Introduction to Krylov Complexity

Abstract: Understanding operator growth is central to probing the dynamics of complex quantum systems. The earlier geometric approach of Nielsen’s provides a foundational framework for quantifying complexity, however it relies on subjective, external definitions of distance. This talk transitions from these early geometric definitions to the more robust framework of Krylov complexity, which utilises the Lanczos algorithm to offer an intrinsic, basis-independent measure of operator evolution. Moreover, within the holographic dictionary, this boundary evolution finds a physical interpretation via the size-momentum correspondence. We will explore how an operator’s growth along the boundary Krylov chain maps to the radial proper momentum of an infalling particle in the bulk geometry. Finally, applying this duality, we will discuss recent results utilising the proper momentum relation to evaluate Krylov complexity.

condensed mattergeneral relativity and quantum cosmologyHEP - theorymathematical physicsdifferential geometry

Audience: advanced learners

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Europe and Pacific Theoretical Physics

Series comments: European and Pacific Theoretical Physics is a group that runs online seminars and talks across broad topics in mathematical and theoretical physics. Our goal is to share concepts across continents to bring about a deeper understanding of cutting-edge topics in physics.

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