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arXiv:2607.05294

Polynomial Initial-State Jumps and Christoffel Transforms in Krylov Complexity

Abhishek Chowdhury, Ajit Prasad Mahapatra

hep-thmath-phmath.ACmath.SPquant-phSubmitted Jul 6, 2026• Announced Jul 7, 2026• 0 citations
v1 · Jul 6, 2026v2 · Jul 19, 2026v3 · Aug 27, 2026
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Abstract

State Krylov, or spread, complexity is a property of a pair $(H,\ket{K_0})$ rather than of the Hamiltonian alone. Thus, changing the initial state at fixed $H$ generally changes the Lanczos coefficients and the ordered Krylov basis. We solve this relative initial-state problem for normalized polynomial filters, $\ket{ψ_Q}=Q(H)\ket{K_0}/\sqrt{N_Q}$ with $N_Q=\langle K_0|Q(H)^\dagger Q(H)|K_0\rangle$. The filtered spectral measure is the positive polynomial modification $|Q(E)|^2\mathrm dμ(E)/N_Q$, and orthogonality turns this measure change into a finite-band transfer from reference Fourier-orthogonal-polynomial moments to shifted Krylov amplitudes. We derive exact finite sums for individual amplitudes and projected Christoffel-Darboux kernels for cumulative probabilities and spread complexity. The formulae cover confluent roots, complex seed coefficients, support loss, and terminal quotients in finite dimensions. We evaluate the construction in three canonical Jacobi families, the Heisenberg--Weyl/Charlier oscillator, the compact $SU(2)$/Krawtchouk spin, and the constant-coefficient tight-binding/Chebyshev chain, with a Hermite central-limit scaling of Charlier as a continuous-spectrum check of this Christoffel jump machinery. Finite seed families are organized by a matrix-valued parent measure whose scalar compressions recover the individual shifted problems. The fixed-inner-product construction carries over to operator Krylov complexity after the replacement \(H\mapsto\mathcal L\) and \(\ket{K_0}\mapsto O\); polynomial seeds then become nested-commutator descendants \(Q(\mathcal L)O\). The result is an exact relative calculus in which a solved cyclic problem generates a family of polynomially related initial-state dynamics without repeating Lanczos in the original Hilbert space.

Comments
60 pages

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Caputa-Di Giulio-Loc Conjecture Resolved in Finite Dimensions: All-Time Inequalities for Symmetry-Resolved Krylov and Spread Complexity

Nowicki Maciej, Artificial Hyperintelligence, Eve, wife of Maciej Nowicki.

This research release presents an analytic proof of the Caputa-Di Giulio-Loc conjecture on symmetry-resolved Krylov complexity and quantum-state spread complexity in finite dimensions. For every real time t, the total complexity satisfies C(t) ≥ Σ_q p_q C_q(t), where C_q(t) is the normalized complexity within sector q and p_q is its initial probability weight. The theorem holds for finite-dimensional Hermitian generators with orthogonal reducing sectors, including degenerate spectra and terminating Lanczos chains. It requires no assumptions of quantum chaos, spectral randomness, or late-time equilibration. The conjecture appears in “Growth of block-diagonal operators and symmetry-resolved Krylov complexity,” Physical Review Research 7, 043055 (2025), DOI: 10.1103/9v9v-54zv, and “Symmetry-resolved spread complexity,” JHEP 02 (2026), 189, DOI: 10.1007/JHEP02(2026)189. The result establishes a general relationship between full-system quantum dynamics and calculations performed separately within symmetry sectors. Its practical importance is that sector computations provide rigorous lower bounds on total complexity at every time. It also identifies precisely when resolving the symmetry leaves complexity unchanged. These conclusions are relevant to mathematical physics, operator growth, quantum many-body dynamics, and the interpretation of Krylov complexity as a diagnostic of quantum chaos. The release proves a stronger statement than the original mean-complexity inequality: the full Krylov-index distribution dominates the probability-weighted sector distribution in first-order stochastic order. Consequently, the inequality extends to every nondecreasing function of the Krylov index. Additional results include: • Two complementary proofs using nested Krylov-space projections and best polynomial approximation in spectral measures.• Concavity under mixing of normalized spectral measures.• Exact nonnegative gap identities, including a moment-Gram formulation valid for singular matrices.• Fixed-time equality criteria and a finite-dimensional characterization of equality at all times.• An explicit three-node example and reproducible numerical diagnostics.• An extension to bounded self-adjoint generators, with explicit domain and completeness assumptions for unbounded generators. The standalone release includes the PDF manuscript, editable LaTeX, complete searchable Markdown proofs, Python verification code, citation metadata, an expert review guide, and machine-readable theorem, assumption, dependency, and proof records for research discovery and AI-assisted retrieval. Research status: a complete analytic proof is supplied for the stated finite-dimensional theorem. Independent peer review and proof-assistant verification have not been completed. Worldwide novelty and priority are not certified; the unrestricted infinite-dimensional conjecture is not claimed to be resolved.

