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

The Fourier Transform Approach to Quantum Coding