From Extraspecial Two-Groups To GHZ States

In this paper we explore natural connections among extraspecial 2-groups, almost-complex structures, unitary representations of the braid group and the Greenberger-Horne-Zeilinger (GHZ) states. We first present new representations of extraspecial 2-groups in terms of almost-complex structures and use them to derive new unitary braid representations as extensions of representations of the extraspecial 2-groups by the symmetric group. A few subtleties related to the correspondence between the unitary braid representations and the GHZ states (particularly those for an odd number of qubits) are clarified. We also discuss Yang–Baxterization of the new braid group representations and unitary evolution of the GHZ states. Our study suggests that the unitary braiding quantum gates may play an important role, through extraspecial 2-groups, in quantum error correction and topological quantum computing. PACS numbers: 02.10.Kn, 03.65.Ud, 03.67.Lx MSC 2000 numbers: 81P68 (Primary) 20F36, 20C35, 81R05 (Secondary)