Operator complexity: a journey to the edge of Krylov space

Heisenberg time evolution under a chaotic many-body Hamiltonian $H$ transforms an initially simple operator into increasingly complex one, as it spreads over Hilbert space. Krylov complexity, or `K-complexity', quantifies this growth with respect to special basis, generated by successive nested commutators the operator. In work we study of K-complexity in finite-entropy systems for scales greater than scrambling $t_s>\log (S)$. We prove rigorous bounds on well associated Lanczos sequence and, using refined parallelized algorithms, undertake detailed numerical these quantities SYK$_4$ model, which is maximally chaotic, and compare results SYK$_2$ integrable. While former saturates bound, latter stays exponentially below it. discuss what extent generic feature distinguishing between vs. integrable systems.