Permutationally Invariant Codes for Quantum Error Correction

A permutationally invariant n-bit code for quantum error correction can be realized as a subspace stabilized by the non-Abelian group S_n. The code corresponds to bases for the trivial representation, and all other irreducible representations, both those of higher dimension and orthogonal bases for the trivial representation, are available for error correction. A number of new (non-additive) binary codes are obtained, including two new 7-bit codes and a large family of new 9-bit codes. It is shown that the degeneracy arising from permutational symmetry facilitates the correction of certain types of two-bit errors. The correction of two-bit errors of the same type is considered in detail, but is shown not to be compatible with single-bit error correction using 9-bit codes.