Generalized Krylov Complexity

• Generalized Krylov Complexity is defined as a measure of how an operator or state spreads over a dynamically generated Krylov basis, extending beyond standard closed-system evolution.
• It generalizes the traditional framework by modifying key ingredients such as evolution generators, basis construction algorithms, and interpretations, thus addressing scenarios from time-dependent Hamiltonians to open and curved-space systems.
• This approach offers practical insights into operator growth, synthesis costs, and quantum geometric structures by linking complexity measures with observable phenomena like particle production and localization.

Generalized Krylov complexity denotes a family of extensions of the standard Krylov-complexity construction, whose baseline form measures the spreading of an operator or state over a Lanczos-generated Krylov basis. In the standard setting one studies Heisenberg evolution under a time-independent Hamiltonian in a closed system, but the literature extends the framework to continuum quantum field theory, time-dependent and curved backgrounds, open-system Lindbladian evolution, density matrices and subregions, Floquet dynamics, multi-generator control problems, and information geometry, while retaining the central idea that complexity is the average position of a wavefunction on an emergent Krylov chain or an appropriate generalization of that chain (Adhikari et al., 2022, 2207.13603, Alishahiha et al., 2022, Nizami et al., 2023).

1. Standard framework and baseline definition

The standard operator formulation starts from Heisenberg evolution

$$\partial_t \mathcal{W}(t)= i[H,\mathcal{W}(t)], \qquad \mathcal{W}(t)=e^{iHt}\mathcal{W}e^{-iHt},$$

with Liouvillian superoperator

$$\mathcal{L}_X Y=[X,Y], \qquad \mathcal{L}\equiv \mathcal{L}_H.$$

Repeated action of $\mathcal{L}$ generates a Krylov subspace, and the Lanczos algorithm turns the non-orthogonal sequence $\{\mathcal{L}^n\mathcal{W}\}$ into an orthonormal Krylov basis $\{|\mathcal{W}_n)\}$. In that basis, the Liouvillian is tridiagonal,

$$\mathcal{L}|\mathcal{W}_n)=b_{n+1}|\mathcal{W}_{n+1})+b_n|\mathcal{W}_{n-1}),$$

with Lanczos coefficients $b_n$. Expanding the evolved operator as

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

the amplitudes obey a discrete Schrödinger-like equation and define the standard Krylov complexity

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

This is the average position of a fictitious particle on the Krylov chain, and large $K(t)$ indicates deep spreading in operator space (Adhikari et al., 2022).

This baseline definition already has two equivalent readings. In operator language it is a measure of operator growth under the Liouvillian. In state language it is the expectation value of a “position operator” on the Krylov chain. Generalized Krylov complexity preserves this structure but changes one or more of the underlying ingredients: the generator of evolution, the object being evolved, the basis-construction algorithm, or the geometric or physical interpretation of the resulting “position.”

2. Principal directions of generalization

In the literature, the expression “generalized Krylov complexity” does not denote a single canonical modification. It refers instead to several extensions that alter different layers of the construction.

One line of work generalizes the dynamics. Closed-system Hamiltonian evolution is replaced by Lindbladian evolution in open systems, or by periodic Floquet evolution in driven systems, or by explicitly time-dependent Hamiltonians in cosmology and other non-stationary backgrounds. In these settings the standard Lanczos recursion is supplemented either by non-Hermitian terms, generalized Lanczos coefficients, or Arnoldi coefficients (2207.13603, Nizami et al., 2023).

A second line generalizes the object whose complexity is measured. The universal density-matrix formulation writes Krylov complexity as

$$\mathcal{L}_X Y=[X,Y], \qquad \mathcal{L}\equiv \mathcal{L}_H.$$0

with $$\mathcal{L}_X Y=[X,Y], \qquad \mathcal{L}\equiv \mathcal{L}_H.$$1, thereby placing state and operator Krylov complexities on the same footing and extending the notion to mixed states, reduced density matrices, subregions, and Krylov mutual complexity (Alishahiha et al., 2022).

A third line generalizes the space in which spreading occurs. In continuum QFT, the Krylov basis can coincide with the Fock basis for generalized coherent states, so that Krylov complexity equals the average particle number and inherits a direct volume interpretation. In analog quantum simulation, the relevant structure is a block Krylov hierarchy generated by a set of native Hamiltonians, and the complexity is weighted by layer depth rather than by a single one-dimensional site index (Adhikari et al., 2022, Zhang et al., 8 May 2026).

A fourth line generalizes the interpretation. In coherent-state and information-geometric approaches, operator growth becomes geodesic motion on a phase-space or Fubini–Study manifold, and Krylov complexity becomes proportional to a volume. In the most explicit two-mode realizations, this relation is promoted to a generalized “complexity = volume” statement, both for closed and open systems (Caputa et al., 2021, Zhai et al., 2024).

