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A Universal Operator Growth Hypothesis

Daniel E. Parker, Xiangyu Cao, Alexander Avdoshkin, Thomas Scaffidi, Ehud Altman

arXiv:1812.08657v5cond-mat.stat-mechcond-mat.str-elhep-thnlin.CDquant-ph

TL;DR

The paper addresses how to characterize and compute the growth of complex operators under Hamiltonian dynamics. It proposes asymptotically linear Lanczos growth governed by α, connects 2α to Krylov-complexity growth and Lyapunov bounds, and applies the recursion framework to diffusion calculations.

  • Problem

    Operator complexity is difficult to compute because operator expansions can acquire exponentially many nonzero coefficients, while diffusion coefficients depend on the full continued-fraction structure.

    Source: §III · p. 3, B. Hydrodynamical Phenomenology · p. 13

  • Method

    The paper hypothesizes bn = αn + γ + o(1) in nonintegrable systems, with a logarithmic correction in 1d, and combines this framework with recursion-based Green’s-function approximations.

    Source: C. Numerical Diffusion Coefficients · p. 14, A. Discussion · p. 17

  • Results

    The paper identifies 2α as the exponential growth rate of Krylov-complexity and as an upper bound on the Lyapunov exponent when it is well-defined.

    Source: A. Discussion · p. 17

  • Takeaways & Limitations

    The growth rate α serves as a central quantity linking operator complexity, Lyapunov growth, and the analysis of diffusion through continued fractions.

    Source: A. Discussion · p. 17

    The classical OTOC bound remains conjectural because classical OTOCs are not q-complexities, and the growth rate is not a diagnostic of classical chaos.

    Source: §1 · pp. 11–12

Abstract

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We present a hypothesis for the universal properties of operators evolving under Hamiltonian dynamics in many-body systems. The hypothesis states that successive Lanczos coefficients in the continued fraction expansion of the Green's functions grow linearly with rate $\alpha$ in generic systems, with an extra logarithmic correction in 1d. The rate $\alpha$ --- an experimental observable --- governs the exponential growth of operator complexity in a sense we make precise. This exponential growth even prevails beyond semiclassical or large-$N$ limits. Moreover, $\alpha$ upper bounds a large class of operator complexity measures, including the out-of-time-order correlator. As a result, we obtain a sharp bound on Lyapunov exponents $\lambda_L \leq 2 \alpha$, which complements and improves the known universal low-temperature bound $\lambda_L \leq 2 \pi T$. We illustrate our results in paradigmatic examples such as non-integrable spin chains, the Sachdev-Ye-Kitaev model, and classical models. Finally we use the hypothesis in conjunction with the recursion method to develop a technique for computing diffusion constants.

I. INTRODUCTION

The paper proposes universal operator growth in generic non-integrable Hamiltonian systems, using Lanczos coefficients and the recursion method to connect operator dynamics with complexity, chaos, and transport.

III. PRELIMINARIES: THE RECURSION METHOD

  • The recursion method maps operator dynamics to a particle on a semi-infinite chain whose hopping amplitudes are the Lanczos coefficients b_n.The Lanczos algorithm produces the coefficients and Krylov basis, while Heisenberg evolution becomes a discrete Schrödinger equation for the chain wavefunction.Source: §II · p. 2, §III · pp. 3–4

II. SYNOPSIS

  • Generic non-integrable systems are hypothesized to have asymptotically linear Lanczos coefficients, with a logarithmic correction in one dimension; α is the central growth rate.The hypothesis is supported by numerical evidence, SYK calculations, physical arguments, and the relation between α and high-frequency spectral decay.Source: §II · p. 2
  • Under the hypothesis, the chain wavefunction delocalizes exponentially at rate α, providing a precise description of the irreversible growth of operator complexity.The expected position of the wavefunction supplies an upper bound for q-complexities, including out-of-time-order correlators.Source: §II · p. 2, Figure 1 · p. 3
  • The hypothesis is tested in SYK and classical tops models, while the recursion method is extended into a semi-analytical technique for computing diffusion coefficients from Green’s-function poles.The finite-temperature generalization remains associated with open questions, although the authors provide evidence for a bound that can imply and improve the universal thermal chaos bound.Source: §II · p. 2