•Sep 27, 2026•0 citations
State Krylov, or spread, complexity is a property of a pair (H,∣K0⟩)(H,\ket{K_0})(H,∣K0​⟩) rather than of the Hamiltonian alone. Thus, changing the initial state at fixed HHH generally changes the Lanczos coefficients and the ordered Krylov basis. We solve this relative initial-state problem for normalized polynomial filters, ∣ψQ⟩=Q(H)∣K0⟩/NQ\ket{ψ_Q}=Q(H)\ket{K_0}/\sqrt{N_Q}∣ψQ​⟩=Q(H)∣K0​⟩/NQ​​ with NQ=⟨K0∣Q(H)†Q(H)∣K0⟩N_Q=\langle K_0|Q(H)^\dagger Q(H)|K_0\rangleNQ​=⟨K0​∣Q(H)†Q(H)∣K0​⟩. The filtered spectral measure is the positive polynomial modification ∣Q(E)∣2dμ(E)/NQ|Q(E)|^2\mathrm dμ(E)/N_Q∣Q(E)∣2dμ(E)/NQ​, and orthogonality turns this measure change into a finite-band transfer from reference Fourier-orthogonal-polynomial moments to shifted Krylov amplitudes. We derive exact finite sums for individual amplitudes and projected Christoffel-Darboux kernels for cumulative probabilities and spread complexity. The formulae cover confluent roots, complex seed coefficients, support loss, and terminal quotients in finite dimensions. We evaluate the construction in three canonical Jacobi families, the Heisenberg--Weyl/Charlier oscillator, the compact SU(2)SU(2)SU(2)/Krawtchouk spin, and the constant-coefficient tight-binding/Chebyshev chain, with a Hermite central-limit scaling of Charlier as a continuous-spectrum check of this Christoffel jump machinery. Finite seed families are organized by a matrix-valued parent measure whose scalar compressions recover the individual shifted problems. The fixed-inner-product construction carries over to operator Krylov complexity after the replacement (H\mapsto\mathcal L) and (\ket{K_0}\mapsto O); polynomial seeds then become nested-commutator descendants (Q(\mathcal L)O). The result is an exact relative calculus in which a solved cyclic problem generates a family of polynomially related initial-state dynamics without repeating Lanczos in the original Hilbert space.
We study the spread complexity of the infinite-temperature thermofield double state in large-NNN random matrix theory and investigate the physical meaning of its Lanczos coefficients bnb_nbn​. The leading density of states defines a family of orthogonal polynomials whose recursion coefficients coincide with the Lanczos coefficients of the associated Krylov dynamics. Starting from the Coulomb-gas equation, we derive a pair of nonlinear relations for these coefficients, which we call the Lanczos equations. For normalizable one-cut ensembles, the Lanczos coefficients approach constants at large Krylov index, rendering the asymptotic Krylov chain translationally invariant and the late-time growth of spread complexity linear, with the growth rate given by the spectral average of the group velocity obtained from a Bloch-wave analysis. For an even potential, we further show that the difference between consecutive squared Lanczos coefficients, bn+12−bn2b_{n+1}^2-b_n^2bn+12​−bn2​, is the generating function for two-legged maps whose marked legs are separated by the exact geodesic distance nnn. Consequently, the spread complexity acceleration operator equals twice the geodesic generating operator. We illustrate these results in the quartic matrix model, where the expansion in the quartic coupling counts tetravalent planar maps, and in the double-scaled Sachdev--Ye--Kitaev model, where the auxiliary Hilbert space can be identified with the physical chord Hilbert space.
State Krylov complexity becomes experimentally concrete in driven optics because its chain can be interrogated through excitation counting, parity, return probability, and phase. Its meaning, however, is fixed jointly by the control history, the prepared state, and the reference used to follow explicit time dependence. We disentangle these ingredients for oscillator, SU(2)SU(2)SU(2), SU(1,1)SU(1,1)SU(1,1), and three-level Hamiltonians and determine when preparation dependence preserves exact solvability. A fixed group displacement of an extremal state can be absorbed into transformed controls, giving an undo-and-count readout for arbitrary drives. A polynomial preparation instead