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Journal of Computational and Theoretical Nanoscience, 7 (2010) 1759-1770
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Unitary reflection groups for quantum fault tolerance
Michel Planat1, Maurice R. Kibler2
(2010)

This paper explores the representation of quantum computing in terms of unitary reflections (unitary transformations that leave 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, {\it J Phys A: Math Theor} {\bf 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 $B_3$ and $G_2$ (for single qubits), $D_5$ and $A_4$ (for two qubits), $E_7$ and $E_6$ (for three qubits), the complex reflection groups $G(2^l,2,5)$ and groups No $9$ and $31$ in the Shephard-Todd list. The relevant fault tolerant subsets of the Clifford groups (the Bell groups) are generated by the Hadamard gate, the $\pi/4$ phase gate and an entangling (braid) gate [Kauffman L~H and Lomonaco S~J 2004 {\it New J. of Phys.} {\bf 6}, 134]. Links to the topological view of quantum computing, the lattice approach and the geometry of smooth cubic surfaces are discussed.
1 :  FEMTO-ST - Franche-Comté Électronique Mécanique, Thermique et Optique - Sciences et Technologies
2 :  IPNL - Institut de Physique Nucléaire de Lyon
Physique/Physique Quantique

Mathématiques/Physique mathématique

Physique/Physique mathématique

Mathématiques/Théorie des groupes
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