Pith. sign in

REVIEW 2 major objections 5 minor 66 references

Krylov complexity is the leading derivative of a Loschmidt amplitude under an angular Hamiltonian deformation, and is upper-bounded by the volume of the resulting two-dimensional quantum geometry.

Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

T0 review · grok-4.5

2026-07-14 08:16 UTC pith:HFREJQEM

load-bearing objection Clean algebraic rewrite of Krylov complexity as a Loschmidt derivative plus a volume bound; soft spots are the non-rigorous approximate inequalities, not the core claims. the 2 major comments →

arxiv 2607.10921 v1 pith:HFREJQEM submitted 2026-07-12 hep-th cond-mat.stat-mechquant-ph

Krylov Complexity from Loschmidt Amplitude

Debarghya Chakraborty This is my paper
classification hep-th cond-mat.stat-mechquant-ph
keywords Krylov complexityLoschmidt amplitudeLanczos coefficientsspectral propagatorFubini-Study geometryoperator growthquantum chaos
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper shows that Krylov complexity—the mean position of a time-evolving state or operator in the special orthonormal basis generated by the Lanczos algorithm—can be rewritten as the leading φ-derivative of a Loschmidt amplitude. That amplitude is the overlap between a state evolved with the original Hamiltonian and the same state evolved with a Hamiltonian rotated by an angle φ in the Krylov plane. The full distribution of complexity moments is encoded in a spectral propagator for this rotation. The same algebraic structure yields a second Loschmidt amplitude, now in Euclidean time τ, whose τ-derivative is exactly the time derivative of Krylov complexity. How strongly the τ-perturbation grows with Lanczos index supplies a classification of complexity dynamics; when the Lanczos coefficients grow slowly enough the perturbation may be truncated to finite rank. The pair of trajectories (original and φ-deformed) span a two-dimensional submanifold of Hilbert space. Its Fubini-Study volume up to time t always upper-bounds Krylov complexity, with equality when the relevant operators close a finite-dimensional algebra. The construction therefore ties three classical diagnostics—complexity growth, Loschmidt decay, and geometric volume—into a single analytic framework, and produces model-independent bounds relating the survival amplitude to Krylov variance and inverse-participation ratio.

Core claim

Krylov complexity equals the leading φ-derivative of the Loschmidt amplitude G(t,φ)=⟨0|e^{iMt}e^{-iM(φ)t}|0⟩, where M(φ) is the Hamiltonian rotated by the Krylov number operator; the same complexity is always upper-bounded by the volume of the two-dimensional (t,φ) geometry induced by the Fubini-Study metric.

What carries the argument

The φ-rotated Hamiltonian M(φ)=M cos φ + B sin φ + L_0(1-cos φ), together with the associated spectral Poisson kernel K(E1,E2,φ) that propagates the Krylov number operator in the energy basis; its φ-derivatives generate every moment of complexity, while the volume of the induced Fubini-Study geometry bounds the first moment.

Load-bearing premise

The paper relies on a physically motivated but non-rigorous exponentiated bound that the Loschmidt amplitude stays above a Gaussian of width set by the Krylov variance; this is verified only on selected models and can fail near zeros of the autocorrelation.

What would settle it

Compute both Krylov complexity and the Fubini-Study volume of the (t,φ) geometry for a model whose complexity algebra does not close (for example constant or slowly growing Lanczos coefficients); if complexity systematically exceeds the claimed volume bound, the central geometric claim is false.

Watch this falsifier — get emailed when new claim-graph text bears on it.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

2 major / 5 minor

Summary. The paper expresses Krylov complexity as the leading φ-derivative of a Loschmidt amplitude G(t,φ)=⟨0|e^{iMt}e^{-iM(φ)t}|0⟩ generated by the Krylov number operator (Eqs. 24a–26), and shows that the full complexity distribution is encoded in a spectral Poisson kernel K(E1,E2,φ). A second Loschmidt amplitude with a Euclidean τ-deformation yields the time derivative of complexity (Eq. 37); the growth of the associated operator F is used to classify dynamics and to justify finite-rank truncations when Lanczos coefficients grow slowly. The two-parameter family |Ψ(t,φ)⟩ induces a Fubini-Study geometry whose volume upper-bounds complexity via the Robertson-Schrödinger relation (Eqs. 52–54). Exact solutions for complexity algebras and constant-b sequences are recovered, and numerical checks on DSSYK and a finite Ising chain support the truncation error and approximate IPR relations.

