Abstract Error Groups Via Jones Unitary Braid Group Representations at q=i

In this paper, we classify a type of abstract groups by the central products of dihedral groups and quaternion groups. We recognize them as abstract error groups which are often not isomorphic to the Pauli groups in the literature. We show the corresponding nice error bases equivalent to the Pauli error bases modulo phase factors. The extension of these abstract groups by the symmetric group are finite images of the Jones unitary representations (or modulo a phase factor) of the braid group at q=i or r=4. We hope this work can finally lead to new families of quantum error correction codes via the representation theory of the braid group.