Operator complexity: a journey to the edge of Krylov space

By E. Rabinovici, A. Sánchez-Garrido, R. Shir, J. Sonner. Journal of High Energy Physics, 2020

191 citations recorded by the source corpus.

Heisenberg time evolution under a chaotic many-body Hamiltonian H transforms an initially simple operator into an increasingly complex one, as it spreads over Hilbert space. Krylov complexity, or ‘K-complexity’, quantifies this growth with respect to a special basis, generated by H by successive nested commutators with the operator. In this work we study the evolution of K-complexity in finite-entropy systems for time scales greater than the scrambling time ts> log(S). We prove rigorous bounds on K-complexity as well as the associated Lanczos sequence and, using refined parallelized algorithms, we undertake a detailed numerical study of these quantities in the SYK4 model, which is maximally chaotic, and compare the results with the SYK2 model, which is integrable. While the former saturates the bound, the latter stays exponentially below it. We discuss to what extent this is a generic feature distinguishing between chaotic vs. integrable systems.

Subfields: Atomic and Molecular Physics, and Optics, Molecular Biology, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/4d6da6633297fc6a18e5c108c22053b7904b2323

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Citations in the corpus

Krylov complexity from integrability to chaos (2022, methodology): “The Lanczos algorithm and K-complexity computations were performed using the codes we developed in [5] and [7].”
Krylov complexity in conformal field theory (2021, result): “At the level of deﬁnition it is not re-lated to circuit complexity, but a number of recent works [10–12] found qualitative agreement between the behavior of K O ( t ) with the behavior of circuit and holographic complexities [13].”
Quantum chaos and the complexity of spread of states (2022, unknown): “A similar phenomenon was noted for operator growth complexity in [48, 53, 55].”
Krylov complexity in saddle-dominated scrambling (2022, unknown): “This is due to the ﬁnite-dimensional size of the system as argued in previous computations, for example, in SYK model [42] and XXZ model [52].”
Geometry of Krylov complexity (2021, background): “Various aspects of this hypothesis were already investigated in [22, 25–34] and Krylov complexity became a a r X i v : 2109 .”