Generalized Clifford groups and simulation  of associated quantum circuits (pp0106-0126)
          Sean Clark, Richard Jozsa, and Noah Linden
          doi: https://doi.org/10.26421/QIC8.1-2-8
Abstracts: Quantum computations starting with computational basis states and involving only Clifford operations, are classically simulable despite the fact that they generate highly entangled states; this is the content of the Gottesman-Knill theorem. Here we isolate the ingredients of the theorem and provide generalisations of some of them with the aim of identifying new classes of simulable quantum computations. In the usual construction, Clifford operations arise as projective normalisers of the first and second tensor powers of the Pauli group. We consider replacing the Pauli group by an arbitrary finite subgroup G of U(d). In particular we seek G such that G X G has an entangling normaliser. Via a generalisation of the Gottesman-Knill theorem the resulting normalisers lead to classes of quantum circuits that can be classically efficiently simulated. For the qubit case d = 2 we exhaustively treat all finite irreducible subgroups of U(2) and find that the only ones (up to unitary equivalence and trivial phase extensions) with entangling normalisers are the groups generated by X and the nth root of Z for n belongs to N.
Key words:  quantum circuits, Clifford operations