Holography, cellulations and error correcting codes

Abstract: Quantum error correction codes associated with the hyperbolic plane have been explored extensively in the context of the AdS$_3$/CFT$_2$ correspondence. In this paper we initiate a systematic study of codes associated with holographic geometries in higher dimensions, relating cellulations of the spatial sections of the geometries to stabiliser codes. We construct analogues of the HaPPY code for three-dimensional hyperbolic space (AdS$_4$), using both absolutely maximally entangled (AME) and non-AME codes. These codes are based on uniform regular tessellations of hyperbolic space but we note that AME codes that preserve the discrete symmetry of the polytope of the tessellation do not exist above two dimensions. We also explore different constructions of stabiliser codes for hyperbolic spaces in which the logical information is associated with the boundary and discuss their potential interpretation. We explain how our codes could be applied to interesting classes of holographic dualities based on gravity-scalar theories (such as JT gravity) through toroidal reductions of hyperbolic spaces.