Abstract: There is a rich connection between classical and quantum codes and holographic correspondence connecting 2d CFTs and abelian 3d Chern-Simons theories. In the 3d language the codes emerge as a way to parametrize condensable anyons. Upon condensation 3d topological field theory gives rise to 2d CFT at the boundary. This provides a way to construct 2d CFTs from codes - the so called "code CFTs." This construction of code CFT has a natural interpretation in terms of a CSS quantum code (defined in terms of the original classical code, defining the CFT). From the holographic point of view a particularly interesting question is that of an ensemble of codes and associated code CFTs. This ensemble of boundary theories is holographically dual to ``Chern-Simons gravity", a topological field theory that sums over 3d topologies. From the quantum information point of view, this relation is evaluating average over the stabilizer states, invariant under the action of (a subgroup) of Clifford group. This provides an explicit connection between models of quantum gravity and contemporary problems of quantum information theory.
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