论文标题
操作员的复杂性：到克里洛夫空间边缘的旅程
Operator complexity: a journey to the edge of Krylov space
论文作者
论文摘要
在混乱的多体汉密尔顿$ h $下，海森伯格时代的演变将最初简单的操作员转变为日益复杂的操作员，因为它在希尔伯特（Hilbert）的空间上传播。 Krylov的复杂性或“ K-Complexity”量化了这种增长相对于特殊基础的增长，该增长由$ h $由连续的嵌套换向器与操作员产生。在这项工作中，我们研究了有限渗透系统中K-复合物的演变，以大于争夺时间$ t_s> \ log（s）$的时间尺度。我们证明对K复杂性以及相关的Lanczos序列具有严格的界限，并且使用精制的并行化算法，我们对Syk $ _4 $模型中的这些数量进行了详细的数值研究，该模型是最大混乱的，并将结果与SYK $ _2 $模型进行了比较。前者饱和界限，而后者则在其下方保持指数。我们在多大程度上讨论这是区分混乱与可集成系统的通用特征。
Heisenberg time evolution under a chaotic many-body Hamiltonian $H$ transforms an initially simple operator into an increasingly complex one, as it spreads over Hilbert space. Krylov complexity, or `K-complexity', quantifies this growth with respect to a special basis, generated by $H$ by successive nested commutators with the operator. In this work we study the evolution of K-complexity in finite-entropy systems for time scales greater than the scrambling time $t_s>\log (S)$. We prove rigorous bounds on K-complexity as well as the associated Lanczos sequence and, using refined parallelized algorithms, we undertake a detailed numerical study of these quantities in the SYK$_4$ model, which is maximally chaotic, and compare the results with the SYK$_2$ model, which is integrable. While the former saturates the bound, the latter stays exponentially below it. We discuss to what extent this is a generic feature distinguishing between chaotic vs. integrable systems.
