发表机构
Asia Pacific Center for Theoretical Physics; Pohang University of Science and Technology; Instituto de Física Teórica UAM/CSIC（亚太理论物理中心; 浦项科技大学; 马德里自治大学/西班牙国家科学研究委员会理论物理研究所）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该综述介绍克雷洛夫复杂度作为量子混沌等的诊断工具，阐述其过冲特征、相关模型及全息描述，还给出数值探索用的Mathematica笔记本并展望未来方向。
AI 中文摘要
克雷洛夫态复杂度（又称扩散复杂度）已成为表征量子混沌、信息扩散及多体动力学的精准且通用的诊断工具。该方法基于Lanczos算法构建，以最优基定理为基础，为研究量子动力学提供了可靠的谱学窗口。核心主题是混沌系统中观测到的特征过冲现象：混沌演化会使态在克雷洛夫链中延伸得比可积系统更深，随后弛豫至平衡态，形成复杂度峰值。该行为与谱统计的随机矩阵普适类相关，在量子台球、量子自旋链及SYK模型的变体等多种模型中均有体现。本文还讨论了爱因斯坦引力中克雷洛夫复杂度的全息描述，最后概述了未来研究方向与开放问题，包括含时系统及量子场论形式化。文中提供了一个Mathematica笔记本，用于对不同模型的克雷洛夫复杂度与谱统计进行数值探索。
英文摘要
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.
Comments39 pages, 29 figures, 1 Mathematica notebook; v2: added references and corrected typos; Invited review for the special issue "Krylov Complexity Across Scales: From Quantum Many-Body Systems to Black Holes" for the Journal of Physics A: Mathematical and Theoretical
