Krylov State Complexity in Quantum Dynamics

• Krylov state complexity is a measure that quantifies how a quantum state or operator spreads in the recursively generated Krylov basis using the Lanczos process.
• It captures universal features of operator growth, linking exponential growth rates to quantum chaos, integrability, and localization phenomena across diverse models.
• Extensions to open systems and holographic models generalize its application to non-Hermitian dynamics and geometric interpretations in quantum gravity.

Krylov state complexity is a quantitatively precise and dynamically intrinsic measure of operator or state growth in quantum many-body systems. Originally motivated by themes in quantum chaos, quantum computation, and holography, it formalizes how the support of an evolving operator or state expands in a specially constructed orthonormal basis—called the Krylov basis—generated via repeated action of the evolution generator (Hamiltonian or Liouvillian) starting from a simple reference. Unlike geometric complexities, Krylov complexity is unambiguously defined by the quantum dynamics and the initial data, and encodes universal features of operator spreading, integrability, and localization phenomena across models ranging from conformal field theory to open quantum dynamics and cosmological scenarios.

1. Mathematical Definition and Basis Construction

Krylov state complexity, often denoted $\mathcal{C}(t)$ or $K(t)$, quantifies the spread of a time-evolved state or operator in the Krylov basis. For a quantum system with Hamiltonian $H$ and reference operator $\mathcal{O}$ (or state $|\psi_0\rangle$), the Krylov basis $\{|\mathcal{O}_n\rangle\}$ is constructed recursively via the Lanczos (or Gram–Schmidt) process:

$|\mathcal{O}_0\rangle = \mathcal{O}\,,$

$$|\mathcal{O}_{n+1}\rangle = [H, |\mathcal{O}_n\rangle] - b_{n-1}^2 |\mathcal{O}_{n-1}\rangle,$$

with the inner product typically defined through a thermal two-point function for operators or the usual Hilbert space inner product for states. The coefficients $b_n$ (Lanczos coefficients) are fixed by orthogonality.

The time-dependent operator (or state) is expanded as

$$\mathcal{O}(t) = \sum_n i^n\, \phi_n(t)\, |\mathcal{O}_n\rangle,$$

and Krylov complexity is given by the expectation value of the "Krylov number operator": $K(t) = \sum_{n=0}^{\infty} n |\phi_n(t)|^2.$ This construction is universal and can be adapted to operator, state, and mixed-state (density matrix) evolution; for operators, the Liouvillian replaces the Hamiltonian.

2. Krylov Complexity as a Probe of Operator Growth and Chaos

Krylov complexity captures the effective "size" of an operator or state: how far it has "spread" in the Krylov basis chain under evolution. In closed quantum systems, rapid (often exponential) growth at early times is linked to chaotic operator dynamics. The conjecture that this growth rate $\lambda_K$ bounds the Lyapunov exponent from out-of-time-ordered correlators (OTOCs), formally: $\lambda_{OTOC} \leq \lambda_K,$ has motivated its use as a chaos probe. In 2d CFTs, free field theories, and holographic models, Krylov complexity grows as

$$K(t) = 1 + 2\Delta \sinh^2\left(\frac{\pi t}{\beta}\right),$$

with exponential rate $\lambda_K = 2\pi/\beta$ matching the universal chaos bound of Maldacena, Shenker, and Stanford (MSS), regardless of whether the theory is actually chaotic (Dymarsky et al., 2021). Thus, exponential growth is determined by the singularity structure of the thermal two-point function rather than being an exclusive signature of chaos.

In cosmological models, especially in the inflationary (de Sitter) patch, the Krylov complexity of evolving (squeezed) modes tracks the average particle number: $K = \sinh^2 r_k$ with $r_k$ the squeezing parameter; linear growth of the Lanczos coefficients signals operator growth with a Lyapunov exponent set by cosmological parameters (Adhikari et al., 2022).

3. Distinction from Other Complexity and Chaos Quantifiers

Although Krylov complexity reflects operator spreading, it is fundamentally different from circuit/Nielsen complexity. Unlike those measures, which satisfy the triangle inequality and can be interpreted as geodesic distances in unitary space, Krylov complexity fails the triangle inequality for general evolutions (Aguilar-Gutierrez et al., 2023). This is traced to its construction as an average "site" occupation on an emergent one-dimensional chain, lacking a cost-minimizing or metric structure.