Significance. If the algebraic and geometric identities hold, the work supplies a parameter-free dictionary between Krylov complexity, Loschmidt echoes, and a concrete two-dimensional quantum geometry. The volume bound and the spectral-propagator representation are model-independent and recover known closed-form results (Appendix C, constant-b RMT). The finite-rank truncation of F (Eq. 40) and the Frobenius-norm bound (Eq. 47) give practical, falsifiable tools for systems whose Lanczos coefficients grow slowly. These contributions are of clear interest for quantum chaos, operator growth, and holographic complexity.

major comments (2)
  1. Section IV, Eqs. (60)–(64): the exponentiated bound |G(t,φ)|≳exp(-1/2(Δn(t))²φ²) and the consequent IPR estimates are presented as physically motivated but non-rigorous. They fail near zeros of C(2t) and are only verified numerically on selected models. Because they are used to constrain the complexity distribution, the manuscript should either (i) state clearly that they are heuristic estimates outside the rigorous core, or (ii) supply a controlled error estimate (e.g., after the proposed time-smearing) that quantifies the domain of validity.
  2. Section III.B and Appendix A.2: the late-time asymptotic for all moments under constant Lanczos coefficients (Eqs. 44, A19) relies on a factorized low-frequency ansatz for ñ(E,ω). While the leading power is checked numerically for k=1–3,6, a sharper statement of the error incurred by discarding the mean-energy dependence of A2(E) would strengthen the claim that the entire distribution is obtained from the spectral function alone.
minor comments (5)
  1. Figure 1 caption and surrounding text: the visual guide is helpful, but the right-hand panel (τ-rotation of the Krylov basis) is only sketched; a short explicit formula linking B(τ) to the Toda flow would improve readability.
  2. Eq. (41) and Fig. 2: the truncation error bound is stated for a_n=0; a one-sentence remark on the nonzero-a_n case (already mentioned later) would avoid ambiguity.
  3. Appendix B, Eq. (B3): the approximate kernel is central to the late-time claims; a brief numerical comparison of the integrated moments against exact φ_n(t) for a non-constant bounded Φ(E) would make the accuracy claim more concrete.
  4. Notation: the same letter K is used for the Poisson kernel and, occasionally, for Krylov complexity; a consistent distinction (e.g., K vs. ⟨n̂⟩) would reduce minor confusion.
  5. References: the recent reviews [2,3] and the adiabatic-gauge-potential literature [25,43] are appropriately cited; a pointer to the original Loschmidt-echo Lyapunov papers [21–23] already appears, but a short sentence linking the present nonlocal φ-deformation to those semiclassical results would help non-specialist readers.

Circularity Check

0 steps flagged

No significant circularity: central identities follow by construction from the Krylov algebra and Fubini-Study metric without fitted parameters or load-bearing self-citation.

full rationale

The paper's strongest claims are algebraic identities, not empirical predictions. Krylov complexity is defined as the mean of the number operator n̂ on the Krylov chain (Eq. 17). The commutation relations [n̂, L±] = ±L± (Eq. 20) immediately imply the Heisenberg evolution M(ϕ) = e^{i n̂ ϕ} M e^{-i n̂ ϕ} (Eq. 23). Differentiating the resulting Loschmidt amplitude G(t, ϕ) = ⟨0|e^{i M t} e^{-i M(ϕ) t}|0⟩ with respect to ϕ at ϕ = 0 recovers ⟨n̂(t)⟩ by definition (Eqs. 24a–26). The volume bound ⟨n̂(t)⟩ ≤ (1/π) Vol_t (Eq. 54) is likewise an immediate consequence of the Robertson–Schrödinger uncertainty relation applied to the Fubini–Study metric components (Eqs. 51–53); equality holds precisely when the complexity algebra closes, which is verified by direct computation in Appendix C rather than assumed. The spectral-propagator representation (Eq. 27) and the τ-deformation that yields ∂t ⟨n̂⟩ (Eq. 37) are likewise rearrangements of the same operators. Model checks (DSSYK Lanczos sequence (D4), SL(2,R) coherent states (C20), constant-b RMT) use previously published sequences without re-fitting any free parameter to the complexity data being “predicted.” The only non-rigorous statements—the exponentiated bound |G| ≳ exp(−½(Δn)^{2} ϕ^{2}) (Eq. 60) and the consequent IPR estimates (Eqs. 63–64)—are explicitly labeled as physically motivated hypotheses and are not used to derive the two central claims. No uniqueness theorem is imported from the author’s prior work, no ansatz is smuggled via self-citation, and no fitted input is renamed a prediction. The derivation chain is therefore self-contained and non-circular.

