Lanczos Coefficients in Quantum Dynamics

• Lanczos coefficients are recurrence parameters in Krylov subspace methods that simplify spectral analysis and operator evolution in quantum systems.
• They enable efficient continued-fraction representations of Green’s functions and approximate spectral densities using robust numerical algorithms.
• Their growth patterns reflect underlying physics such as quantum chaos or integrability, aiding in diagnostics of many-body behavior and convergence.

Lanczos coefficients are the recurrence parameters appearing in the three-term tridiagonalization of an operator—typically a matrix, Liouvillian, or moment-permuted Hamiltonian—in a Krylov (orthogonalized) basis. They underlie the continued-fraction expansion of Green’s functions, form the bridge to orthogonal polynomial theory, and encode key features of spectral densities, correlation functions, and operator complexity in quantum systems.

1. Definition and Origin of Lanczos Coefficients

The classical Lanczos algorithm constructs an orthonormal basis in a Krylov subspace by iterated application of an operator. For a Hermitian matrix $A$, starting from a normalized vector $v_0$, the recursion is

$$A v_j = \beta_{j-1} v_{j-1} + \alpha_j v_j + \beta_j v_{j+1},$$

with $\alpha_j = \langle v_j, A v_j \rangle$ and $$\beta_j = \|A v_j - \alpha_j v_j - \beta_{j-1} v_{j-1}\|$$. Analogously, for a Liouvillian $\mathcal{L}$ acting on operators $\mathcal{O}_n$, the recursion is

$$b_{n+1} \mathcal{O}_{n+1} = \mathcal{L} \mathcal{O}_n - b_n \mathcal{O}_{n-1}.$$

Here, the $\alpha_j$ and $\beta_j$ (or, more commonly, the single off-diagonal sequence $v_0$0) are the Lanczos (or Jacobi) coefficients. For orthogonal polynomials $v_0$1 with respect to measure $v_0$2, similar recursions hold: $v_0$3 and explicit formulas are available for classical weights, but generic measures require numerical schemes (Liu et al., 2021).

2. Structural Properties and Computational Algorithms

The Lanczos coefficients reflect the measure or operator’s moments and can be computed either explicitly (for classical cases) or via iterative/numerical techniques. Stabilized versions of the Lanczos algorithm with reorthogonalization are standard for discrete measures or finite matrices, while continuous or hybrid measures use predictor-corrector or Stieltjes-type approaches. For general measures,

• Purely discrete: Stabilized Lanczos iteration [Step 4 and 5 in (Liu et al., 2021)].
• Purely continuous: Predictor–corrector algorithm.
• Mixed measure: Predictor–corrector–Lanczos hybrid (PCL).

Orthogonality defects and successive changes in $v_0$4 are used as convergence diagnostics. In quantum computing, block Lanczos algorithms extend the method to resolve degeneracies and treat non-Hermitian evolutions by bi-orthogonal bases (Baker, 2021, Bhattacharya et al., 2023).

3. Connection to Continued Fractions and Functional Representation

Lanczos coefficients parameterize continued-fraction representations of Green’s functions and autocorrelation functions. The retarded Green’s function for operator $v_0$5 and Liouvillian $v_0$6 is expressed as (Pinna et al., 30 Apr 2025): $v_0$7 which recasts the entire spectral problem in terms of a one-dimensional tridiagonal system governed by $v_0$8. This enables highly efficient computation and asymptotic analysis of global spectral features, spectral densities, and moments (Pinna et al., 30 Apr 2025, Qu, 17 Dec 2025).

Such continued fractions also underlie recursion methods for quantum autocorrelation functions, where the Laplace or Fourier transforms are analytic functions determined by the sequence of $v_0$9. Corrections, truncations, and “stitching” approximations quantify finite-size errors or incomplete knowledge of the underlying coefficients (Pinna et al., 30 Apr 2025, Füllgraf et al., 21 Mar 2025).

4. Physical Interpretation and Asymptotic Growth

The growth of Lanczos coefficients encodes deep physical and statistical properties:

• Quantum Many-body Systems: In chaotic, non-integrable models, $$A v_j = \beta_{j-1} v_{j-1} + \alpha_j v_j + \beta_j v_{j+1},$$0 typically exhibits asymptotically linear growth, $$A v_j = \beta_{j-1} v_{j-1} + \alpha_j v_j + \beta_j v_{j+1},$$1, with possible sublinear corrections ($$A v_j = \beta_{j-1} v_{j-1} + \alpha_j v_j + \beta_j v_{j+1},$$2 in one dimension) (Wang et al., 2024, Pinna et al., 30 Apr 2025).
• Operator Growth Hypothesis (OGH): Linear or nearly linear growth of $$A v_j = \beta_{j-1} v_{j-1} + \alpha_j v_j + \beta_j v_{j+1},$$3 is conjectured as a hallmark of quantum chaos in the thermodynamic limit.
• Free (quadratic) Models: The Anderson impurity model demonstrates that diverse asymptotic forms of $$A v_j = \beta_{j-1} v_{j-1} + \alpha_j v_j + \beta_j v_{j+1},$$4 (constant, $$A v_j = \beta_{j-1} v_{j-1} + \alpha_j v_j + \beta_j v_{j+1},$$5, linear) can all arise in an exactly solvable, integrable context (Füllgraf et al., 19 Jan 2026).
• Finite-size Scaling: For finite systems, the late-time scaling and fluctuations of $$A v_j = \beta_{j-1} v_{j-1} + \alpha_j v_j + \beta_j v_{j+1},$$6 (or the ratios $$A v_j = \beta_{j-1} v_{j-1} + \alpha_j v_j + \beta_j v_{j+1},$$7) control hydrodynamical relaxation, zero-mode behavior, and saturation plateaux of autocorrelation functions. Specific scaling conjectures relate these ratios to system size and the presence or absence of conserved quantities (Capizzi et al., 23 Jul 2025).
• Open Systems: In Lindbladian evolution, bi-Lanczos coefficients (off-diagonals and pure imaginary diagonals) reflect both operator scrambling and dissipation. Early-time behavior may distinguish chaos and integrability, but late-time fluctuations induced by dissipation universalize the saturation of K-complexity (Bhattacharya et al., 2023).

