Computational complexity theory

Chemistry

Hilbert space

Artificial intelligence

Eigenvalues and eigenvectors

Biochemistry

Physics

Operator (biology)

Integrable system

Pure mathematics

Mathematics

Gene

Lanczos resampling

Quantum mechanics

Scrambling

Chaotic

Computer science

Algorithm

Invertible matrix

Discrete mathematics

Transcription factor

Repressor

Applied mathematics

Journal of High Energy Physics

A bstract 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 &gt; 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.

Operator complexity: a journey to the edge of Krylov space