Quantum chaos and complexity in Krylov space

The dynamics of a quantum system is confined to a subspace of the state or operator space, the Krylov space, which is often smaller than the full Hilbert/operator space.

Krylov space methods therefore provide a compact and computationally efficient description of quantum evolution, which is particularly useful for nonequilibrium phenomena in many-body systems with large Hilbert spaces. In this talk, I will discuss a notion of complexity defined in Krylov space and its use as a probe of operator growth, quantum chaos, information scrambling, and state evolution in both closed and open quantum systems. Illustrative examples include quantum spin chains, Sachdev–Ye–Kitaev models, and random matrix theory. The talk is based on selected topics from the review article arXiv:2405.09628 (Physics Reports) and more recent developments.