Axiom & Free-Parameter Ledger

0 free parameters · 4 axioms · 2 invented entities

The central claims rest on the standard algebraic structure of the Krylov basis, the Fubini-Study metric, and classical asymptotics of orthogonal polynomials. No free parameters are fitted to obtain the main identities; model-specific parameters (q, h, α, γ) appear only in illustrative examples. The spectral propagator and the (t,ϕ) geometry are new constructs introduced for the analysis.

axioms (4)
  • standard math The Krylov number operator satisfies [n̂,L±]=±L± and B=i[n̂,M] is the conjugate momentum (Eqs. 20–21).
    Direct consequence of the definition of the Lanczos basis; used throughout Sections III–IV.
  • standard math The Fubini-Study metric on the projective Hilbert space induces a Riemannian metric on any smooth submanifold of states (Eqs. 48–49).
    Standard quantum information geometry; applied to the (t,ϕ) family in Section IV.
  • domain assumption For spectral densities supported on a finite interval the orthonormal polynomials admit the Szegő asymptotics (B1)–(B2).
    Classical result of orthogonal-polynomial theory; used to construct the approximate kernel (B3) and late-time moments.
  • standard math When the Frobenius norm of F is finite, Cauchy–Schwarz yields the complexity bound (47).
    Elementary inequality applied to the spectral representation of F; assumes only existence of ||F||.
invented entities (2)
  • Spectral propagator (Poisson kernel) K(E1,E2,ϕ) no independent evidence
    purpose: Encodes the full distribution of Krylov complexity moments via its ϕ-derivatives (Eqs. A1, 27).
    Defined in Appendix A; reduces to known Mehler/Hardy–Hille kernels for special cases but is new as a general object for complexity.
  • Complexity geometry (t,ϕ) with metric components (51) no independent evidence
    purpose: Provides a Riemannian volume that upper-bounds Krylov complexity and relates geodesic deviation to Krylov variance.
    Generalizes the coherent-state geometries of Caputa et al. to arbitrary systems; introduced in Section IV.

pith-pipeline@v1.1.0-grok45 · 38287 in / 2767 out tokens · 30648 ms · 2026-07-14T08:16:34.337338+00:00 · methodology

0 comments
read the original abstract

Krylov complexity is a powerful diagnostic of quantum dynamics, with clear connections to other measures of quantum chaos and operator growth. One such measure is the Loschmidt amplitude, defined as the overlap of initially identical states evolved under two slightly different Hamiltonians. Its decay in certain systems is controlled by the classical Lyapunov exponent. Using the algebraic properties of the Krylov complexity operator, we express Krylov complexity as the derivative of a Loschmidt amplitude whose perturbation is parameterized by an angular variable $\phi$. This formulation allows us to define a spectral propagator that encodes the entire complexity distribution, which we characterize for specific types of systems. We study the two-dimensional quantum geometry spanned by time and $\phi$ where the original and deformed trajectories reside, demonstrating that Krylov complexity is upper-bounded by its volume. We also express the time derivative of Krylov complexity in terms of a distinct Loschmidt amplitude. Depending on the growth of the Lanczos coefficients, the perturbation term in this amplitude can be truncated. We propose that the strength of this perturbation provides a classification scheme for Krylov complexity dynamics and relate it to the $\phi$-derivative of the spectral propagator. Using this analytical framework, we derive general relations between the time-dependence of the survival amplitude and Krylov space measures.