I. INTRODUCTION

  • Operator growth is difficult because a simple operator develops exponentially many basis components, motivating a statistical description in terms of increasingly large operator baths.The recursion method compresses this growth into how operator baths connect and how the operator moves irreversibly away from simple operators.Source: §I · p. 2, §III · p. 3

III. PRELIMINARIES: THE RECURSION METHOD

  • The chain representation preserves the autocorrelation function: its wavefunction component ϕ_0(t) equals C(t), and the Lanczos coefficients fully determine the autocorrelation.The coefficients can equivalently be viewed as hopping amplitudes connecting operators of different Krylov complexity.Source: §III · p. 4

IV. THE HYPOTHESIS

The hypothesis proposes asymptotically maximal, linear Lanczos growth in generic non-integrable systems, with logarithmic corrections in one dimension, and supports this through bounds, analytic examples, and numerical evidence. The growth rate α is observable through spectral functions and is realized in SYK and interacting spin models.

IV. THE HYPOTHESIS

  • Linear Lanczos growth is the proposed universal behavior for generic non-integrable systems in dimensions greater than one, with logarithmic corrections in one dimension.The growth rate α > 0 characterizes this asymptotic form and is treated as a universal quantity.Source: §IV · p. 4

A. Upper Bounds

  • Lanczos coefficients cannot grow faster than linearly, because the spectral-function tail bound enforces an exponent δ ≤ 1.Linear growth therefore represents the maximal allowed asymptotic behavior.Source: A. Upper Bounds · pp. 4–5
  • The growth rate α is equivalent to the exponential high-frequency decay rate of the spectral function and can therefore be measured through linear-response experiments.The equivalence is independent of dimension, temperature, and whether the system is quantum or classical; nuclear magnetic resonance measurements have reported exponential decays in CaF2.Source: A. Upper Bounds · p. 5, Figure 3 · p. 5
  • The rate α is bounded by the model’s local bandwidth and geometry, satisfying α ≤ π/2κ.This bound reflects the natural energy scale of the Lanczos coefficients without reducing α to bandwidth alone.Source: A. Upper Bounds · p. 5

B. Analytical Evidence

  • Analytic spin-model results show that the linear upper bound is tight, supporting the hypothesis that chaotic systems generically achieve it.A theorem relating operator moments to Lanczos asymptotics establishes linear growth for a realistic two-dimensional spin model [38].Source: B. Analytical Evidence · p. 6
  • SYK Lanczos coefficients closely follow the universal form, and in the large-q limit satisfy α = J, showing tight linear growth in a canonical quantum-chaotic model.The evidence comes from disorder-averaged moments and their large-N self-averaging properties; SYK’s special disordered, large-N, zero-dimensional setting is not required for linear growth.Source: B. Analytical Evidence · p. 6

C. The Special Case d = 1

In one dimension, non-integrable systems exhibit asymptotically sublinear Lanczos growth with a logarithmic correction, while the resulting Krylov wavefunction still spreads stretched-exponentially. More generally, linear Lanczos growth in higher dimensions produces exponential Krylov-complexity growth at rate 2α.

C. The Special Case d = 1

  • Non-integrable spin chains appear asymptotically linear after integrability breaking, although logarithmic corrections remain difficult to resolve numerically.The onset shifts to smaller Krylov index as interaction strength increases, while the asymptotic slope depends only weakly on that strength.Source: C. The Special Case d = 1 · pp. 6–7
  • Lanczos coefficients in one-dimensional non-integrable systems follow n/ln n growth rather than strictly linear growth, reflecting a geometric constraint.The hypothesis proposes the maximal geometrically allowed asymptotic form, with logarithmic correction parameters A and α bounded by the local bandwidth.Source: C. The Special Case d = 1 · pp. 6–7