3. Time-dependent, curved-spacetime, and observer-dependent formulations

A concrete generalization to an explicitly time-dependent background is provided by cosmological perturbations in de Sitter space. There the physical setting is a spatially flat FLRW background with conformal time $$\mathcal{L}_X Y=[X,Y], \qquad \mathcal{L}\equiv \mathcal{L}_H.$$2, time-dependent scale factor $$\mathcal{L}_X Y=[X,Y], \qquad \mathcal{L}\equiv \mathcal{L}_H.$$3, and effective sound speed $$\mathcal{L}_X Y=[X,Y], \qquad \mathcal{L}\equiv \mathcal{L}_H.$$4. The relevant states are cosmological two-mode squeezed states rather than thermal equilibrium states, and the $$\mathcal{L}_X Y=[X,Y], \qquad \mathcal{L}\equiv \mathcal{L}_H.$$5 sector forms an $$\mathcal{L}_X Y=[X,Y], \qquad \mathcal{L}\equiv \mathcal{L}_H.$$6 algebra. In this setting the Krylov basis is the two-mode Fock basis,

$$\mathcal{L}_X Y=[X,Y], \qquad \mathcal{L}\equiv \mathcal{L}_H.$$7

the Lanczos coefficients are

$$\mathcal{L}_X Y=[X,Y], \qquad \mathcal{L}\equiv \mathcal{L}_H.$$8

and the single-mode Krylov complexity is

$$\mathcal{L}_X Y=[X,Y], \qquad \mathcal{L}\equiv \mathcal{L}_H.$$9

The same quantity equals the mean particle number,

$\mathcal{L}$0

so the generalized Krylov complexity becomes a direct probe of cosmological particle production and, in the paper’s words, is tied to volume growth. The corresponding Lyapunov exponent is

$\mathcal{L}$1

and for $\mathcal{L}$2 in de Sitter one obtains $\mathcal{L}$3. Since the Lanczos coefficients grow linearly with $\mathcal{L}$4, the construction realizes generalized Krylov complexity in a curved, time-dependent background while preserving the standard operator-growth logic (Adhikari et al., 2022).

An observer-dependent variant appears in non-inertial quantum systems. For uniformly accelerating observers, the Klein–Gordon symplectic form singles out an $\mathcal{L}$5 sector, and the Rindler pair-number basis naturally forms the Krylov basis. With

$\mathcal{L}$6

the action on the Krylov basis $\mathcal{L}$7 is

$\mathcal{L}$8

so that

$\mathcal{L}$9

The generalized Bogoliubov coefficients are

$\{\mathcal{L}^n\mathcal{W}\}$0

with $\{\mathcal{L}^n\mathcal{W}\}$1, and the Krylov complexity is exactly the mean number of correlated Rindler pairs,

$\{\mathcal{L}^n\mathcal{W}\}$2

The dynamics separates into “hyperbolic Krylov spreading, critical growth, and bounded Krylov-space motion,” and in the detuning-dominated regime the packet remains exponentially confined to low Krylov levels, producing “the localization of Krylov complexity” (Ma et al., 29 Jun 2026).

Time dependence also raises a structural issue. In the geometric construction based on dynamical symmetries, a single Krylov complexity corresponds to the “height” of the evolving operator on a coherent-state manifold. For time-dependent Liouvillian superoperators, however, the motion can proceed in directions invisible to one chosen Krylov axis. In that case, “a single Krylov complexity is no longer sufficient,” and “multiple Krylov complexity may be exploited jointly to fully describe operator dynamics” (Lv et al., 2023).

4. Open, mixed, subregion, and driven extensions

In open quantum systems coupled to a Markovian bath, Hamiltonian evolution is replaced by Lindbladian evolution. The open-system construction in Krylov space keeps the Krylov basis defined from the Hamiltonian commutator $\{\mathcal{L}^n\mathcal{W}\}$3, but expands the operator under the full Lindbladian. The resulting amplitudes satisfy a non-Hermitian tight-binding equation,

$\{\mathcal{L}^n\mathcal{W}\}$4

Because norm is not conserved, the generalized open-system Krylov complexity is defined by

$\{\mathcal{L}^n\mathcal{W}\}$5

For local Hermitian jump operators, the diagonal dissipative coefficients $\{\mathcal{L}^n\mathcal{W}\}$6 are non-negative and grow linearly with $\{\mathcal{L}^n\mathcal{W}\}$7 at intermediate $\{\mathcal{L}^n\mathcal{W}\}$8, before saturating. The effective non-Hermitian Krylov chain then develops localized edge modes, and the long-time behavior is controlled by those modes: dissipation suppresses growth and drives $\{\mathcal{L}^n\mathcal{W}\}$9 to a finite saturation value much smaller than in the closed system (2207.13603).