Moreover, while OTOCs and entanglement generation show universal signatures of chaos largely independent of the initial condition in the fully chaotic regime, Krylov complexity is acutely sensitive to the initial state's spread in the energy eigenbasis. Its saturation values are monotonically related to the inverse participation ratio (IPR), highlighting a dependence on the initial condition even in fully chaotic regions (PG et al., 5 Mar 2025).

4. Krylov Complexity in Open, Localized, and Specialized Systems

The generalization of Krylov complexity to open quantum systems proceeds by replacing the Hamiltonian with a Lindbladian, leading to an effective non-Hermitian tight-binding chain for Krylov amplitudes (2207.13603). Dissipation introduces non-Hermitian terms whose strengths increase with Krylov index $n$, selectively damping "more complex" operators and leading to the emergence of localized edge modes that saturate complexity at finite values much smaller than in closed systems.

In the context of localization, such as the quantum kicked rotor, Krylov complexity distinguishes between quantum anti-resonance, classical-induced localization, dynamical localization, and power-law localization by characteristic differences in the long-time behavior and saturation of complexity, as well as in the scaling and variance of the Arnoldi coefficients (Kannan et al., 30 Mar 2025).

In $\mathcal{PT}$-symmetric and non-Hermitian oscillator systems, Krylov complexity reveals signatures of distinct dynamical phases. Under balanced loss/gain and $\mathcal{PT}$-symmetric conditions, it remains bounded and oscillatory, whereas in the symmetry-broken phase it grows without bound, correlating with the emergence of ill-defined vacua (Beetar et al., 2023).

5. Unified, Generalized, and Geometric Extensions

Krylov complexity has been unified into a density matrix-based framework that naturally extends to subregion, mixed-state, and mutual complexity contexts (Alishahiha et al., 2022). In this framework, complexity is written as a trace over the density matrix and a label operator, reconciling state and operator complexity and connecting—via the double-pole structure of the label operator’s spectrum—to the linear late-time growth associated with holographic complexity.

A significant extension is to generalized Krylov complexity under arbitrary unitary transformations generated by multiple (possibly noncommuting) symmetry generators. The Krylov basis is then a layered network of orthogonal "blocks," each labeled by the total "order" (often weighted) of generator insertions, rather than a simple chain (Astaneh et al., 31 Jul 2025). This generalization allows the framework to compare (via block weights) with geometric/cost-based complexities (e.g., Nielsen complexity), with explicit formulas in both unweighted and weighted cases for diverse symmetry groups (Abelian and non-Abelian).

6. Holographic and Geometric Interpretation

In models where a holographic dual is available, Krylov complexity takes on a precise geometric meaning. In double-scaled SYK (DSSYK), the Krylov basis constructed via chord diagrams is mapped to bulk length eigenstates in JT gravity (Rabinovici et al., 2023, Ambrosini et al., 19 Dec 2024). The expectation value of the “chord number operator” (i.e., Krylov complexity) corresponds directly to geometric quantities, such as the length of a wormhole. In the large-$N$ (triple-scaling) limit, the time evolution of Krylov complexity is governed by classical equations for a particle in a Morse or Liouville potential, closely mirroring the holographic growth of wormhole interiors.

This correspondence generalizes to operator insertions (matter-chord states), where the Krylov complexity tracks the scrambling dynamics and matches the total chord length in the semiclassical limit. The effective Krylov Hamiltonian is recast as a Morse or Liouville quantum mechanics, with scrambling time and operator size encoded in its parameters (Ambrosini et al., 19 Dec 2024).

7. Further Directions and Limitations

While Krylov complexity encodes rich features of operator growth, spread, and even localization, its interpretation as a chaos quantifier is subtle. (i) Universality of the exponential growth rate in field theories implies exponential Krylov growth cannot, on its own, diagnose chaos (Dymarsky et al., 2021). (ii) Strong sensitivity to the initial state/operator—via the IPR and the initial expansion in the energy eigenbasis—confounds its use as a universal, basis-independent measure of chaos (PG et al., 5 Mar 2025). (iii) Averaged Krylov complexities—even over many initial conditions—do not reliably distinguish chaotic from regular regimes, in contrast to OTOC or entanglement plateau behaviors.

Nevertheless, Krylov complexity continues to serve as a powerful and insightful tool in contexts where operator growth, localization mechanisms, open system dynamics, and holographic dualities require a nuanced and operationally clear quantifier of complexity. The continuous extension to generalized unitaries and the block-decomposed orthogonalization scheme establishes its flexibility and relevance for quantum information theory, many-body chaos, and quantum gravity.