V. EXPONENTIAL GROWTH OF COMPLEXITIES

  • The paper cautions that its complexity notion is not equivalent to circuit complexity, and no unique definition of operator complexity is established.This scope distinction is important when interpreting K-complexity as an operational measure of operator growth.Source: §V · p. 7

A. Exponential Growth in the Semi-infinite Chain

  • In dimensions above one, linear Lanczos growth yields exponential wavefunction spreading and maximal K-complexity growth with rate 2α.The continuum Dirac equation gives characteristic spreading x proportional to e^(±2αt), whereas sublinear growth produces power-law spreading.Source: A. Exponential Growth in the… · pp. 7–8
  • For the exact semi-infinite-chain solution, each fixed-site amplitude peaks near t ∼ ln n and then decays exponentially as probability escapes toward n → ∞.The delocalization length grows as ξ(t) ∼ e^(2αt), while the one-dimensional logarithmic correction changes spreading to stretched exponential.Source: A. Exponential Growth in the… · p. 8
  • The Krylov-complexity identifies the wavefunction’s expected position on the semi-infinite chain, whose larger-index basis operators form a progressively less-local hierarchy.This complexity is distinct from circuit complexity and reflects increasing nonlocality and computational difficulty in the Krylov basis.Source: A. Exponential Growth in the… · p. 8

B. A Bound on Complexity Growth

The paper defines q-complexities as positive quadratic observables whose Liouvillian dynamics changes complexity in a controlled way, encompassing operator size and OTOCs. K-complexity bounds their growth, implying λL ≤ 2α under the stated quantum conditions, with near saturation in q-SYK but weaker classical results.

B. A Bound on Complexity Growth

  • K-complexity provides an upper bound for every q-complexity, including operator size and certain OTOCs.The q-complexity framework uses a positive-semidefinite superoperator with bounded Liouvillian changes and low initial complexity.Source: B. A Bound on Complexity Growth · p. 9
  • OTOCs qualify as q-complexities when the Hamiltonian and initial operator are sufficiently local and the on-site Hilbert-space dimension is finite.Operator size is another example, and OTOCs can often be bounded by or reduced to operator size.Source: B. A Bound on Complexity Growth · p. 9
  • In one dimension, the logarithmic correction changes the resulting q-complexity bound from exponential to stretched-exponential growth.The growth remains faster than any power law but slower than any exponential.Source: B. A Bound on Complexity Growth · pp. 9–10

VI. GROWTH RATE AS A BOUND ON CHAOS

  • For infinite-temperature quantum systems satisfying the operator-growth hypothesis, the Lyapunov exponent obeys λL ≤ 2α.The result follows by combining the q-complexity bound with exponential K-complexity growth.Source: §VI · p. 10

A. SYK Model

  • In q-SYK, α is nearly equal to λL/2 across the model family and agrees exactly in the q → ∞ limit, showing that the bound is tight.The comparison is presented in Table I, with α computed numerically and λL taken from prior results.Source: A. SYK Model · p. 10, Table I · p. 10
  • The bound is especially close to saturation in all-to-all models such as SYK, whereas spatial locality generally makes OTOC growth slower than exponential.In finite-dimensional systems, operator complexity can grow exponentially even when operator size is constrained by Lieb-Robinson bounds.Source: A. SYK Model · p. 10

B. Classical Chaos

  • For classical systems, the paper proves only a weaker bound because the quantum argument does not carry over fully, while numerical evidence supports a stronger conjecture.The classical case is presented as a numerical case study rather than a complete proof.Source: B. Classical Chaos · pp. 10–11

1. A (Weaker) Bound on Classical Chaos

The paper proves a weaker classical chaos bound for polynomial few-spin models and tests the conjectured bound in the Feingold–Peres model, where it becomes tight near maximal chaos.