Figures

Figures reproduced from arXiv: 2607.10921 by Debarghya Chakraborty.

Figure 1
Figure 1. Figure 1: FIG. 1 [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2 [PITH_FULL_IMAGE:figures/full_fig_p010_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3 [PITH_FULL_IMAGE:figures/full_fig_p012_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4 [PITH_FULL_IMAGE:figures/full_fig_p016_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5 [PITH_FULL_IMAGE:figures/full_fig_p017_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6: We plot IPR [PITH_FULL_IMAGE:figures/full_fig_p026_6.png] view at source ↗

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

66 extracted references · 5 canonical work pages

  1. [1]

    Just the 11 existence of|| F ||constrains the asymptotic growth of complexity through (43), as shown below in (47) and through the analysis of the kernel in Appendix A 3

    Since the Frobenius norm is unitarily invariant:|| ˜F ||=|| F ||, whenever well-defined, it provides a sum-rule for the matrix elements ˜F jk. Just the 11 existence of|| F ||constrains the asymptotic growth of complexity through (43), as shown below in (47) and through the analysis of the kernel in Appendix A 3. Whenever Φ(E) is bounded i.e Φ(E) = 0 for|E...

  2. [2]

    As long as the measure is even, meaning thata n continue to vanish,g tϕ = 0 and the metric is diagonal

    obtained the two-dimensional maximally symmetric spaces of constant curvature in the case of operator complex- ity, where all thea n vanish. As long as the measure is even, meaning thata n continue to vanish,g tϕ = 0 and the metric is diagonal. From (28),(21), combined with theϕisometry, it simply follows thatF ϕt is the time derivative of Krylov complexi...

  3. [3]

    Krylov- winding

    TheSL(2, R) toy model (orange) is specified by the Lanczos coefficients (C20), where both theb n anda n coefficients have been perturbed by independent positive random variables with mean 0.25. For theSL(2, R) computationα= 1, γ= 0.1, h= 1 2.(c) Illustration of (61)in spin chain:Ising chain with same parameters and initial operator as above used. This amp...

    2024

  4. [4]

    D. E. Parker, X. Cao, A. Avdoshkin, T. Scaffidi, and E. Altman, A universal operator growth hypothesis, Physical Review X9, 10.1103/physrevx.9.041017 (2019)

    doi:10.1103/physrevx.9.041017 2019

  5. [5]

    Rabinovici, A

    E. Rabinovici, A. S´ anchez-Garrido, R. Shir, and J. Sonner, Krylov complexity (2025), arXiv:2507.06286 [hep-th]

    Pith/arXiv arXiv 2025

  6. [6]

    Nandy, A

    P. Nandy, A. S. Matsoukas-Roubeas, P. Mart´ ınez-Azcona, A. Dymarsky, and A. del Campo, Quantum dynamics in krylov space: Methods and applications, Physics Reports1125-1128, 1–82 (2025)

    2025

  7. [8]

    S.-K. Jian, B. Swingle, and Z.-Y. Xian, Complexity growth of operators in the syk model and in jt gravity, Journal of High Energy Physics2021, 10.1007/jhep03(2021)014 (2021)

    doi:10.1007/jhep03(2021)014 2021

  8. [9]

    Caputa and S

    P. Caputa and S. Datta, Operator growth in 2d cft, Journal of High Energy Physics2021, 10.1007/jhep12(2021)188 (2021)

    doi:10.1007/jhep12(2021)188 2021

  9. [10]

    A. Kar, L. Lamprou, M. Rozali, and J. Sully, Random matrix theory for complexity growth and black hole interiors, Journal of High Energy Physics2022, 10.1007/jhep01(2022)016 (2022)

    doi:10.1007/jhep01(2022)016 2022

  10. [11]

    J. Kim, J. Murugan, J. Olle, and D. Rosa, Operator delocalization in quantum networks, Physical Review A105, 10.1103/physreva.105.l010201 (2022)

    doi:10.1103/physreva.105.l010201 2022

  11. [12]

    Alishahiha and S

    M. Alishahiha and S. Banerjee, A universal approach to krylov state and operator complexities, SciPost Physics15, 10.21468/scipostphys.15.3.080 (2023)

    doi:10.21468/scipostphys.15.3.080 2023

  12. [13]

