The Fourier Transform Approach to Quantum Coding

Abstract: Quantum codes are subspaces of the state space of a quantum system that are used to protect quantum information. Some common classes of quantum codes are stabilizer (or additive) codes, non-stabilizer (or non-additive) codes obtained from stabilizer codes, and Clifford codes. We analyze these in the framework of the Fourier inversion formula on a group algebra, the group in question being a subgroup of the error group considered. We study other possible code spaces that may be obtained via such an approach, obtaining codes that are the direct sums of translates of Clifford codes, and more general codes obtained from idempotents in the transform domain. We derive necessary and sufficient conditions for error detection by direct sums of translates of Clifford codes, and provide an example using an error group with non-Abelian index group.