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.
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, pr...
Johanna Erdmenger, Kyoung-Bum Huh, Hyun-Sik Jeong et al.· 2 citations· ⚡1
We study symmetry-resolved Krylov complexity in finite-dimensional chaotic quantum many-body systems. When both the Hamiltonian and the initial operator commute with a conserved charge, the operator dynamics decomposes into independent symmetry sectors, each with its own Krylov chain. We show that, after saturation, th...
Jayashis Das, Suman Das, Juan F. Pedraza et al.· 2 citations
We investigate the bulk-boundary correspondence in the SYK model from a quantum information perspective. The SYK model describes a system of Majorana fermions with random all-to-all interactions, whose disorder average-typically taken over a Gaussian ensemble-admits a dual description in terms of JT gravity in the larg...
Chen-Te Ma, Jeff Murugan, Masaki Tezuka· 0 citations
Understanding and coherently controlling the properties of interacting quantum many-body systems is a central challenge in non-equilibrium quantum physics. While in the past decades, a wide range of many-body Hamiltonians have been introduced to study quantum chaos, atypical eigenstates, and quantum resources, systemat...
Prasant Mallik, A. Sil, Sudipto Singha Roy· 0 citations
Universal statistical laws are a hallmark of quantum chaos, usually associated with the loss of spatial structure. Here we show, theoretically and experimentally, that universality emerges already at shallow depths in chaotic quantum circuits. We derive universal distributions of output-probability fluctuations and tes...
Arman Sauliere, Guglielmo Lami, Hugo L'oio et al.· 0 citations
These notes develop the diagnostics of chaos from their classical definitions through to holography, in six lectures given at the ST4 Workshop, Chennai Mathematical Institute, in 2026. No prior exposure to chaos theory is assumed. Lectures 1 and 2 define classical chaos geometrically and build the instruments that meas...
B. Shukla· 0 citations
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