Chaos and Krylov Complexity

Explore the intricate relationship between chaos and Krylov complexity in quantum systems through this comprehensive lecture. Delve into the concept of Krylov complexity as a measure of wavefunction spread in the Krylov basis, constructed using the Hamiltonian and an initial state. Examine the long-standing debate surrounding the connection between Krylov complexity growth and Hamiltonian chaos. Investigate the evolution of maximally entangled states in the Krylov basis for both chaotic and non-chaotic systems. Discover that linear growth and saturation of Krylov complexity are not necessarily indicative of chaos. Analyze the universal rise-slope-ramp-plateau behavior in transition probability from initial state to Krylov basis, characteristic of chaos in the Hamiltonian spectrum. Understand how the long ramp in transition probability contributes to the late-time peak of Krylov complexity in chaotic systems. Compare this behavior with non-chaotic systems, which exhibit different transition probability patterns. Gain insights into which features of wavefunction time evolution in Krylov space truly characterize chaos, helping to clarify this complex topic in quantum mechanics.