Krylov Basis in Numerical & Quantum Dynamics

• Krylov basis is a set of linearly independent vectors spanning a subspace generated from repeated actions of a linear operator on a seed vector, used in both numerical linear algebra and quantum theory.
• It transforms high-dimensional evolution problems into structured banded recurrences, enabling efficient iterative solvers, complexity diagnostics, and reduced-order modeling.
• Methods like Lanczos and Arnoldi utilize Krylov bases to reveal system dynamics, aiding in the analysis of localization, thermalization, and spectral properties in various physical settings.

Searching arXiv for papers on “Krylov basis” and related uses, to support a comprehensive article. A Krylov basis is a basis of a Krylov subspace, namely the span generated by repeated action of a linear operator on a seed vector. In numerical linear algebra, for $A\in\mathbb{R}^{n\times n}$ and $r_0\in\mathbb{R}^n$, the canonical construction is $$\mathcal{K}_m(A,r_0)=\mathrm{span}\{r_0,Ar_0,A^2r_0,\dots,A^{m-1}r_0\}$$, and a Krylov basis is any linearly independent set spanning this space; in practice one often uses orthonormal vectors or $A$-conjugate directions (Li et al., 2024). In quantum theory, the same idea appears in state space and operator space: one starts from a state $|\psi_0\rangle$ or an operator $\mathcal{O}$, acts repeatedly with the Hamiltonian $H$ or Liouvillian $\mathcal{L}=[H,\cdot]$, and orthonormalizes the resulting sequence by Lanczos or Arnoldi to obtain a dynamically adapted basis (Mück et al., 2022). Across these settings, the Krylov basis converts a high-dimensional evolution problem into a structured banded recurrence, typically tridiagonal, and thereby underlies iterative solvers, reduced models, complexity diagnostics, and several recent constructions in quantum dynamics (Choudhary et al., 27 Mar 2026).

1. Algebraic definition and basic variants

The basic Krylov subspace for a matrix $A$ and starting vector $r_0$ is

$r_0\in\mathbb{R}^n$0

A Krylov basis of $r_0\in\mathbb{R}^n$1 is any set of $r_0\in\mathbb{R}^n$2 linearly independent vectors whose span is exactly $r_0\in\mathbb{R}^n$3. In preconditioned iterative methods one instead works with $r_0\in\mathbb{R}^n$4, and the basis may be chosen as orthonormal Arnoldi or Lanczos vectors, or as $r_0\in\mathbb{R}^n$5-conjugate directions in conjugate gradients (Li et al., 2024).

For quantum states, the analogous construction begins from a Hamiltonian $r_0\in\mathbb{R}^n$6 and an initial normalized state $r_0\in\mathbb{R}^n$7. The Krylov space of order $r_0\in\mathbb{R}^n$8 is

$r_0\in\mathbb{R}^n$9

where $$\mathcal{K}_m(A,r_0)=\mathrm{span}\{r_0,Ar_0,A^2r_0,\dots,A^{m-1}r_0\}$$0 is the number of linearly independent vectors obtained before the sequence closes (Choudhary et al., 27 Mar 2026). For operators in the Heisenberg picture, the relevant generator is the Liouvillian $$\mathcal{K}_m(A,r_0)=\mathrm{span}\{r_0,Ar_0,A^2r_0,\dots,A^{m-1}r_0\}$$1, and one starts from the orbit $$\mathcal{K}_m(A,r_0)=\mathrm{span}\{r_0,Ar_0,A^2r_0,\dots,A^{m-1}r_0\}$$2 in operator Hilbert space (Mück et al., 2022).

The operator-space formulation requires an inner product. In the operator-growth literature reviewed in the supplied works, examples include the Hilbert–Schmidt inner product

$$\mathcal{K}_m(A,r_0)=\mathrm{span}\{r_0,Ar_0,A^2r_0,\dots,A^{m-1}r_0\}$$3

at infinite temperature, abstract inner products for which the Liouvillian is Hermitian, and Wightman-type norms in quantum field theory (Mück et al., 2022). Once the inner product is fixed, the Krylov basis is determined by the pair consisting of the generator and the seed state or seed operator.