1. A (Weaker) Bound on Classical Chaos

  • λL ≤ 2α holds for finite-dimensional nonlinear classical systems with linear Lanczos growth, extending the recursion-method framework from quantum to classical dynamics.The authors establish this bound generally for classical few-spin models and expect a rigorous proof beyond the presented argument.Source: §1 · p. 11
  • Classical OTOCs are not generally q-complexities, so the stronger conjectured chaos bound remains unproven in the classical case.The obstruction arises because the relevant classical operator-space condition can fail when spin representations approach the classical limit.Source: §1 · p. 11
  • Linear Lanczos growth implies high-frequency spectral decay, providing quantitative support for using power-spectrum decay as an empirical probe of deterministic chaos.The paper contrasts this high-frequency relation with long-time relaxation, which can be governed by Ruelle resonances and is not automatically equivalent.Source: §1 · pp. 11–12
  • The growth rate α bounds but does not diagnose classical chaos, since integrable systems can have nonzero α despite lacking chaotic dynamics.Consequently, the bound is generally not tight for classical integrable systems.Source: §1 · p. 12

2. Numerical Case Study

  • In the Feingold–Peres coupled-tops model, α ≥ λL/2 across the explored parameter region and equality holds near c ≈ 0, where the model is maximally chaotic.Quantum Lanczos coefficients converge to the classical sequence as s increases, while growth appears discontinuous at the non-interacting points c = ±1.Source: §2 · p. 12, Figure 6 · p. 12

VII. APPLICATION TO HYDRODYNAMICS

The paper applies the recursion method and the operator-growth hypothesis to Green’s functions and hydrodynamic transport. By matching exact low-order Lanczos data to an analytically solvable asymptotic tail, it extracts diffusion coefficients in favorable cases.

A. Continued Fraction Expansion: Brief Review

  • The continued-fraction recursion converts the Green’s-function problem into a half-chain quantum-particle problem whose hopping coefficients are the Lanczos coefficients.Finite truncation is generally unreliable because the full analytic structure depends on all coefficients and pole locations shift with each added fraction.Source: A. Continued Fraction Expansion: Brief… · p. 13

B. Hydrodynamical Phenomenology

  • Hydrodynamic behavior is encoded in Green’s-function singularities: a pole quadratic in q gives diffusion, while additional cuts can produce power-law decay and frequency-dependent transport.Computing this singularity structure is difficult because it depends on the complete Lanczos sequence.Source: B. Hydrodynamical Phenomenology · p. 13
  • Linear asymptotics of Lanczos coefficients alone do not determine long-time autocorrelation decay, because identical leading growth can accompany exponential or power-law relaxation.Subleading corrections, including alternating tails, are needed to distinguish these behaviors.Source: B. Hydrodynamical Phenomenology · pp. 13–14

C. Numerical Diffusion Coefficients

  • The method stitches exactly computed low-n Lanczos coefficients to a universal large-n asymptotic solution, producing a meromorphic approximation to the Green’s function.The resulting pole structure yields the dispersion relation needed for transport calculations.Source: C. Numerical Diffusion Coefficients · p. 14
  • The hypothesis is sometimes sufficient to recover emergent hydrodynamics, but its leading-order asymptotics provide no general guarantee for transport or long-time correlations.The authors defer broader tests and a fuller analysis of the algorithm’s accuracy to subsequent work.Source: C. Numerical Diffusion Coefficients · p. 14
  • D = 3.3(5) is obtained for energy diffusion in the chaotic Ising model by fitting the leading pole to z = iDq^2/2 + O(q^4).The value agrees with other methods to within a few percent, despite using a linear fit that omits the one-dimensional logarithmic correction.Source: C. Numerical Diffusion Coefficients · pp. 14–15

VIII. FINITE TEMPERATURE

The paper extends the recursion method to finite temperature by changing the operator inner product, then formulates a finite-temperature chaos-bound conjecture. This conjecture would imply and potentially tighten the universal λL ≤ 2πT bound, but remains unproven below infinite temperature.

