Approximate Quantum Error Protection with Chiral Topological Order Edges

It is well motivated to search for naturally occurring many-body systems that can robustly store quantum information. Non-chiral topological phases with gapped edges, such as the Toric code and the surface code, are well-known examples. Here we show that even the chiral edges of two-dimensional topologically ordered systems—usually thought to be gapless and fragile—can realize an approximate quantum error-correcting code with a quasi-local decoder. Using the chiral conformal field theory (CFT) description of the edge and the entanglement properties of the bulk, we identify a natural logical subspace spanned by edge-CFT primary and descendant states. We then prove the existence of a locality-preserving completely positive trace-preserving (CPTP) recovery map for localized noise. Importantly, we show that our code on a cylindrical geometry exhibits enhanced robustness compared to recently studied 1+1D CFT codes under weak noise. Our construction can be viewed as a hybrid between topological and CFT codes, revealing new connections between gapless edge physics and quantum information protection.