Analytic Spread Complexity from Level Statistics: From Chaos to Integrability
Abstract
Spread complexity has emerged as a useful probe of quantum chaos, yet the microscopic spectral origin of its characteristic finite-time peak remains incompletely understood. We develop an analytic framework that relates spread complexity directly to local spectral statistics. Starting from an energy-space representation of the Krylov kernel, we show that the kernel is approximately banded, leading to a rapidly convergent diagonal expansion dominated by nearby levels in the ordered spectrum. Motivated by this structure, we propose an approximate kernel-universality hypothesis: after unfolding, the Krylov kernel is well approximated by that of a uniform lattice. Combining this universal kernel with local spectral statistics yields a simple analytic expression for spread complexity in terms of the Fourier transforms of the k-th nearest-neighbour spacing distributions. In particular, at leading order, the finite-time peak is controlled by the Fourier transform of the nearest-neighbour spacing distribution. The resulting framework describes both chaotic random-matrix ensembles and the integrable Poisson limit, identifies the spectral origin of the complexity peak and its late-time behaviour, and provides a direct connection between Krylov dynamics and spectral statistics.
Related
- Analytic level statistics of the quantum Rabi model
- Distance to integrability and complexity
- Equations of state of hydrodynamic type and particle statistics of a Dyson gas in an analytic confining potential
- Holographic Krylov spread complexity for a confining multi-charge AdS soliton
- Spread Complexity for Local Operator Quenches with Conserved Momentum
BibTeX
@misc{basu2026analytic,
title = {{Analytic Spread Complexity from Level Statistics: From Chaos to Integrability}},
author = {Pallab Basu and Suman das and Bigboy Madlala},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2608.07412},
eprint = {2608.07412},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.07412},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Analytic Spread Complexity from Level Statistics: From Chaos to Integrability
AU - Pallab Basu
AU - Suman das
AU - Bigboy Madlala
PY - 2026
DA - 2026/09/17
PB - arXiv
DO - 10.48550/arxiv.2608.07412
UR - https://arxiv.org/abs/2608.07412
AB - Spread complexity has emerged as a useful probe of quantum chaos, yet the microscopic spectral origin of its characteristic finite-time peak remains incompletely understood. We develop an analytic framework that relates spread complexity directly to local spectral statistics. Starting from an energy-space representation of the Krylov kernel, we show that the kernel is approximately banded, leading to a rapidly convergent diagonal expansion dominated by nearby levels in the ordered spectrum. Motivated by this structure, we propose an approximate kernel-universality hypothesis: after unfolding, the Krylov kernel is well approximated by that of a uniform lattice. Combining this universal kernel with local spectral statistics yields a simple analytic expression for spread complexity in terms of the Fourier transforms of the k-th nearest-neighbour spacing distributions. In particular, at leading order, the finite-time peak is controlled by the Fourier transform of the nearest-neighbour spacing distribution. The resulting framework describes both chaotic random-matrix ensembles and the integrable Poisson limit, identifies the spectral origin of the complexity peak and its late-time behaviour, and provides a direct connection between Krylov dynamics and spectral statistics.
ER -