A distinct extension replaces the evolving vector by a density matrix. In the universal density-matrix formulation, Krylov complexity is

$\{|\mathcal{W}_n)\}$0

This places state and operator Krylov complexities on the same footing. Operator complexity is obtained by mapping the operator to a Choi state in a doubled Hilbert space, while mixed-state and subregion Krylov complexities are obtained by using reduced density matrices and reduced label operators. The same framework defines Krylov mutual complexity,

$\{|\mathcal{W}_n)\}$1

and thereby extends Krylov complexity beyond pure states and full systems (Alishahiha et al., 2022).

Periodic driving leads to a further generalization. For Floquet systems one works with the Floquet operator

$\{|\mathcal{W}_n)\}$2

and constructs the Krylov basis from the orbit $\{|\mathcal{W}_n)\}$3 or its operator analogue. Because $\{|\mathcal{W}_n)\}$4 is unitary rather than Hermitian, the appropriate recursion is Arnoldi rather than Lanczos. The state and operator Krylov complexities retain the same mean-position form, but the recursion data are now Arnoldi coefficients $\{|\mathcal{W}_n)\}$5 rather than a single Lanczos sequence. In kicked systems, especially the toral quantum kicked rotor, Arnoldi coefficients, Krylov complexity, and Krylov subspace dimension vary sharply between weak and strong coupling, providing a natural driven-system extension of the standard framework (Nizami et al., 2023).

In the quantum kicked rotor with cylindrical boundary conditions, state Krylov complexity and Arnoldi coefficients probe several localization mechanisms. The long-time behavior of K-complexity and the wavefunction evolution on the Krylov chain distinguish quantum anti-resonance, classical-induced localization, dynamical localization, and power-law localization. In that setting, “K-complexity not only indicates the degree of localization, but surprisingly also of the nature of localization” (Kannan et al., 30 Mar 2025).

5. Geometric and volumetric interpretations

A major strand of the literature reformulates Krylov complexity geometrically. When the Liouvillian belongs to a Lie algebra and can be written as a sum of ladder operators,

$\{|\mathcal{W}_n)\}$6

Heisenberg evolution can be identified with a displacement operator acting on a manifold of generalized coherent states. The manifold carries a natural Fubini–Study information metric, and the resulting trajectories are geodesics. In this geometry, Krylov complexity is proportional to a phase-space volume. For the explicit $\{|\mathcal{W}_n)\}$7, $\{|\mathcal{W}_n)\}$8, and Heisenberg–Weyl examples analyzed in the paper, one finds

$\{|\mathcal{W}_n)\}$9

with the geometry respectively hyperbolic, spherical, and flat. In this sense, generalized Krylov complexity is the volume swept by a classical geodesic in coherent-state phase space (Caputa et al., 2021).

In continuum QFT, a complementary volume interpretation arises when the Krylov basis matches the Fock basis. For generalized coherent states such as the two-mode squeezed vacuum,

$$\mathcal{L}|\mathcal{W}_n)=b_{n+1}|\mathcal{W}_{n+1})+b_n|\mathcal{W}_{n-1}),$$0

the probabilities in the Fock basis are identical to the Krylov probabilities, and therefore

$$\mathcal{L}|\mathcal{W}_n)=b_{n+1}|\mathcal{W}_{n+1})+b_n|\mathcal{W}_{n-1}),$$1

The paper shows this explicitly for two-mode squeezed states, Heisenberg–Weyl coherent states, Perelomov $$\mathcal{L}|\mathcal{W}_n)=b_{n+1}|\mathcal{W}_{n+1})+b_n|\mathcal{W}_{n-1}),$$2 coherent states, and large-$$\mathcal{L}|\mathcal{W}_n)=b_{n+1}|\mathcal{W}_{n+1})+b_n|\mathcal{W}_{n-1}),$$3 $$\mathcal{L}|\mathcal{W}_n)=b_{n+1}|\mathcal{W}_{n+1})+b_n|\mathcal{W}_{n-1}),$$4 coherent states. In lattice QFT, this yields the heuristic relation “Krylov complexity = average particle number = volume,” and in the UV-dominated early-time regime of the free scalar field,

$$\mathcal{L}|\mathcal{W}_n)=b_{n+1}|\mathcal{W}_{n+1})+b_n|\mathcal{W}_{n-1}),$$5

This parallels the volume scaling of holographic “complexity = volume,” but the paper treats it as a heuristic rather than a theorem (Adhikari et al., 2022).

