Geometry of Krylov complexity
By Paweł Caputa, Javier M. Magán, Dimitrios Patramanis. Physical Review Research, 2021
199 citations recorded by the source corpus.
We develop a geometric approach to operator growth and Krylov complexity in many-body quantum systems governed by symmetries. We start by showing a direct link between a unitary evolution with the Liouvillian and the displacement operator of appropriate generalized coherent states. This connection maps operator growth to a purely classical motion in phase space. The phase spaces are endowed with a natural information metric. We show that, in this geometry, operator growth is represented by geodesics and Krylov complexity is proportional to a volume. This geometric perspective also provides two novel avenues towards computation of Lanczos coefficients and sheds new light on the origin of their maximal growth. We describe the general idea and analyze it in explicit examples among which we reproduce known results from the Sachdev-Ye-Kitaev model, derive operator growth based on SU(2) and Heisenberg-Weyl symmetries, and generalize the discussion to conformal field theories. Finally, we use techniques from quantum optics to study operator evolution with quantum information tools such as entanglement and Renyi entropies, negativity, fidelity, relative entropy and capacity of entanglement.
Subfields: Atomic and Molecular Physics, and Optics, Nuclear and High Energy Physics, Statistical and Nonlinear Physics
View Paper: https://www.semanticscholar.org/paper/a2caa25e6cdf077cac614f7e3848f04303fd79b3
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In Atomic and Molecular Physics, and Optics
In Nuclear and High Energy Physics
In Statistical and Nonlinear Physics
Citations in the corpus
- Quantum chaos and the complexity of spread of states (2022, unknown): “The link with coherent states allows us to geometrize the notion of spread complexity following [56, 58].”
- Krylov complexity from integrability to chaos (2022, methodology): “In addition to the behavior of K-complexity for given individual quantum systems, such as the SYK model [3, 5, 6], 2D CFTs [17, 18], and more general symmetry-based Hamiltonian systems [19, 20], it is interesting and important to categorize the possible Krylov phenomenologies according to more unive”