VIII. FINITE TEMPERATURE

  • Finite-temperature Lanczos analysis changes only the operator inner product, leaving the Krylov subspaces and recursion structure otherwise unchanged.Different physically motivated inner products produce different thermal correlation functions, including standard and Wightman correlators.Source: §VIII · p. 15, A. Choice of Inner Product · p. 15

A. Choice of Inner Product

  • The Wightman inner product adds temperature-dependent high-frequency spectral decay through thermal suppression, enabling comparison with finite-temperature chaos bounds.At infinite temperature, the standard and Wightman inner products coincide.Source: A. Choice of Inner Product · p. 16

B. Bound on Chaos

  • The proposed finite-temperature extension λL,T ≤ 2αT is conjectural, although exact q-SYK results suggest it is plausible and tight.The conjecture is motivated by matching the Wightman regularization to the finite-temperature OTOC scheme.Source: B. Bound on Chaos · p. 16
  • If valid, the finite-temperature conjecture would be tighter than λL,T ≤ 2πT and would reduce to the universal bound in the low-temperature limit.The low-temperature equivalence follows from αT/(πT) approaching one as β becomes large.Source: B. Bound on Chaos · p. 16

C. SYK Model

The SYK analysis supports the operator-growth hypothesis and shows that the Wightman growth rate saturates the conjectured chaos bound across temperatures. The discussion emphasizes α as a central link between operator complexity, spectral decay, and Lyapunov growth, while identifying unresolved finite-temperature and non-thermalizing cases.

C. SYK Model

  • The Wightman growth rate saturates λL(T) ≤ 2α(T) at all temperatures in large-q SYK, whereas λL(T) ≤ 2πT saturates only at zero temperature.The exact saturation is shown in Fig. 8.Source: Figure 8 · p. 17, C. SYK Model · p. 17
  • Exact SYK calculations support linear Lanczos growth at finite temperature, with the growth rate increasing as temperature decreases.This conclusion is supported by numerical high-temperature expansion and exact large-q calculations.Source: C. SYK Model · p. 17

A. Discussion

  • The coefficient α determines spectral decay, Krylov-complexity growth, q-complexity bounds, and the Lyapunov bound λL ≤ 2α.These relations connect two-point spectral information with operator complexity and OTOC growth.Source: A. Discussion · p. 17
  • K-complexity can grow exponentially in quantum systems beyond semiclassical or large-N limits, unlike chaos measures whose growth may saturate in deeply quantum regimes.The paper contrasts temporal structure growth measured by K-complexity with progressively finer phase-space structure measured by chaos.Source: A. Discussion · p. 17

B. Outlook

  • The hypothesis remains unresolved for integrable systems, scars, many-body localization, standard finite-temperature inner products, and rigorous mathematical foundations, especially in 1d.The authors describe these as open directions rather than established failures of the hypothesis.Source: B. Outlook · p. 18

Appendix A: Brief Review of the Recursion Method

The recursion method relates operator autocorrelations, moments, Green functions, spectral functions, and Lanczos coefficients through complementary representations. Its nonlinear moment-to-Lanczos mapping makes asymptotic Lanczos growth especially informative about correlation analyticity and spectral decay.

Appendix A: Brief Review of the Recursion Method

  • The recursion method provides five equivalent representations of operator dynamics, with Lanczos coefficients related nonlinearly to moments and the other quantities.Moments, autocorrelation, Green functions, and spectral functions are linearly related, while the Lanczos transformation is non-linear.Source: Appendix A · p. 22

1. From Moments to Lanczos Coefficients

  • Moments determine Lanczos coefficients through Hankel-matrix determinants and an iterative recursion that computes b1 through bn.The procedure is efficient but can suffer large numerical instabilities from repeated floating-point divisions.Source: §1 · p. 22

2. From Lanczos Coefficients to Moments

  • Lanczos coefficients determine moments through weighted sums over Dyck paths, yielding lower and upper bounds that reformulate linear coefficient growth as moment growth.The number of Dyck paths is governed by Catalan numbers.Source: §2 · p. 22
  • Linear Lanczos growth places autocorrelation singularities at ±iπ/(2α), giving convergence radius π/(2α) and exponential spectral decay.This establishes the asymptotic connection between Lanczos growth, analytic structure, and the high-frequency spectral tail.Source: §2 · p. 22

