There is a rich web of connections between classical error-correcting codes, Euclidean lattices, and chiral conformal field theories.  I will explain an analogous relationship between quantum error-correcting codes, Lorentzian lattices, and nonchiral CFTs arising from toroidal compactifications of strings.  Specifically, I will show how quantum stabilizer codes are embedded in the operator algebras of associated Narain CFTs.   With respect to “code” CFTs, the constraints of modular invariance reduce to simple algebraic equations.  

Solving these equations provides many examples of physically distinct CFTs with the same spectrum; we also find many solutions that do not correspond to any code CFT, and likely do not correspond to any CFT at all.  This exposes a crucial limitation of the conformal bootstrap program – that in general, a solution of the bootstrap equations implies neither the existence nor the uniqueness of a corresponding CFT.