Holographic non-computers

We introduce the notion of holographic non-computer as a system which exhibits parametrically large delays in the growth of complexity, as calculated within the Complexity-Action proposal. Some known examples of this behavior include extremal black holes and near-extremal hyperbolic black holes. Generic black holes in higher-dimensional gravity also show non-computing features. Within the $1/d$ expansion of General Relativity, we show that large-$d$ scalings which capture the qualitative features of complexity, such as a linear growth regime and a plateau at exponentially long times, also exhibit an initial computational delay proportional to $d$. While consistent for large AdS black holes, the required `non-computing' scalings are incompatible with thermodynamic stability for Schwarzschild black holes, unless they are tightly caged.