| This paper explores the representation of quantum computing in terms of unitary reflections (unitary transformations that leaves invariant a hyperplane of a vector space). The symmetries of qubit systems are found to be supported by Euclidean real reflections (i.e., Coxeter groups) or by specific imprimitive reflection groups, introduced (but not named) in a recent paper [Planat M and Jorrand Ph 2008, J Phys A: Math Theor 41, 182001]. The automorphisms of multiple qubit systems are found to relate to some Clifford operations once the corresponding group of reflections is identified. For a short list, one may point out the Coxeter systems of type B3 and G2 (for single qubits), D5 and A4 (for two qubits), E7 and E6 (for three qubits), and the complex reflection groups G(2l, 2, 5). The relevant fault tolerant groups of reflections (the Bell groups) are generated, as subgroups of the Clifford groups, by the Hadamard gate, the $\pi$/4 phase gate and an entangling (braid) gate [Kauffman L H and Lomonaco S J 2004 New J. of Phys. 6, 134]. Links to the topological view of quantum computing, to the lattice approach and to the geometry of smooth cubic surfaces are discussed. |
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