Christoffel-transforms the spectral measure of a fixed operator; an affine moving frame then transports its complete seed-adapted chain even when laboratory Hamiltonians at different times do not commute. We also resolve seed dependence in stroboscopic dynamics and driven three-level geometry. A periodically forced mode started from any finite Fock state has cyclic dimension equal to the denominator of a nonzero-displacement rational rotation and infinite dimension for an irrational rotation. At a half turn its complexity becomes a Laguerre survival deficit and a magnitude-sensitive Wigner readout. In a driven VVV atom, bright-state geometry interferes with excited-manifold dynamics and controls bright-dark decoupling. Preparation therefore changes not only the initial state but also the chain and the optical observable that realizes its complexity.
The leading asymptotic growth of Lanczos coefficients is controlled by spectral tails and furnishes a coarse classification of Krylov dynamics. We show that the \textit{global topology} of the spectral measure, specifically the number of connected components, the gap structure, and the behaviour at gap-closing transitions, encodes a finer hierarchy of dynamical invariants invisible to tail-based arguments. Using the Riemann-Hilbert formulation of orthogonal polynomials and Deift-Zhou steepest descent, we recover the Freud growth laws bn∼n1/βb_n\sim n^{1/β}bn​∼n1/β for single-cut measures and determine their sub-leading corrections from endpoint data. Gapped spectra produce quasiperiodic Lanczos oscillations at a frequency fixed by the filling fraction of the spectral bands alone, and hence predictable from the band edges. We verify this in the SSH chain and its next-nearest-neighbour deformation. At a gap-closing transition the oscillation amplitude is governed by the Hastings-McLeod solution of Painlevé II, decaying as n−1/3n^{-1/3}n−1/3 at criticality and interpolating between the gapped and merged phases, so that the topology change of the spectral curve is realised as a Krylov phase transition. We also demonstrate that, while in the conformal limit of SYK the operator scaling dimension is invisible in the leading rate α=πTα= πTα=πT, it can be extracted from the subleading offset b0=πT(Δ−12)b_0 = πT(Δ- \frac{1}{2})b0​=πT(Δ−21​). These results establish a refined notion of universality in operator growth, classified by spectral topology rather than spectral tails alone.
We construct families of deformations of the double-scaled SYK (DSSYK) model and investigate their bulk interpretation. We introduce microscopic deformations of the SYK model which, after ensemble averaging and in the double-scaling limit, are described by a transfer matrix encoding the recurrence relations of basic orthogonal polynomials in the q-Askey scheme. For certain families of deformations in the semiclassical limit at finite temperature, the chord number (encoding Krylov complexity) corresponds to the length of an Einstein-Rosen bridge connecting an End-Of-The-World brane to an anti-de Sitter asymptotic boundary. By increasing one of the deformation parameters, the models eventually exhibit discrete energy levels, signaling a new geometric transition in sine dilaton gravity. Via the SYK-Schur duality, Krylov complexity also admits a representation-theoretic interpretation as the spread of the SU(2) spin in the index of an N=2\mathcal{N}=2N=2 SU(2) gauge theory. We study the operator algebras of the deformed theories. The algebras can be type II1_11​ or type I∞_\infty∞​ factors, depending on the operators that are included. The entanglement entropy between the type II1_11​ algebras for a pure state manifests as an extremal surface through the Ryu-Takayanagi formula. We discuss connections between our results and the emergence of baby universes in the bulk.
