Boundary Maps in Holography: The precise mapping between Krylov complexity (boundary measure) and bulk (gravity-side) observables is being explored beyond solvable JT/SYK scenarios (Rabinovici et al., 2023).
• Universality and the Role of Inner Products: In QFT, the universality of asymptotic Lanczos growth and its relation to chaos depends on regularization choices and the specific inner product used in operator space (Avdoshkin et al., 2022).
• Entropic and Information-theoretic Implications: The connection of K-complexity to other entropic measures (e.g., operator entanglement, K-complexity entropy) remains under investigation.

Krylov complexity thus stands at a nexus of quantum information, operator dynamics, random matrix theory, and gravitational holography, offering a unified, mathematically grounded framework for understanding operator growth, chaos, and the emergence of complexity in both microscopic quantum systems and their holographic duals (Rabinovici et al., 8 Jul 2025, Adhikari et al., 2022, Avdoshkin et al., 2022, Rabinovici et al., 2023).