Holographic complexity: braneworld gravity versus the Lloyd bound

We explore the complexity equals volume proposal for planar black holes in
anti-de Sitter (AdS) spacetime in 2+1 dimensions, with an end of the world (ETW) brane
behind the horizon. We allow for the possibility of intrinsic gravitational dynamics in
the form of Jackiw-Teitelboim (JT) gravity to be localized on the brane. We compute
the asymptotic rate of change of volume complexity analytically and obtain the full time
dependence using numerical techniques. We fnd that the inclusion of JT gravity on the
brane leads to interesting efects on time dependence of holographic complexity. We identify
the region in parameter space (the brane location and the JT coupling) for which the rate of
change of complexity violates the Lloyd bound. In an equivalent description of the model in
terms of an asymptotically AdS wormhole, we connect the violation of the Lloyd bound to
the violation of a suitable energy condition in the bulk that we introduce. We also compare
the Lloyd bound constraints to previously derived constraints on the bulk parameters in this
model that are based on bounds on entanglement growth in the dual CFT state.