    Caputa, N

    P. Caputa, N. Gupta, S. S. Haque, S. Liu, J. Murugan, and H. J. R. Van Zyl, Spread complexity and topological transitions in the kitaev chain, Journal of High Energy Physics2023, 10.1007/jhep01(2023)120 (2023). 19

    doi:10.1007/jhep01(2023)120 2023

  13. [14]

    Erdmenger, S.-K

    J. Erdmenger, S.-K. Jian, and Z.-Y. Xian, Universal chaotic dynamics from krylov space, Journal of High Energy Physics 2023, 10.1007/jhep08(2023)176 (2023)

    doi:10.1007/jhep08(2023)176 2023

  14. [15]

    M. P. Heller, J. Papalini, and T. Schuhmann, Krylov spread complexity as holographic complexity beyond jackiw-teitelboim gravity, Physical Review Letters135, 10.1103/spcr-jgm6 (2025)

    work page doi:10.1103/spcr-jgm6 2025

  15. [16]

    Nandy, T

    P. Nandy, T. Pathak, Z.-Y. Xian, and J. Erdmenger, Krylov space approach to singular value decomposition in non- hermitian systems, Physical Review B111, 10.1103/physrevb.111.064203 (2025)

    doi:10.1103/physrevb.111.064203 2025

  16. [17]

    S. E. Aguilar-Gutierrez, H. A. Camargo, V. Jahnke, K.-Y. Kim, and M. Nishida, Krylov operator complexity in holographic cfts: Smeared boundary reconstruction and the dual proper radial momentum, Physical Review D112, 10.1103/6bgg-vglp (2025)

    work page doi:10.1103/6bgg-vglp 2025

  17. [18]

    Ermakov and O

    I. Ermakov and O. Lychkovskiy, Symbolic recursion method for strongly correlated fermions in two and three dimensions (2026), arXiv:2512.23678 [cond-mat.str-el]

    arXiv 2026

  18. [19]

    Maldacena, S

    J. Maldacena, S. H. Shenker, and D. Stanford, A bound on chaos, Journal of High Energy Physics2016, 10.1007/jhep08(2016)106 (2016)

    doi:10.1007/jhep08(2016)106 2016

  19. [20]

    Balasubramanian, P

    V. Balasubramanian, P. Caputa, J. M. Magan, and Q. Wu, Quantum chaos and the complexity of spread of states, Physical Review D106, 10.1103/physrevd.106.046007 (2022)

    doi:10.1103/physrevd.106.046007 2022

  20. [21]

    Peres, Stability of quantum motion in chaotic and regular systems, Phys

    A. Peres, Stability of quantum motion in chaotic and regular systems, Phys. Rev. A30, 1610 (1984)

    1984

  21. [22]

    Wisniacki, Loschmidt echo, Scholarpedia7, 11687 (2012)

    A. Wisniacki, Loschmidt echo, Scholarpedia7, 11687 (2012)

    2012

  22. [23]

    Gorin, T

    T. Gorin, T. Prosen, T. H. Seligman, and M. ˇZnidariˇ c, Dynamics of loschmidt echoes and fidelity decay, Physics Reports 435, 33–156 (2006)

    2006

  23. [24]

    R. A. Jalabert and H. M. Pastawski, Environment-independent decoherence rate in classically chaotic systems, Physical Review Letters86, 2490–2493 (2001)

    2001

  24. [25]

    Jacquod, P

    P. Jacquod, P. Silvestrov, and C. Beenakker, Golden rule decay versus lyapunov decay of the quantum loschmidt echo, Physical Review E64, 10.1103/physreve.64.055203 (2001)

    doi:10.1103/physreve.64.055203 2001

  25. [26]

    Jacquod, I

    P. Jacquod, I. Adagideli, and C. W. J. Beenakker, Decay of the loschmidt echo for quantum states with sub-planck-scale structures, Physical Review Letters89, 10.1103/physrevlett.89.154103 (2002)

    work page doi:10.1103/physrevlett.89.154103 2002

  26. [28]