An information-geometric refinement of this idea is the generalized CV conjecture. For Hermitian two-mode Hamiltonians, including open systems encoded through a generalized Lanczos algorithm,

$$\mathcal{L}|\mathcal{W}_n)=b_{n+1}|\mathcal{W}_{n+1})+b_n|\mathcal{W}_{n-1}),$$6

the proposal is that the Fubini–Study volume equals $$\mathcal{L}|\mathcal{W}_n)=b_{n+1}|\mathcal{W}_{n+1})+b_n|\mathcal{W}_{n-1}),$$7 times the Krylov complexity,

$$\mathcal{L}|\mathcal{W}_n)=b_{n+1}|\mathcal{W}_{n+1})+b_n|\mathcal{W}_{n-1}),$$8

For the closed two-mode squeezed state,

$$\mathcal{L}|\mathcal{W}_n)=b_{n+1}|\mathcal{W}_{n+1})+b_n|\mathcal{W}_{n-1}),$$9

and for the open two-mode squeezed state built from second kind Meixner polynomials,

$b_n$0

with exactly the same expression for $b_n$1. The resulting equalities are exact in the examples studied and provide explicit information-geometric realizations of generalized Krylov complexity in both closed and open two-mode systems (Zhai et al., 2024).

6. QFT asymptotics, block constructions, and conceptual boundaries

In QFT, generalized Krylov complexity has been used to study how field-theoretic scales, UV regularization, and lattice discretization modify operator growth. In Lifshitz-type Dirac field theories with dynamical critical exponent $b_n$2, the Wightman power spectrum determines the moments and hence the Lanczos coefficients. The paper finds that in the presence of a hard UV cutoff, Krylov complexity shows “an initial exponential growth followed by a linear regime,” whereas in the lattice model “due to the finite Krylov basis, Krylov complexity saturates rather than growing indefinitely.” The dependence is controlled by the Lifshitz scaling parameter

$b_n$3

and the analysis exhibits a sharp continuum–lattice distinction (Imani et al., 10 Jun 2025).

A broader QFT survey shows that asymptotic Lanczos behavior can go beyond previously observed universality. Free massive bosons on flat space exhibit persistent staggering between even and odd $b_n$4, while compact-space examples such as CFTs on spheres and thermal AdS display a two-slopes asymptotic with different even and odd linear branches. The same work argues that the generalized MSS-type inequality

$b_n$5

is satisfied in all the cases considered, but also emphasizes that “the relation between the growth of Lanczos coefficients and chaos may only hold for the sufficiently late, truly asymptotic regime governed by the physics at the UV cutoff.” It further shows “scenarios when Krylov complexity in quantum field theory behaves qualitatively differently from the holographic complexity” (Avdoshkin et al., 2022).

Another extension replaces the single-seed Krylov chain by a multi-generator block Krylov hierarchy. In analog quantum simulators with native Hamiltonians $b_n$6, one defines the initial subspace

$b_n$7

generates higher blocks by repeated commutators with all $b_n$8, and decomposes a target Hamiltonian as

$b_n$9

With layer weights

$$|\mathcal{W}(t))=\sum_n i^n \phi_n(t)|\mathcal{W}_n),$$0

the generalized Krylov complexity is

$$|\mathcal{W}(t))=\sum_n i^n \phi_n(t)|\mathcal{W}_n),$$1

This weighting is motivated by the layer circuit complexity

$$|\mathcal{W}(t))=\sum_n i^n \phi_n(t)|\mathcal{W}_n),$$2

so $$|\mathcal{W}(t))=\sum_n i^n \phi_n(t)|\mathcal{W}_n),$$3 estimates synthesis cost rather than operator growth under a single Hamiltonian. In the systems studied, including Rydberg atom arrays, the generalized Krylov complexity of a target operation is a strong predictor of the minimum time required for its realization (Zhang et al., 8 May 2026).

At the opposite extreme, one recent result shows that ordinary operator Krylov complexity can itself be complete. The Taylor expansion

$$|\mathcal{W}(t))=\sum_n i^n \phi_n(t)|\mathcal{W}_n),$$4

admits an explicit recursive algorithm that reconstructs the Lanczos coefficients from the coefficients $$|\mathcal{W}(t))=\sum_n i^n \phi_n(t)|\mathcal{W}_n),$$5. In that sense, “Krylov complexity contains the entire information about the dynamics of a quantum operator,” just as the Lanczos coefficients, the return amplitude, and the spectral density do. The same paper argues that no analogous recursive algorithm can exist for spread complexity without additional dynamical input, because spread complexity alone does not separate diagonal and off-diagonal recursion data (Mück, 27 May 2026).

This suggests that generalized Krylov complexity is best understood as a structured umbrella rather than a single canonical definition. The common core is the Krylov idea itself: construct a basis adapted to dynamics, represent evolution as spreading in that basis, and read complexity from the induced distribution. What changes from one generalization to another is the generator, the basis algorithm, the underlying state space, and the physical interpretation of the resulting “distance” in Krylov space.