The Extensions of the Generalized Quadrangle of Order (3, 9)

AbstractIt is shown that there is only one extension of GQ(3, 9) namely the one admitting the sporadic simple groupMcLas a flag-transitive automorphism group. The proof depends on a computer calculation

On highly regular strongly regular graphs

In this paper we unify several existing regularity conditions for graphs, including strong regularity, $k$-isoregularity, and the $t$-vertex condition. We develop an algebraic composition/decomposition theory of regularity conditions. Using our theoretical results we show that a family of non rank 3 graphs known to satisfy the $7$-vertex condition fulfills an even stronger condition, $(3,7)$-regularity (the notion is defined in the text). Derived from this family we obtain a new infinite family of non rank $3$ strongly regular graphs satisfying the $6$-vertex condition. This strengthens and generalizes previous results by Reichard.Comment: 29 page

On hyperovals of polar spaces

We derive lower and upper bounds for the size of a hyperoval of a finite polar space of rank 3. We give a computer-free proof for the uniqueness, up to isomorphism, of the hyperoval of size 126 of H(5, 4) and prove that the near hexagon E-3 has up to isomorphism a unique full embedding into the dual polar space DH(5, 4)

Doily as Subgeometry of a Set of Nonunimodular Free Cyclic Submodules

It is shown that there exists a particular associative ring with unity of order 16 such that the relations between nonunimodular free cyclic submodules of its two-dimensional free left module can be expressed in terms of the structure of the generalized quadrangle of order two. Such a doily-centered geometric structure is surmised to be of relevance for quantum information.Comment: 5 pages, 3 figure

On group theory for quantum gates and quantum coherence

Finite group extensions offer a natural language to quantum computing. In a nutshell, one roughly describes the action of a quantum computer as consisting of two finite groups of gates: error gates from the general Pauli group P and stabilizing gates within an extension group C. In this paper one explores the nice adequacy between group theoretical concepts such as commutators, normal subgroups, group of automorphisms, short exact sequences, wreath products... and the coherent quantum computational primitives. The structure of the single qubit and two-qubit Clifford groups is analyzed in detail. As a byproduct, one discovers that M20, the smallest perfect group for which the commutator subgroup departs from the set of commutators, underlies quantum coherence of the two-qubit system. One recovers similar results by looking at the automorphisms of a complete set of mutually unbiased bases.Comment: 10 pages, to appear in J Phys A: Math and Theo (Fast Track Communication

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 $B_3$ 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.Comment: new version for the Journal of Computational and Theoretical Nanoscience, focused on "Technology Trends and Theory of Nanoscale Devices for Quantum Applications