    Pandey, P

    M. Pandey, P. W. Claeys, D. K. Campbell, A. Polkovnikov, and D. Sels, Adiabatic eigenstate deformations as a sensitive probe for quantum chaos, Physical Review X10, 10.1103/physrevx.10.041017 (2020)

    doi:10.1103/physrevx.10.041017 2020

  27. [29]

    Szeg¨ o,Orthogonal Polynomials, Vol

    G. Szeg¨ o,Orthogonal Polynomials, Vol. 23 (American Mathematical Society, Providence, RI, 1959)

    1959

  28. [30]

    Caputa, J

    P. Caputa, J. M. Magan, and D. Patramanis, Geometry of Krylov complexity, Phys. Rev. Res.4, 013041 (2022), arXiv:2109.03824 [hep-th]

    Pith/arXiv arXiv 2022

  29. [31]

    M¨ uck and Y

    W. M¨ uck and Y. Yang, Krylov complexity and orthogonal polynomials, Nuclear Physics B984, 115948 (2022)

    2022

  30. [32]

    Murugan and H

    J. Murugan and H. J. R. van Zyl, A Schwinger-Keldysh Formulation of Semiclassical Operator Dynamics (2026), arXiv:2602.02106 [quant-ph]

    arXiv 2026

  31. [33]

    Bhattacharjee, P

    B. Bhattacharjee, P. Nandy, and T. Pathak, Krylov complexity in large q and double-scaled syk model, Journal of High Energy Physics2023, 10.1007/jhep08(2023)099 (2023)

    doi:10.1007/jhep08(2023)099 2023

  32. [34]

    Chakraborty and D

    D. Chakraborty and D. Rosa, Revival Dynamics from Equilibrium States: Scars from Chords in SYK (2025), arXiv:2512.16836 [cond-mat.str-el]

    arXiv 2025

  33. [35]

    H. W. Lin, The bulk hilbert space of double scaled syk, Journal of High Energy Physics2022, 10.1007/jhep11(2022)060 (2022)

    doi:10.1007/jhep11(2022)060 2022

  34. [36]

    Rabinovici, A

    E. Rabinovici, A. S´ anchez-Garrido, R. Shir, and J. Sonner, A bulk manifestation of Krylov complexity, JHEP08, 213, arXiv:2305.04355 [hep-th]

    Pith/arXiv arXiv

  35. [37]

    Xu, On chord dynamics and complexity growth in double-scaled SYK, JHEP06, 259, arXiv:2411.04251 [hep-th]

    J. Xu, On chord dynamics and complexity growth in double-scaled SYK, JHEP06, 259, arXiv:2411.04251 [hep-th]

    Pith/arXiv arXiv

  36. [38]

    S. E. Aguilar-Gutierrez, Building the holographic dictionary of the DSSYK from chords, complexity & wormholes with matter, JHEP10, 221, arXiv:2505.22716 [hep-th]

    arXiv

  37. [39]

    Bueno, J

    P. Bueno, J. M. Magan, and C. S. Shahbazi, Complexity measures in QFT and constrained geometric actions, JHEP09, 200, arXiv:1908.03577 [hep-th]

    Pith/arXiv arXiv 1908

  38. [40]

    H. W. Lin and L. Susskind, Complexity geometry and schwarzian dynamics, Journal of High Energy Physics2020, 10.1007/jhep01(2020)087 (2020)

    work page doi:10.1007/jhep01(2020)087 2020

  39. [41]

    Balasubramanian, P

    V. Balasubramanian, P. Caputa, and J. Sim´ on, Variations on a theme of krylov (2025), arXiv:2511.03775 [hep-th]

    arXiv 2025

  40. [42]

    Angelinos, D

    N. Angelinos, D. Chakraborty, and A. Dymarsky, Temperature dependence in Krylov space, J. Phys. A59, 235204 (2026), arXiv:2508.19233 [hep-th]

    Pith/arXiv arXiv 2026

  41. [43]

    Rabinovici, A

    E. Rabinovici, A. S´ anchez-Garrido, R. Shir, and J. Sonner, Operator complexity: a journey to the edge of krylov space, Journal of High Energy Physics2021, 10.1007/jhep06(2021)062 (2021)

    doi:10.1007/jhep06(2021)062 2021

  42. [44]

