Krylov-Ehrenfest Time Defines Limit of Classical-Quantum Complexity

Researchers at the University of Geneva and University of Southampton develop and study a classical Lanczos algorithm by replacing quantum commutators with Poisson brackets, allowing us to define Krylov complexity using the symplectic structure of phase space. This approach demonstrates that classical Krylov complexity accurately approximates quantum complexity during early-time chaotic dynamics, offering a characteristic scale for these systems. We define a Krylov-Ehrenfest time, which quantifies the eventual divergence of classical and quantum complexities. Applying this framework to the Lipkin-Meshkov-Glick and Feingold-Peres models, collective spin systems exhibiting distinct chaotic behaviors, the analysis shows that saddle points enhance the K-complexity saturation value…