Simple family of nonadditive quantum codes

Abstract

Most known quantum codes are additive, meaning the code can be described as the simultaneous eigenspace of an Abelian subgroup of the Pauli group. While in some scenarios such codes are strictly suboptimal, very little is understood about how to construct nonadditive codes with good performance. Here we present a family of distance 2 nonadditive quantum codes for all odd block lengths n, that has a particularly simple form. Our codes detect single qubit errors (or correct single qubit erasures) while encoding a higher dimensional space than is possible with an additive code or, for n 11, any previous codes. We exhibit the encoding circuits and automorphism group for our codes as well. © 2007 The American Physical Society.