    Balasubramanian, J

    V. Balasubramanian, J. M. Magan, and Q. Wu, Quantum chaos, integrability, and late times in the Krylov basis, Phys. Rev. E111, 014218 (2025), arXiv:2312.03848 [hep-th]

    Pith/arXiv arXiv 2025

  43. [45]

    Chakraborty,Topics in Quantum Information, Complexity and Chaos, Ph.D

    D. Chakraborty,Topics in Quantum Information, Complexity and Chaos, Ph.D. thesis, University of Kentucky (2024)

    2024

  44. [46]

    Kolodrubetz, D

    M. Kolodrubetz, D. Sels, P. Mehta, and A. Polkovnikov, Geometry and non-adiabatic response in quantum and classical systems, Physics Reports697, 1–87 (2017)

    2017

  45. [47]

    Caputa, B

    P. Caputa, B. Chen, R. W. McDonald, J. Sim´ on, and B. Strittmatter, Spread complexity rate as proper momentum, Phys. Rev. D113, L041901 (2026)

    2026

  46. [48]

    M. V. Berry, Quantal phase factors accompanying adiabatic changes, Proc. Roy. Soc. Lond. A392, 45 (1984). 20

    1984

  47. [49]

    D. J. Thouless, M. Kohmoto, M. P. Nightingale, and M. den Nijs, Quantized hall conductance in a two-dimensional periodic potential, Phys. Rev. Lett.49, 405 (1982)

    1982

  48. [50]

    Simon, Holonomy, the quantum adiabatic theorem, and berry’s phase, Phys

    B. Simon, Holonomy, the quantum adiabatic theorem, and berry’s phase, Phys. Rev. Lett.51, 2167 (1983)

    1983

  49. [51]

    Qi, Y.-S

    X.-L. Qi, Y.-S. Wu, and S.-C. Zhang, Topological quantization of the spin hall effect in two-dimensional paramagnetic semiconductors, Physical Review B74, 10.1103/physrevb.74.085308 (2006)

    work page doi:10.1103/physrevb.74.085308 2006

  50. [52]

    H¨ ornedal, N

    N. H¨ ornedal, N. Carabba, A. S. Matsoukas-Roubeas, and A. del Campo, Ultimate speed limits to the growth of operator complexity, Communications Physics5, 10.1038/s42005-022-00985-1 (2022)

    doi:10.1038/s42005-022-00985-1 2022

  51. [53]

    Ozawa and B

    T. Ozawa and B. Mera, Relations between topology and the quantum metric for chern insulators, Physical Review B104, 10.1103/physrevb.104.045103 (2021)

    doi:10.1103/physrevb.104.045103 2021

  52. [54]

    Perugu, B

    R. Perugu, B. Kobrin, M. O. Flynn, and T. Scaffidi, Krylov winding and emergent coherence in operator growth dynamics, Phys. Rev. Lett.136, 150402 (2026)

    2026

  53. [55]

    Barb´ on, E

    J. Barb´ on, E. Rabinovici, R. Shir, and R. Sinha, On the evolution of operator complexity beyond scrambling, Journal of High Energy Physics2019, 10.1007/jhep10(2019)264 (2019)

    doi:10.1007/jhep10(2019)264 2019

  54. [56]

    Deift, T

    P. Deift, T. Kriecherbauer, K. T.-R. McLaughlin, S. Venakides, and X. Zhou, Strong asymptotics of orthogonal polynomials with respect to exponential weights, Communications on Pure and Applied Mathematics52, 1491 (1999)

    1999

  55. [57]

    Deift, T

    P. Deift, T. Kriecherbauer, K. T.-R. McLaughlin, S. Venakides, and X. Zhou, A riemann–hilbert approach to asymptotic questions for orthogonal polynomials, Journal of Computational and Applied Mathematics133, 47 (2001)

    2001

  56. [58]

    Ismail,Interplay Between Computational Techniques and Quantum Theory: Advancing Quantum Chaos and Quantum Error Correction, Ph.D

    R. Ismail,Interplay Between Computational Techniques and Quantum Theory: Advancing Quantum Chaos and Quantum Error Correction, Ph.D. thesis, Kentucky University (2025)

