Curator's Take
AI Commentary
This article shows that the often‑studied Krylov‐space measure of state complexity is directly tied to the geometric Berry phase in adiabatic spin dynamics, revealing a new bridge between operator growth and quantum geometry. By demonstrating that even simple single‑qubit evolutions can exhibit harmonic oscillations of Krylov complexity set by the external field strength and Berry curvature, the work provides a concrete diagnostic for tracking nontrivial adiabatic pathways that could inform error‑resilient gate design and Hamiltonian engineering. The result also suggests that monitoring Krylov complexity may become a practical tool for probing geometric phases in larger quantum processors, although extending the analysis beyond idealized two‑level systems will be essential to gauge its scalability.
— Mark Eatherly
Summary
The connection between Krylov complexity and Berry phase in adiabatic dynamics is investigated under the instantaneous eigenstate basis of a slowly evolving spin system. Adiabatic dynamics force the Krylov complexity to vanish if the initial Krylov basis is an instantaneous eigenstate of the Hamiltonian. Nevertheless, we demonstrate that the Krylov complexity will be nonvanishing if the initial Krylov basis is a superposition state rather than an eigenstate. Time evolution of Krylov complexity will behave periodically or quasi-periodically depending on the controlling parameters. In particular, for a single qubit system with constant parameters, the Krylov complexity will oscillate harmonically in time with the frequency relevant to the combination of the strength of the external field and the geometric Berry phase.