Appendix B: Moments and Lanczos Coefficients in the SYK Model

  • In SYK, disorder-averaged moments can be computed diagrammatically and then converted into Lanczos coefficients using the recursion method.The general SYK calculation uses diagrammatic moment expansions and requires arbitrary-precision arithmetic at large moment order.Source: Appendix B · p. 23

1. General Method

The paper develops a diagrammatic and recursive method for computing SYK moments and Lanczos coefficients, then solves the large-q limit analytically. The resulting finite-temperature growth rate supports the conjectured chaos bound and its saturation in SYK.

1. General Method

  • The SYK moment calculation uses diagrammatic Schwinger-Dyson relations and computes moments in polynomial time and space in the expansion order.The resulting truncated Green function supplies moments for recursive Lanczos-coefficient extraction.Source: §1 · pp. 23–24
  • For mixed q = 2 and q = 4 SYK, the linear growth rate depends only weakly on the interaction strength J near the free limit, with logarithmic dependence fitting vanishing-J data.The q = 2 component is non-interacting, while q = 4 supplies the thermalizing interaction.Source: §1 · p. 24, Figure 9 · p. 24

2. Large-q limit

  • The large-q solution gives Lanczos coefficients with asymptotically linear growth, supporting the universal operator-growth hypothesis.The corresponding semi-infinite-chain probability distribution matches the operator-size distribution in the large-q model.Source: §2 · pp. 24–25
  • At finite temperature with the Wightman inner product, the large-q SYK growth rate saturates the conjectured Lyapunov bound at every temperature, unlike the universal 2πT bound.The comparison is shown in Fig. 8 and follows from the finite-temperature large-q solution.Source: §2 · p. 25
  • The standard inner product produces a different temperature dependence: its growth rate approaches Jπ/2 at low temperature and J at high temperature.The authors distinguish this behavior from the Wightman-inner-product result.Source: §2 · p. 25

Appendix C: Numerical Details for 1d Spin Chains

The spin-chain implementation represents translation-invariant Pauli-string operators compactly and applies the Liouvillian without finite-size diagonalization. It enables Lanczos calculations but leaves physical-space operator distributions strongly model-dependent.

Appendix C: Numerical Details for 1d Spin Chains

  • The thermodynamic-limit spin-chain calculation stores translation-invariant operators as hash maps of Pauli strings and applies the Liouvillian through commutator rules.Only three operators need to be stored during the Lanczos algorithm, but memory is limited by the exponential number of Pauli strings.Source: Appendix C · pp. 25–26
  • Translation invariance preserves wavevector under Liouvillian evolution, allowing operators to be represented using strings starting at a single reference site.Phase factors track shifts caused by commutators while maintaining the chosen wavevector.Source: Appendix C · p. 25
  • Pauli strings are encoded by binary vectors and digits over Z2, making commutator operations suitable for matrix-free computation.This representation uses the identity iY = ZX to reduce the computational cost of commutators.Source: Appendix C · pp. 25–26
  • The physical-space string-length distribution broadens with Krylov index, but its mean and variance are model-dependent and do not directly translate semi-infinite-chain growth into spatial spreading.This model dependence is identified as a limitation of interpreting Krylov-space dynamics physically.Source: Appendix C · p. 26

Appendix D: A Family of Exact Solution with Linear Growth

This appendix derives an exact solution for a one-dimensional quantum-mechanics recursion problem with Lanczos coefficients exhibiting linear-growth structure, using orthogonal-polynomial methods and su(1,1) symmetry.