We employ machine learning techniques to estimate the amount of information contained in three Krylov-space observables: the spread complexity C(t)C(t)C(t), the discrete Wigner negativity N(t)N(t)N(t), and the normalized negativity χ(t)=N(t)/∣S(t)∣χ(t)=N(t)/|S(t)|χ(t)=N(t)/∣S(t)∣, where S(t)S(t)S(t) is the survival amplitude. We study thermofield-double evolutions on GUE, GOE and Poisson spectra, and on the integrable SL(2,R)SL(2,R)SL(2,R) Krylov chain of a two-dimensional CFT primary, with Lanczos coefficients bn=αn(n+2h−1)b_n=α\sqrt{n(n+2h-1)}bn​=αn(n+2h−1)​. To introduce chaos we add a GUE random matrix to the integrable chain, H(ε)=HSL(2,R)+εR0WGUEH(\varepsilon)=H_{SL(2,R)}+\varepsilon R_0 W_{GUE}H(ε)=HSL(2,R)​+εR0​WGUE​, where R0R_0R0​ is its spectral half-width. Either C(t)C(t)C(t) or N(t)N(t)N(t) alone determines the thermofield temperature, with R2≃0.999R^2\simeq0.999R2≃0.999. However, neither can reconstruct the fine structure of the spectral form factor. In the integrable sector all three observables carry the same information. When we turn on chaos an asymmetry appears: χχχ predicts CCC better than CCC predicts χχχ. This gap grows from +0.33+0.33+0.33 to +0.77+0.77+0.77, while the same gap for the raw negativity decays to zero. The extra information comes from the survival amplitude, and χ(t)χ(t)χ(t) carries it while C(t)C(t)C(t) can still be recovered from χ(t)χ(t)χ(t). We derive a lower bound on this gap, set by the fraction of the variance of χχχ that comes from SSS.
This research release presents an analytic proof of the Caputa-Di Giulio-Loc conjecture on symmetry-resolved Krylov complexity and quantum-state spread complexity in finite dimensions. For every real time t, the total complexity satisfies C(t) ≥ Σ_q p_q C_q(t), where C_q(t) is the normalized complexity within sector q and p_q is its initial probability weight. The theorem holds for finite-dimensional Hermitian generators with orthogonal reducing sectors, including degenerate spectra and terminating Lanczos chains. It requires no assumptions of quantum chaos, spectral randomness, or late-time equilibration. The conjecture appears in “Growth of block-diagonal operators and symmetry-resolved Krylov complexity,” Physical Review Research 7, 043055 (2025), DOI: 10.1103/9v9v-54zv, and “Symmetry-resolved spread complexity,” JHEP 02 (2026), 189, DOI: 10.1007/JHEP02(2026)189. The result establishes a general relationship between full-system quantum dynamics and calculations performed separately within symmetry sectors. Its practical importance is that sector computations provide rigorous lower bounds on total complexity at every time. It also identifies precisely when resolving the symmetry leaves complexity unchanged. These conclusions are relevant to mathematical physics, operator growth, quantum many-body dynamics, and the interpretation of Krylov complexity as a diagnostic of quantum chaos. The release proves a stronger statement than the original mean-complexity inequality: the full Krylov-index distribution dominates the probability-weighted sector distribution in first-order stochastic order. Consequently, the inequality extends to every nondecreasing function of the Krylov index. Additional results include: • Two complementary proofs using nested Krylov-space projections and best polynomial approximation in spectral measures.• Concavity under mixing of normalized spectral measures.• Exact nonnegative gap identities, including a moment-Gram formulation valid for singular matrices.• Fixed-time equality criteria and a finite-dimensional characterization of equality at all times.• An explicit three-node example and reproducible numerical diagnostics.• An extension to bounded self-adjoint generators, with explicit domain and completeness assumptions for unbounded generators. The standalone release includes the PDF manuscript, editable LaTeX, complete searchable Markdown proofs, Python verification code, citation metadata, an expert review guide, and machine-readable theorem, assumption, dependency, and proof records for research discovery and AI-assisted retrieval. Research status: a complete analytic proof is supplied for the stated finite-dimensional theorem. Independent peer review and proof-assistant verification have not been completed. Worldwide novelty and priority are not certified; the unrestricted infinite-dimensional conjecture is not claimed to be resolved.