5. Error Analysis, Approximations, and Numerical Significance

Various truncation and approximation schemes rely on the rapid convergence properties of continued-fraction expansions with smoothly-growing $$A v_j = \beta_{j-1} v_{j-1} + \alpha_j v_j + \beta_j v_{j+1},$$8:

• Stitching Approximation: Completing the continued fraction beyond a finite set of known $$A v_j = \beta_{j-1} v_{j-1} + \alpha_j v_j + \beta_j v_{j+1},$$9 with an asymptotically-matching tail yields robust approximations. The rate of convergence depends on the decay of staggered subleading terms in the sequence; error terms scale as $\alpha_j = \langle v_j, A v_j \rangle$0 (best case, e.g., purely smooth spectral densities) to $\alpha_j = \langle v_j, A v_j \rangle$1 when subleading staggering decays only logarithmically (Pinna et al., 30 Apr 2025).
• Lanczos-Pascal Truncation: For chaotic systems with smooth, near-linear $\alpha_j = \langle v_j, A v_j \rangle$2, as few as 10–30 coefficients suffice to recover temporal autocorrelators and spectral areas to high precision; the method reduces the problem to a small set of damped oscillations (Füllgraf et al., 21 Mar 2025).

In finite-dimensional settings, strong concentration theorems show that $\alpha_j = \langle v_j, A v_j \rangle$3 Lanczos steps yield coefficients overwhelmingly close to deterministic medians for “bulk” spectral features, with $\alpha_j = \langle v_j, A v_j \rangle$4 controlling convergence to limiting Jacobi parameters (Garza-Vargas et al., 2019).

6. Mapping to Orthogonal Polynomials and Random Matrix Theory

In random matrix theory and the theory of orthogonal polynomials, the Lanczos coefficients are precisely the recurrence coefficients $\alpha_j = \langle v_j, A v_j \rangle$5 and $\alpha_j = \langle v_j, A v_j \rangle$6 (for off-diagonals and diagonals, respectively). In the large-$\alpha_j = \langle v_j, A v_j \rangle$7 continuum limit, one finds the mapping

$\alpha_j = \langle v_j, A v_j \rangle$8

where $\alpha_j = \langle v_j, A v_j \rangle$9. The density of states, moments, and Krylov dynamics computed via the Jacobi (tridiagonal) matrix directly coincide whether generated from the Lanczos approach or from orthogonal polynomial theory (Qu, 17 Dec 2025).

Explicit models, for example the Gaussian Unitary Ensemble, yield algebraic forms for $$\beta_j = \|A v_j - \alpha_j v_j - \beta_{j-1} v_{j-1}\|$$0, the Hermite polynomial recurrence coefficients, and the Wigner semicircle law. Krylov state amplitudes, operator spread, and survival probabilities can all be expressed and analyzed in this unified framework.

7. Applications and Limitations

Lanczos coefficients underlie efficient algorithms for:

• Spectral density estimation and continued-fraction representations of Green’s functions
• Quantum many-body correlation functions, including autocorrelators at infinite temperature
• Extraction of hydrodynamical transport coefficients (e.g., diffusion constants via infinite product formulas over $$\beta_j = \|A v_j - \alpha_j v_j - \beta_{j-1} v_{j-1}\|$$1) (Pinna et al., 30 Apr 2025)
• Efficient time-evolution strategies for quantum systems, both in classical and quantum computing contexts (block Lanczos methods for excited states and non-Hermitian evolution) (Baker, 2021)
• Assessment of chaos, integrability, and hydrodynamic scales in both closed and open (dissipative) quantum systems (Füllgraf et al., 19 Jan 2026, Bhattacharya et al., 2023, Capizzi et al., 23 Jul 2025)

However, the mere growth rate of $$\beta_j = \|A v_j - \alpha_j v_j - \beta_{j-1} v_{j-1}\|$$2 cannot distinguish chaos from integrability in general: quadratic models (non-interacting, integrable) can be tuned to display constant, $$\beta_j = \|A v_j - \alpha_j v_j - \beta_{j-1} v_{j-1}\|$$3, or even linear growth in $$\beta_j = \|A v_j - \alpha_j v_j - \beta_{j-1} v_{j-1}\|$$4, and the Markovian limit in such cases always yields simple exponential decay of correlators regardless of $$\beta_j = \|A v_j - \alpha_j v_j - \beta_{j-1} v_{j-1}\|$$5 structure (Füllgraf et al., 19 Jan 2026). In open systems, dissipation-induced fluctuations in the Krylov chain wash out initial chaos/integrability signatures at late times (Bhattacharya et al., 2023).

Lanczos coefficients thus serve as a unifying language and computational toolset across numerical linear algebra, quantum dynamics, random matrix theory, orthogonal polynomials, and statistical physics, but with important model-specific caveats for physical interpretation.