Operator-size structure

  • The Pauli-string size distribution in the Krylov vectors decreases rapidly after its peak but remains supported on [0, ⌊n/2⌋ + 2].This distribution is shown for Hamiltonian H1 with the parameters and initial operator used in Fig. 4.Source: Figure 10 · p. 26

Exact solution

  • The recursion problem maps to a one-dimensional quantum-mechanics system with high su(1,1) symmetry, enabling an exact solution through Meixner orthogonal polynomials.The construction uses a tridiagonal Liouvillian and the associated three-term polynomial recursion.Source: Appendix D · p. 26

Orthogonal-polynomial construction

  • The polynomials Q_n are monic orthogonal polynomials whose recursion determines the Krylov problem and whose coefficients can be expressed through inverse-polynomial generating functions.The appendix relates Q_n to Meixner polynomials of the second kind and uses their generating function to calculate transformation coefficients.Source: Appendix D · p. 27

Exact family

  • For b_n = α√(n(n − 1 + η)), a time rescaling yields the general exact solution corresponding to Eq. (25).The η = 1 case is known, while the general family had not previously been applied to the recursion method according to the appendix.Source: Appendix D · p. 27

Appendix E: Derivation of the q-Complexity Bound

This appendix proves a geometric upper bound on q-complexity by comparing the Krylov basis with a basis whose growth is controlled by a dilation property.

Bound construction

  • The dilation property implies that q-complexity is bounded by Krylov complexity, yielding (Q)t ≤ C(n)t with C = 2M.The eigenbasis of Q is dilated by at most a factor C relative to the Krylov basis.Source: Appendix E · p. 27

Support growth

  • A finite-support condition in the Q basis remains controlled under Liouvillian evolution, so Krylov vectors generated after n steps occupy Q-basis components only up to order Cn.The argument uses induction from the bounded action of L on Q-basis support.Source: Appendix E · p. 27

Spectral comparison

  • Projectors onto large spectral values in the Krylov and Q bases convert the support comparison into an inequality between their complexity distributions.The proof combines projector relations with an integration-by-parts identity.Source: Appendix E · pp. 27–28

Consequences

  • The resulting bound also controls the growth rate of higher moments of the q-complexity super-operator.The appendix identifies an application in the paper’s discussion of q-complexity.Source: Appendix E · p. 28

Appendix F: Geometric Origin of the Upper Bounds

This appendix derives geometric upper bounds on Lanczos growth by counting nonzero Liouvillian paths, obtaining linear growth in general dimensions and a sharper sublinear bound in one dimension.

Combinatorial reduction

  • Bounding the number of admissible paths reduces the moment estimate to a combinatorial problem organized by the operator’s endpoint growth.The sets P_n and P_n,ℓ classify paths according to the resulting support size.Source: Appendix F · pp. 28–29

General-dimensional bound

  • Lanczos coefficients can grow at most linearly in any dimension because the number and norm of nonzero Liouvillian paths reproduce the moments associated with linear growth.The argument assumes local Hamiltonian terms with uniformly bounded norm and uses Stirling asymptotics.Source: Appendix F · p. 28

One-dimensional geometry

  • In one dimension, operator support expands only at its left and right edges, sharply restricting the sequences of Liouvillian terms that contribute.Nonzero paths must satisfy the interval constraints L_k ≤ x_k ≤ R_k, with the support size determined by the endpoint distance.Source: Appendix F · pp. 28–29

Asymptotic counting

  • The dominant support size occurs at ℓ = n/W(n), with W(n) = ln n − ln ln n + o(1) asymptotically.This saddle-point estimate controls the resulting path-counting bound.Source: Appendix F · p. 29

Sublinear growth

  • The one-dimensional moment bound grows more slowly than the moments generated by any linear Lanczos growth rate, forcing the coefficients to be sublinear.The moment comparison alone does not yield a useful bound on each individual coefficient.Source: Appendix F · p. 29

Maximal 1d growth

  • The maximal one-dimensional asymptotic growth is b_n = a e^{W(n)}, which saturates the moment upper bound when A ≤ 4E/e.This identifies the correct asymptotic behavior allowed by the geometric bound.Source: Appendix F · p. 29
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