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.
Krylov complexity, as a novel measure of operator complexity under Heisenberg evolution, exhibits many interesting universal behaviors and also bounds many other complexity measures. In this work, we study Krylov complexity
K(t) in Random Matrix Theory (RMT). In large
N limit: (1) For infinite temperature, we analytically show that the Lanczos coefficient
{bn} saturate to constant plateau
n→∞limbn=b, rendering a linear growing complexity
K(t)∼t, in contrast to the exponential-in-time growth in chaotic local systems in thermodynamic limit. After numerically comparing this plateau value
b to a large class of chaotic local quantum systems, we find that up to small fluctuations, it actually bounds the
{bn} in chaotic local quantum systems. Therefore we conjecture that in chaotic local quantum systems after scrambling time, the speed of linear growth of Krylov complexity cannot be larger than that in RMT. (2) For low temperature, we analytically show that
bn will first exhibit linear growth with
n, whose slope saturates the famous chaos bound. After hitting the same plateau
b,
bn will then remain constant. This indicates
K(t)∼e2πt/β before scrambling time
t∗∼O(βlogβ), and after that it will grow linearly in time, with the same speed as in infinite temperature. We finally remark on the effect of finite
N corrections.
A central challenge in diagnosing quantum chaos is to distinguish genuine many-body scrambling from kinematic effects of the spectrum. Krylov state complexity, or spread complexity, has emerged as a powerful diagnostic, with its characteristic growth, peak, and relaxation often taken as signatures of chaos. However, previous work has shown that this criterion can give false positives: saddle-dominated integrable systems may display prominent peaks even without random-matrix level correlations. We argue, based on complementary numerical and analytical evidence, that this ambiguity can be resolved by unfolding the spectrum prior to constructing the ensuing Krylov dynamics. By removing the non-universal smooth density of states while retaining microscopic spectral correlations, unfolding suppresses spurious peaks in integrable systems while preserving the universal spectral signatures of chaotic systems. Analytically, the formulation of the Lanczos iteration in terms of orthogonal polynomials yields an exact complexity kernel with a robust near-diagonal structure whose fine-grained features reflect the underlying spectral correlations. Moreover, for the logarithmic model, unfolding can be performed exactly, mapping the spectrum to a uniform lattice and yielding an analytic spread complexity that removes the false-positive peak. These findings establish unfolded Krylov complexity as a more reliable probe of genuine many-body scrambling.
A bstract We apply a notion of quantum complexity, called “Krylov complexity”, to study the evolution of systems from integrability to chaos. For this purpose we investigate the integrable XXZ spin chain, enriched with an integrability breaking deformation that allows one to interpolate between integrable and chaotic behavior. K-complexity can act as a probe of the integrable or chaotic nature of the underlying system via its late-time saturation value that is suppressed in the integrable phase and increases as the system is driven to the chaotic phase. We furthermore ascribe the (under-)saturation of the late-time bound to the amount of disorder present in the Lanczos sequence, by mapping the complexity evolution to an auxiliary off-diagonal Anderson hopping model. We compare the late-time saturation of K-complexity in the chaotic phase with that of random matrix ensembles and find that the chaotic system indeed approaches the RMT behavior in the appropriate symmetry class. We investigate the dependence of the results on the two key ingredients of K-complexity: the dynamics of the Hamiltonian and the character of the operator whose time dependence is followed.
Krylov state complexity, or spread complexity, has emerged as a sharp and versatile diagnostic of quantum chaos, information spreading, and many-body dynamics. Built from the Lanczos algorithm and grounded in the optimal-basis theorem, Krylov complexity thereby provides a robust spectroscopic window into quantum dynamics. A central theme is the characteristic overshoot observed in chaotic systems: a complexity peak in which chaotic evolution drives the state deeper into the Krylov chain than in integrable systems before relaxing to equilibrium. This behavior, tied to random-matrix universality classes of spectral statistics, is illustrated across a broad range of models, including quantum billiards, quantum spin chains, and variants of the SYK model. We also discuss proposed holographic descriptions of Krylov complexity in Einstein gravity, and conclude by outlining future directions and open problems, including time-dependent systems and quantum-field-theoretic formulations. A Mathematica notebook is provided for numerical exploration of Krylov complexity and spectral statistics across models.
Abstract Krylov complexity measures the spread of the wavefunction in the Krylov basis, which is constructed using the Hamiltonian and an initial state. We investigate the evolution of the maximally entangled state in the Krylov basis for both chaotic and non-chaotic systems. For this purpose, we derive an Ehrenfest theorem for the Krylov complexity, which reveals its close relation to the spectrum. Our findings suggest that neither the linear growth nor the saturation of Krylov complexity is necessarily associated with chaos. However, for chaotic systems, we observe a universal rise-slope-ramp-plateau behavior in the transition probability from the initial state to one of the Krylov basis states. Moreover, a long ramp in the transition probability is a signal for spectral rigidity, characterizing quantum chaos. Also, this ramp is directly responsible for the late-time peak of Krylov complexity observed in the literature. On the other hand, for non-chaotic systems, this long ramp is absent. Therefore, our results help to clarify which features of the wave function time evolution in Krylov space characterize chaos. We exemplify this by considering the Sachdev-Ye-Kitaev model with two-body or four-body interactions.