    2025

  57. [59]

    R. Basu, P. Chowdhury, A. Ganguly, S. Nath, O. Parrikar, and S. Paul, Wigner negativity, random matrices and gravity, JHEP01, 106, arXiv:2506.02110 [hep-th]

    arXiv

  58. [60]

    R. Basu, A. Ganguly, S. Nath, and O. Parrikar, Complexity growth and the krylov-wigner function (2025), arXiv:2402.13694 [hep-th]

    arXiv 2025

  59. [61]

    Avdoshkin and A

    A. Avdoshkin and A. Dymarsky, Euclidean operator growth and quantum chaos, Physical Review Research2, 10.1103/phys- revresearch.2.043234 (2020). Appendix A: Poisson Kernel

    doi:10.1103/phys- 2020

  60. [62]

    The Krylov basis is specified by orthogonal polynomials given in (12)

    General Formalism In this section, having taken the thermodynamic limit first, we consider a continuous energy spectrum which makes the spectral function Φ(E) continuous and normalized it to be consistent with⟨0|0⟩= 1. The Krylov basis is specified by orthogonal polynomials given in (12). In terms of the orthonormal polynomialsp n(E), we define the Poisso...

  61. [63]

    The derivatives∂ k ϕK(E1, E2, ϕ) will typically only be distributional atϕ= 0

    Modeling the kernels If we can directly estimateK(E 1, E2, ϕ) through exact results or asymptotics, then it may be advantageous to compute (27) though saddle point or numerical integration and then take derivatives. The derivatives∂ k ϕK(E1, E2, ϕ) will typically only be distributional atϕ= 0. They might only make sense through a regularization forr. It m...

  62. [64]

    The relevant integral which gives the square of the Frobenius norm is RR | ˜F(E 1, E2)|2dE1dE2

    Frobenius Norm Constraint Now we consider a system where ˜F(E 1, E2) is square-integrable, despite the Lanczos coefficients growing without bound. The relevant integral which gives the square of the Frobenius norm is RR | ˜F(E 1, E2)|2dE1dE2. This can be checked by explicitly plugging in the definition (A8) in terms of matrix elements. Generically, in suc...

  63. [65]

    This matches the asymptotic growth rate predicted by (47), with the additional assumption that lim t→∞ C(t) = 0

    The upper boundχ= 1 2 gives a late-time asymptotic oft 3 2 . This matches the asymptotic growth rate predicted by (47), with the additional assumption that lim t→∞ C(t) = 0. Appendix B: Asymptotics for spectral function with bounded support We now discuss asymptotics for orthogonal polynomials when the spectral function Φ(x) has support only on a finite s...

  64. [66]

    We restrict to the simple case whereMandBare exactly like the position and momentum operator

    Heisenberg Algebra Here we consider the Heisenberg algebra familiar from the harmonic oscillator [a, a †] = 1. We restrict to the simple case whereMandBare exactly like the position and momentum operator. M=α a+a † , B=iα a† −a ,F= 2α 2I.(C38) The time-evolving state|ψ(t)⟩is a coherent state of the harmonic oscillator. This gives φn(t) =e −α2t2/2 αntn √ n...

  65. [67]

    Double-Scaled SYK The SYK model is specified by the disordered Hamiltonian H=i p 2 X 1<j1<j2<...<jp<N Jj1j2...jp ψj1 ψj2 . . . ψjp ,(D1) whereψ k denotes Majorana fermions at sitek, andJ j1,j2...jp is a Gaussian random variable specifying thepbody interaction with the ensemble averaged first and second moments Jj1j2...jp = 0 D J2 j1j2...jp E = 1 4 N p (D2...

  66. [68]

    forward” and “backward

    Finite Ising spin chain For our numerics we used the finite-sized nonintegrable Ising spin chain with the parameters H=− i=6X i=1 σx i σx i+1 + 7X i=1 (0.8σx i + 1.05σz i ) (D6) in terms of the Pauli matricesσ α i at the sitei. The Krylov space dimension isN= 16257. Appendix E: Refined Dyck Paths Here we will describe a combinatorial method for evaluating...