2. Lanczos, Arnoldi, and banded representations

For Hermitian generators, the standard construction is the Lanczos algorithm. In the state formulation one sets

$$\mathcal{K}_m(A,r_0)=\mathrm{span}\{r_0,Ar_0,A^2r_0,\dots,A^{m-1}r_0\}$$4

and then iterates

$$\mathcal{K}_m(A,r_0)=\mathrm{span}\{r_0,Ar_0,A^2r_0,\dots,A^{m-1}r_0\}$$5

$$\mathcal{K}_m(A,r_0)=\mathrm{span}\{r_0,Ar_0,A^2r_0,\dots,A^{m-1}r_0\}$$6

The process stops when $$\mathcal{K}_m(A,r_0)=\mathrm{span}\{r_0,Ar_0,A^2r_0,\dots,A^{m-1}r_0\}$$7, and in the basis $$\mathcal{K}_m(A,r_0)=\mathrm{span}\{r_0,Ar_0,A^2r_0,\dots,A^{m-1}r_0\}$$8 the Hamiltonian is tridiagonal: $$\mathcal{K}_m(A,r_0)=\mathrm{span}\{r_0,Ar_0,A^2r_0,\dots,A^{m-1}r_0\}$$9 (Choudhary et al., 27 Mar 2026). The operator-space recursion has the same structure, with $A$0 replaced by $A$1, and in symmetry-restricted cases the diagonal term may vanish so that only the off-diagonal Lanczos coefficients remain (Mück et al., 2022).

For Floquet systems, the generator is the one-period unitary $A$2, and the natural orthogonalization is Arnoldi rather than Lanczos. Starting from $A$3, one orthonormalizes $A$4 using

$A$5

$A$6

where $A$7 are Arnoldi coefficients (Kannan et al., 30 Mar 2025). In this setting the Krylov basis again defines a one-dimensional chain picture, although the effective matrix structure is no longer restricted to the Hermitian tridiagonal Jacobi form.

The tridiagonal form is not universal if one insists on an algebraically natural basis rather than the strict Lanczos basis. For systems with two-dimensional Schrödinger symmetry, a natural orthonormal basis is the oscillator Fock basis $A$8, because the relevant generators are polynomials of degree one and two in $A$9 and $|\psi_0\rangle$0. In that basis the Liouvillian connects $|\psi_0\rangle$1 to $|\psi_0\rangle$2 and $|\psi_0\rangle$3, producing a pentadiagonal rather than tridiagonal structure (Patramanis et al., 2023). This suggests that the notion of a Krylov basis can be broadened from strict tridiagonalization to symmetry-adapted banded representations when the latter are analytically more transparent.

3. Dynamics in Krylov space and complexity measures

Once the basis is constructed, time evolution becomes a chain problem. For states,

$|\psi_0\rangle$4

and the amplitudes obey

$|\psi_0\rangle$5

(Choudhary et al., 27 Mar 2026). For operators,

$|\psi_0\rangle$6

with a discrete Schrödinger-type equation determined by the Lanczos coefficients of $|\psi_0\rangle$7 (Mück et al., 2022). In both cases, the basis index is interpreted as a site on a semi-infinite or finite Krylov chain.

The most common derived observable is Krylov complexity, or spread complexity,

$|\psi_0\rangle$8

It is the average position of the wavepacket on the Krylov chain (Choudhary et al., 27 Mar 2026). Closely related quantities include the inverse participation ratio

$|\psi_0\rangle$9

and the Shannon entropy of the Krylov distribution (Choudhary et al., 27 Mar 2026). For probability distributions with fixed mean $\mathcal{O}$0, the Shannon entropy is bounded by

$\mathcal{O}$1

a relation used to connect Krylov spreading to entropic quantities (Choudhary et al., 27 Mar 2026).

In several physically important settings, the Krylov basis coincides with a Fock basis. For generalized coherent states, squeezed states, and related constructions, the Fock states are “basically the Krylov basis,” so the occupation-number probabilities $\mathcal{O}$2 coincide with $\mathcal{O}$3, and the mean occupation number equals Krylov complexity: $\mathcal{O}$4 (Adhikari et al., 2022). In the two-mode squeezed vacuum, for example,

$\mathcal{O}$5

while for canonical coherent states,

$\mathcal{O}$6

(Adhikari et al., 2022). This identification is especially useful in free-field and holographic contexts, where particle number has an immediate extensive interpretation.

