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.
Quantum chaotic systems are conjectured to display a spectrum whose fine-grained features (gaps and correlations) are well described by random matrix theory (RMT). We propose and develop a complementary version of this conjecture: quantum chaotic systems display a Lanczos spectrum whose local means and covariances are well described by RMT. To support this proposal, we first demonstrate its validity in examples of chaotic and integrable systems. We then show that for Haar-random initial states in RMTs the mean and covariance of the Lanczos spectrum suffice to produce the full long-time behavior of general survival probabilities including the spectral form factor, as well as the spread complexity. In addition, for initial states with continuous overlap with energy eigenstates, we analytically find the long-time averages of the probabilities of Krylov basis elements in terms of the mean Lanczos spectrum. This analysis suggests a notion of eigenstate complexity, the statistics of which differentiate integrable systems and classes of quantum chaos. Finally, we clarify the relation between spread complexity and the universality classes of RMT by exploring various values of the Dyson index and Poisson distributed spectra.
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.
Krylov complexity has recently emerged as a new paradigm to characterize quantum chaos in many-body systems. However, which features of Krylov complexity are a prerogative of quantum chaotic systems and how they relate to more standard probes, such as spectral statistics or out-of-time-order correlators (OTOCs), remain open questions. Recent insights have revealed that in quantum chaotic systems Krylov state complexity exhibits a distinct peak during time evolution before settling into a well-understood late-time plateau. In this work we propose that this Krylov complexity peak (KCP) is a hallmark of quantum chaotic systems and suggest that its height could serve as an order parameter for quantum chaos. We demonstrate that the KCP effectively identifies chaotic-integrable transitions in two representative quantum-mechanical models at both infinite and finite temperature: the mass-deformed Sachdev-Ye-Kitaev model and the sparse Sachdev-Ye-Kitaev model. Our findings align with established results from spectral statistics and OTOCs while introducing an operator-independent diagnostic for quantum chaos, offering more universal insights and a deeper understanding of the general properties of quantum chaotic systems.
We consider the problem of measuring the spread of quantum states and justify the use of the Krylov basis to measure the spread of quantum states as a Hamiltonian system evolves, resulting in a measure of "spread complexity" akin to Krylov operator complexity but applied to states. In this basis, Hamiltonians become tridiagonal. The time-evolved thermofield double states in chaotic systems shows four regimes of this spread complexity: a linear ramp up to a peak that is exponential in the entropy, followed by a slope down to a plateau. We then derive analytical approximations for the mean of the tridiagonal matrix coefficients from the density of states in an general quantum system, as well as for both the mean and covariance for random matrices with a single-trace potential. Other quantities in the long time limit are shown to be computable as well. This is then applied in two ways: We show in examples that the covariance of the tridiagonal coefficients distinguish integrable and chaotic systems. Essentially, we provide a new set of tools capable of probing the detailed spectral statistics of quantum systems. We also find an application of the mean to infer the late time behavior of double scaled SYK. Lastly, as a somewhat separate topic, we shine a spotlight on the differences between the properties of eigenbasis chaos (e.g. the Eignestate Themalization Hypothesis) and spectral chaos (e.g. the eigenvalue repulsion found in chaotic systems). We show that without any constraints on the system or with a exponentially small tolerance in the constraints, these two properties are unrelated. We then provide a numerically-constructed example that shows that even with exact k-locality it may be possible to construct systems where the two properties do not agree. This suggests that careful considerations of typicality, restrictions, and/or the order of limits, may be needed to fully connect eigenbasis and spectral chaos.
The Yukawa-Sachdev-Ye-Kitaev (YSYK) model, a solvable model of non-Fermi liquids with a proposed cavity-QED realization, couples
N complex fermions to
M bosons of frequency
ω0 through random Yukawa couplings of strength
g, and its dynamics depends only on
R≡ω0/g2/3. For
R≪1 the bosons are slow and act as a static random background in which the fermions are free, while for
R≫1 they are only virtually excited and generate a random four-fermion interaction of rank
M, which is generically chaotic for
M≥2. Using exact diagonalization at infinite temperature, we follow the emergence of random-matrix statistics between these limits with unfolded spectral statistics and the Krylov complexity of the thermofield-double state. For complex couplings the level-spacing distribution evolves from Poisson to Gaussian-unitary-ensemble (GUE) statistics, the spectral form factor develops a correlation hole and a ramp, and the Krylov complexity evaluated on the unfolded spectrum develops a peak above its late-time plateau whose height reaches the GUE value. All diagnostics locate the onset around
R≈0.3, and adding boson modes moves it to smaller
R. This delineates the parameter regime in which a cavity-QED realization would display random-matrix spectral correlations. On the other hand, real couplings give the same crossover towards Gaussian-orthogonal-ensemble (GOE) statistics, and along an interpolation between the two the peak height follows the mean gap ratio, so the peak identifies the symmetry class as well as the onset of chaos. Without unfolding the peak stays about 20
% below its random-matrix value in both classes, which we trace to the non-semicircular density of states, and at larger
R the boson-number bands slow down the spreading of the state in Krylov space.