609 research outputs found
Unitary reflection groups for quantum fault tolerance
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 and (for
single qubits), and (for two qubits), and (for three
qubits), the complex reflection groups 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 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.Comment: new version for the Journal of Computational and Theoretical
Nanoscience, focused on "Technology Trends and Theory of Nanoscale Devices
for Quantum Applications
Clifford group dipoles and the enactment of Weyl/Coxeter group W(E8) by entangling gates
Peres/Mermin arguments about no-hidden variables in quantum mechanics are
used for displaying a pair (R, S) of entangling Clifford quantum gates, acting
on two qubits. From them, a natural unitary representation of Coxeter/Weyl
groups W(D5) and W(F4) emerges, which is also reflected into the splitting of
the n-qubit Clifford group Cn into dipoles Cn . The union of the
three-qubit real Clifford group C+ 3 and the Toffoli gate ensures a orthogonal
representation of the Weyl/Coxeter group W(E8), and of its relatives. Other
concepts involved are complex reflection groups, BN pairs, unitary group
designs and entangled states of the GHZ family.Comment: version revised according the recommendations of a refere
The complex Lorentzian Leech lattice and the bimonster
We find 26 reflections in the automorphism group of the the Lorentzian Leech
lattice L over Z[exp(2*pi*i/3)] that form the Coxeter diagram seen in the
presentation of the bimonster. We prove that these 26 reflections generate the
automorphism group of L. We find evidence that these reflections behave like
the simple roots and the vector fixed by the diagram automorphisms behaves like
the Weyl vector for the refletion group.Comment: 24 pages, 3 figures, revised and proof corrected. Some small results
added. to appear in the Journal of Algebr
Automorphisms and opposition in twin buildings
We show that every automorphism of a thick twin building interchanging the
halves of the building maps some residue to an opposite one. Furthermore we
show that no automorphism of a locally finite 2-spherical twin building of rank
at least 3 maps every residue of one fixed type to an opposite. The main
ingredient of the proof is a lemma that states that every duality of a thick
finite projective plane admits an absolute point, i.e., a point mapped onto an
incident line. Our results also hold for all finite irreducible spherical
buildings of rank at least 3, and as a consequence we deduce that every
involution of a thick irreducible finite spherical building of rank at least 3
has a fixed residue
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