4. Spectral, polynomial, and phase-space structure

The Krylov basis is closely tied to orthogonal polynomial theory. In operator dynamics one may write

$\mathcal{O}$7

where $\mathcal{O}$8 is a degree-$\mathcal{O}$9 polynomial in the Liouvillian. If $H$0 is the spectral measure induced by the seed operator, then the polynomials satisfy

$H$1

and the three-term recurrence

$H$2

in the orthonormal case (Mück et al., 2022). This is the operator-theoretic realization of Favard-type structure for orthogonal polynomials, and it makes the Lanczos coefficients identical to Jacobi parameters. Explicit solvable examples worked out in the supplied literature include classical orthogonal polynomials, polynomials of the Hahn class, and Tricomi–Carlitz polynomials (Mück et al., 2022).

The same basis supports static resolvent diagnostics. For the resolvent-dressed state

$H$3

one defines the normalized distribution

$H$4

called the Krylov distribution (Alishahiha et al., 5 Feb 2026). In the asymptotic analysis summarized in that work, three universal regimes appear: saturation outside spectral support, extensive growth within continuous spectra, and sublinear or logarithmic scaling near spectral edges and quantum critical points (Alishahiha et al., 5 Feb 2026). This extends the chain picture from unitary evolution to inverse-energy response.

A further development is the Krylov–Wigner function, namely the discrete Wigner function defined with respect to the Krylov basis with appropriate phases. In the large-$H$5 limit, the Krylov basis with suitable phase choices minimizes the early-time growth of Wigner negativity, and numerical studies in random matrix theory show three broad regimes for the negativity: a gradual rise for a time of $H$6, a sharp ramp, and saturation close to the upper bound $H$7 (Basu et al., 2024). This suggests that the Krylov basis can serve as a semi-classical phase-space frame for chaotic dynamics.

5. Many-body physics, field theory, and dynamical diagnostics

In free scalar quantum field theory, lattice regularization maps the theory to coupled harmonic oscillators whose normal modes furnish a tensor-product Fock basis. In the cases analyzed in the supplied QFT study, this Fock basis is effectively the Krylov basis, and the total Krylov complexity becomes a sum over momentum modes,

$H$8

with $H$9 (Adhikari et al., 2022). Because the number of oscillators scales as $\mathcal{L}=[H,\cdot]$0, the resulting complexity scales linearly with volume, paralleling the factor that appears in the holographic “complexity = volume” conjecture (Adhikari et al., 2022). In inverted-oscillator sectors one instead finds $\mathcal{L}=[H,\cdot]$1-type growth, and the same framework is used there as a diagnostic of chaotic behavior (Adhikari et al., 2022).

Krylov-space diagnostics also resolve localization and thermalization phenomena. In the quantum kicked rotor, the Arnoldi basis built from the Floquet operator distinguishes four localization scenarios: quantum anti-resonance, classical-induced localization, dynamical localization, and power-law localization. These regimes display distinct long-time behaviors of K-complexity and distinct variations of the Arnoldi coefficients, and the time-averaged K-complexity together with the variance of Arnoldi coefficients distinguishes localization caused by classical regular structures from localization caused by quantum interference (Kannan et al., 30 Mar 2025). In many-body localization, the operator-growth formulation leads to an emergent single-particle hopping problem whose Lanczos coefficients scale asymptotically as $\mathcal{L}=[H,\cdot]$2 but exhibit even–odd alternation and effective randomness; the resulting Krylov-chain problem is localized when initialized on the first site, and Krylov complexity saturates accordingly (Trigueros et al., 2021).

A related proposal for closed non-integrable systems is the Krylov Thermalization Hypothesis. In that framework, typical local operators expressed in the Krylov basis should have a specific tridiagonal form, with all other matrix elements exponentially small, in analogy with ETH. Within this basis, the infinite-time average of Krylov complexity is proposed as a probe of whether thermalization is weak or strong, while the variance of Lanczos coefficients is reported to be less effective (Alishahiha et al., 2024).

The basis also supports direct connections to quantum-information diagnostics. For bipartite systems, the entanglement entropy of the evolved state can be upper bounded in terms of the entanglement of the Krylov basis vectors and the spread complexity; for multipartite systems, the inverse participation ratio in the Krylov basis is bounded by functions of geometric entanglement measures (Choudhary et al., 27 Mar 2026). For a qubit with nondegenerate Hamiltonian, the relation is exact: $\mathcal{L}=[H,\cdot]$3 where $\mathcal{L}=[H,\cdot]$4 is the initial coherence in the energy basis (Choudhary et al., 27 Mar 2026). This makes the dependence of Krylov spreading on initial coherence completely explicit in the two-level case.

