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Unitary Reflection Groups for Quantum Fault Tolerance

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Abstract:

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 [M. Planat and Ph. Jorrand, J. Phys. A: Math. Theor. 41, 182001 (2008)]. 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), the complex reflection groups G(2′,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 /4 phase gate and an entangling (braid) gate [L. H. Kauffman and S. J. Lomonaco, New J. of Phys. 6, 134 (2004)]. Links to the topological view of quantum computing, the lattice approach and the geometry of smooth cubic surfaces are discussed.
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  • Journal of Computational and Theoretical Nanoscience is an international peer-reviewed journal with a wide-ranging coverage, consolidates research activities in all aspects of computational and theoretical nanoscience into a single reference source. This journal offers scientists and engineers peer-reviewed research papers in all aspects of computational and theoretical nanoscience and nanotechnology in chemistry, physics, materials science, engineering and biology to publish original full papers and timely state-of-the-art reviews and short communications encompassing the fundamental and applied research.
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