Unitary Reflection Groups for Quantum Fault Tolerance

作者：Planat Michel^*; Kibler Maurice

来源：Journal of Computational and Theoretical Nanoscience, 2010, 7(9): 1759-1770.

DOI：10.1166/jctn.2010.1541

• Quantum Computing
• Group Extensions
• Pauli and Clifford Groups
• Group of Automorphisms
• Mutually Unbiased Bases

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 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(I), 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 [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.

• 出版日期2010-9