6. Numerical linear algebra, reduced models, and control theory

In numerical linear algebra, Krylov bases are the workhorse of iterative solvers such as conjugate gradients, GMRES, and BiCG. A recent reduced-basis construction for parametric PDEs takes a preconditioned Krylov basis generated for one parameter value and reuses it as a reduced space for the full parameter family (Li et al., 2024). In the symmetric positive-definite case, the reduced conjugate gradient basis method uses PCG search directions $\mathcal{L}=[H,\cdot]$5 as the reduced basis; for nonsymmetric or indefinite problems, the corresponding reduced spaces are built from GMRES residual vectors or BiCG primal and dual sequences (Li et al., 2024). In two-parameter affine families, the key theoretical observation is an invariance of preconditioned Krylov spaces, under which the reduced solution is exactly the $\mathcal{L}=[H,\cdot]$6-step Krylov iterate for every parameter and therefore inherits the classical convergence theory of PCG, GMRES, or BiCG (Li et al., 2024).

The same subspace logic has been imported into shortcuts to adiabaticity. There the relevant generator is not $\mathcal{L}=[H,\cdot]$7 itself but the Liouvillian $\mathcal{L}=[H,\cdot]$8, and the seed is $\mathcal{L}=[H,\cdot]$9. The resulting operator Krylov basis spans the minimal operator subspace in which the adiabatic-gauge-potential problem unfolds, and the counterdiabatic term is expanded exactly in that basis (Takahashi et al., 2023). The Lanczos coefficients obtained during this construction also reveal how higher-body terms enter the counterdiabatic Hamiltonian in many-body models (Takahashi et al., 2023).

For open quantum systems with Lindblad dynamics, the Liouvillian is non-Hermitian, so the appropriate construction is a bi-Lanczos basis with left and right Krylov vectors. After a similarity transformation, the effective tridiagonal representation has symmetric off-diagonal entries $A$0, and Krylov complexity is then computed from the associated chain dynamics (Bhattacharyya et al., 18 Sep 2025). In the damped harmonic oscillator and Caldeira–Leggett model, this basis captures dissipative saturation of Krylov complexity; at the same time, the study argues that Krylov complexity is comparatively insensitive to the onset of decoherence because the Krylov basis does not coincide with the conventional pointer basis used in decoherence theory (Bhattacharyya et al., 18 Sep 2025).

7. Generalizations, natural bases, and the optimality debate

One influential motivation for the Krylov basis in state dynamics is that, for a fixed time interval, the spread and hence K-complexity of the evolving state in the Krylov basis is minimal among all orthonormal bases (Kannan et al., 30 Mar 2025). An analogous large-$A$1 statement has been made for Wigner negativity: with appropriate phases, the Krylov basis minimizes the early-time growth of negativity of the discrete Wigner function (Basu et al., 2024). These results underlie the widespread view that the Krylov basis is an optimal dynamical frame.

That view has recently been challenged. A 2026 study reinterprets the conventional Krylov basis as the basis associated with a first-order approximation of the time-evolution operator and then constructs higher-order generators from higher-order truncations of $A$2. In that framework, an infinite-order generator can be constructed that exhibits smaller spread for arbitrary times, thereby analytically disproving the widely held optimality assumption for conventional Krylov complexity (Čindrak et al., 7 Mar 2026). This suggests that optimality statements are sensitive to the generator and the cost function being minimized.

A different kind of generalization appears in symmetry-adapted constructions. For the Schrödinger algebra, a natural orthonormal basis given by oscillator Fock states leads to a pentadiagonal Liouvillian, not the usual tridiagonal Lanczos form, because the algebra combines linear and quadratic oscillator terms (Patramanis et al., 2023). In the Jacobi group $A$3, where coherent states are physically realized as squeezed states, the simple $A$4 and Heisenberg–Weyl Krylov constructions no longer suffice, and the Lanczos algorithm must be augmented by a refined scheme that mixes several ladder sectors (Haque et al., 2022). These extensions suggest that “Krylov basis” can denote either the strict Lanczos basis or, in a broader but still structured sense, a dynamically natural banded basis adapted to a non-semisimple algebraic setting.

Taken together, the recent literature presents the Krylov basis as both a classical object of numerical linear algebra and a modern organizing principle for quantum dynamics. It is a basis generated by repeated action of a linear operator on a seed, but its contemporary significance lies in what that basis reveals: tridiagonal or banded effective dynamics, orthogonal-polynomial structure, reduced-order approximation spaces, and a unified language for operator growth, localization, thermalization, adiabatic control, and inverse-energy response (Mück et al., 2022).