Krylov complexity and the growth of the black hole interior in 3D gravity

This paper investigates the boundary reconstruction of black hole interior growth in three-dimensional gravity (specifically the BTZ black hole) and asks whether this geometric growth is successfully captured by notions of Krylov complexity. In AdS2/JT gravity, the complexity-volume proposal reduces to the length of an Einstein-Rosen bridge, which is naturally probed by two-point functions of local boundary operators. Moving to AdS3, however, the bulk codimension-one surface is anchored on spatial circles rather than isolated points. To resolve this, the authors propose a boundary and bulk reconstruction of codimension-one observables using spatially smeared non-local operators and Euclidean time averaging. They evaluate these extended correlators using the Chern-Simons formulation of three-dimensional gravity and subsequently extract the corresponding Lanczos data to compute both operator Krylov complexity and Krylov spread complexity.

The authors find that spatially smearing boundary primary operators along the spatial circle, combined with appropriate Euclidean-time integration, allows holographic correlation functions to successfully reproduce the characteristic late-time linear growth predicted by the complexity-volume proposal. Using the bulk Chern-Simons formulation, this boundary observable is represented by a bulk Wilson line with smeared endpoints, extending the known connection between Wilson lines and codimension-two observables to codimension-one objects.

When evaluating Krylov complexity within this setup, a sharp distinction emerges between operator and state notions. Operator Krylov complexity, extracted from the Lanczos data of the smeared operators, successfully reproduces the late-time linear growth of the black hole interior, successfully extending previous connections between operator growth and bulk geometry to AdS3. In contrast, the Krylov spread complexity of the thermofield double state—derived from the semiclassical gravitational partition function—remains approximately quadratic and fails to exhibit the expected linear growth of the bulk volume within the accessible regime.