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This research establishes a classical limit for quantum Krylov complexity by leveraging symplectic phase space structures and Poisson brackets to approximate early-time chaotic dynamics. By defining the "Krylov-Ehrenfest time," it quantifies the specific scale at which classical approximations diverge from true quantum behavior in systems like collective spin models. This theoretical framework provides a rigorous method for analyzing complexity saturation across different energy shells, bridging semiclassical limits with full quantum mechanical descriptions.

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Krylov-Ehrenfest Time Defines Limit of Classical-Quantum Complexity

Quantum ZeitgeistJuly 30, 20265 min read

This research defines a classical limit of quantum Krylov complexity using the symplectic structure of phase space and a Lanczos algorithm.

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…

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