Well-defined, seed-robust Krylov-space diagnostics of quantum chaos: the $N_{\max}\to\infty$ limit of Lanczos statistics, initial operator/state dependence, and universality of the spread-complexity peak

Hashimoto, Murata, Tanahashi and Watanabe (2023) showed numerically, for stadium and Sinai billiards, that the variance of Lanczos coefficients — computed with a finite energy cutoff $N_{\max}$ — correlates with the classical Lyapunov exponent and with quantum level-spacing statistics, while Krylov complexity itself shows no exponential growth. This made Lanczos-coefficient statistics and the early-time peak of spread complexity candidate diagnostics of quantum chaos, but their reliability rests on three unresolved issues raised by that work: the descent regime of the Lanczos sequence used to define the variance disappears as $N_{\max}\to\infty$, so no truncation-independent definition of the statistics exists; the diagnostics depend strongly on the choice of initial operator/state without an established classification; and it is unproven whether the early-time peak of spread complexity is universally a better chaos indicator than late-time saturation. The problem is to resolve these three questions jointly, turning the observed correlations into a well-defined, seed-robust, universally validated Krylov-space criterion for quantum chaos. Substantial partial progress has since accumulated — peak-as-hallmark evidence in spin chains, SYK (Sachdev–Ye–Kitaev, a solvable family of strongly interacting, maximally chaotic quantum models) models and triangular billiards, an inverse-participation-ratio criterion for saturation values, and random-matrix descriptions of Lanczos spectra — but none of the three questions is settled.

Background

Quantum chaos searches for fingerprints of classically chaotic dynamics inside quantum theory. Two standard benchmarks are spectral statistics — the spacings between energy levels of the quantum Hamiltonian follow Wigner–Dyson (random-matrix) statistics in chaotic systems and Poisson statistics in integrable ones — and the classical Lyapunov exponent, which quantifies the exponential divergence of nearby classical trajectories. The classical Lyapunov exponent exists only for systems with a classical limit; for systems without one — SYK-type models and spin chains, for instance — the class-specific Lyapunov benchmark is instead the quantum (out-of-time-order-correlator) Lyapunov exponent, with the level-spacing statistics serving as the spectral benchmark common to all classes.

Krylov complexity is a more recent probe of quantum dynamics. Given a Hamiltonian $H$ and a seed — an operator $O$, paired with the inner product defined by the infinite-temperature (thermal) two-point function, or a state $|\psi\rangle$ — the Lanczos algorithm orthonormalizes the Krylov space spanned by successive actions of $H$ on the seed ($|\psi\rangle$, $H|\psi\rangle$, $H^2|\psi\rangle$, $\dots$) and represents $H$ in this basis as a tridiagonal matrix whose diagonal entries $a_n$ and off-diagonal entries $b_n$ are the Lanczos coefficients. Time evolution becomes hopping of a wavefunction along a one-dimensional "Krylov chain" with these coefficients, and the Krylov (operator) complexity, or spread complexity for states, is the expectation value of the position on that chain, $C(t)=\sum_n n\,|\phi_n(t)|^2$, where $\phi_n(t)$ are the amplitudes in the Krylov basis. For thermofield double states — purifying entangled states of two copies of the system whose evolution encodes finite-temperature dynamics — spread complexity in chaotic systems develops a characteristic early-time peak of height $C_{\max}$ before saturating (Balasubramanian, Caputa, Magan and Wu 2022; Erdmenger, Jian and Xian 2023).

Hashimoto, Murata, Tanahashi and Watanabe (2023) evaluated these quantities numerically for the stadium and Sinai billiards — two-dimensional domains with classically chaotic free motion with specular wall reflections. Krylov complexity showed no exponential growth, but they found that the variance of the Lanczos coefficients, computed over the "descent" portion of the Lanczos sequence after its initial rise, correlates clearly with the classical Lyapunov exponent and with adjacent-level spacing statistics: the more chaotic the billiard, the more regular (lower-variance) its Lanczos coefficients. This proposed Lanczos-coefficient statistics as a measure of quantum chaos. The computation, however, required truncating the infinite energy spectrum at a finite cutoff $N_{\max}$; as $N_{\max}\to\infty$ the Lanczos algorithm no longer terminates and the descent regime used to define the variance disappears, so the authors noted that the variance "measures the response of the system to a regularization $N_{\max}$" — an interpretation whose principled foundation remains to be built. They also observed that late-time saturation values of Krylov complexity are unreliable chaos indicators, while the early-time peak of the state complexity shows some correlation with the classical Lyapunov exponent, and asked how universal this behavior is. The same paper reports that the diagnostics depend on the choice of seed operator or state, without establishing criteria for reliable choices.

Problem Statement

Convert the empirically observed correlations between Krylov-space statistics and classical/spectral chaos indicators, found by Hashimoto, Murata, Tanahashi and Watanabe (JHEP 11 (2023) 040) for stadium and Sinai billiards, into a reliable criterion for quantum chaos by resolving the three interlinked issues that work left open:

1. Well-definedness. Provide a principled, truncation-independent definition — a continuum limit, a renormalization scheme, or a justified replacement statistic — of the Lanczos-coefficient statistics for quantum-mechanical systems with infinite-dimensional Hilbert spaces and dense spectra, such as billiards, and establish whether the correlation of these statistics with classical Lyapunov exponents and with level-spacing statistics survives this definition (or else show that the cutoff-based variance is intrinsically a response to the regularization $N_{\max}$, with that interpretation made precise).

2. Seed dependence. Establish criteria classifying for which choices of initial operator/state the Krylov diagnostics — Lanczos-coefficient statistics, the spread-complexity peak height $C_{\max}$, and the saturation value — reliably distinguish chaotic from integrable dynamics, including whether any averaging or optimization over seeds (for example over ensembles of random operators, or via maximally delocalized seeds) restores a chaos diagnostic. Here "reliably distinguish" is adjudicated against the two standard benchmarks named above: over a specified ensemble of seeds, the diagnostic must maintain a consistent, monotone relation to the class-specific Lyapunov benchmark and to the level-spacing statistics, rather than correlating only for individual favorable seeds.

3. Diagnostic content of the peak. Determine whether the early-time peak $C_{\max}$ of spread complexity is universally a better chaos diagnostic than the late-time saturation value — in particular whether its observed correlation with the class-specific Lyapunov benchmark extends from billiards, SYK-type (Sachdev–Ye–Kitaev-type, solvable strongly interacting models of maximal quantum chaos) models, and spin chains to generic quantum-chaotic systems, and how it behaves in borderline classes such as pseudo-integrable billiards (vanishing Lyapunov exponents with near-random-matrix spectra) and mixed phase space. Here "a better chaos diagnostic" is adjudicated by the same standard: over the same system ensembles and the same specified seed ensemble, the early-time peak $C_{\max}$ must yield a stronger and more transferable monotone correlation with the class-specific Lyapunov benchmark and the level-spacing statistics than the late-time saturation value does.

These three deliverables are each mandatory and are resolved independently — resolving any one of them is progress toward, but not completion of, the task, which is complete only when all three are resolved. Together they support one overarching objective: a Krylov-space criterion for quantum chaos that is well-defined without spectral truncation, robust to (or classified by) the choice of seed, and validated against the class-specific Lyapunov and spectral-statistics benchmarks. This overarching criterion is the conjunction of the three resolutions and is not itself a separate deliverable object, so no additional unified construction beyond items 1–3 is required.

References

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