Chaos and Krylov complexity

The Krylov complexity measures the spread of the wavefunction in the Krylov basis, which is constructed using the Hamiltonian and an initial state. There has been a long-standing debate regarding the relationship between the growth of Krylov complexity and the chaotic nature of the Hamiltonian. In this work (arXiv:2303.12151), we investigate the evolution of the maximally entangled state in the Krylov basis for both chaotic and non-chaotic systems. Our findings suggest that neither the linear growth nor the saturation of Krylov complexity is necessarily associated with chaos. However, for chaotic systems, we observe a universal rise-slope-ramp-plateau behavior in the transition probability from the initial state to a Krylov basis, which is a characteristic of chaos in the spectrum of the Hamiltonian. Additionally, the long ramp in the transition probability is directly responsible for the late-time peak of Krylov complexity observed in previous literature. On the other hand, for non-chaotic systems, the transition probability exhibits a different behavior without the long ramp. Therefore, our results help to clarify which features of the wave function time evolution in Krylov space characterize chaos.

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