Clifford quantum computer and the Mathieu groups

One learned from Gottesman-Knill theorem that the Clifford model of quantum computing \cite{Clark07} may be generated from a few quantum gates, the Hadamard, Phase and Controlled-Z gates, and efficiently simulated on a classical computer. We employ the group theoretical package GAP\cite{GAP} for simulating the two qubit Clifford group $\mathcal{C}_2$. We already found that the symmetric group S(6), aka the automorphism group of the generalized quadrangle W(2), controls the geometry of the two-qubit Pauli graph \cite{Pauligraphs}. Now we find that the {\it inner} group ${Inn}(\mathcal{C}_2)=\mathcal{C}_2/{Center}(\mathcal{C}_2)$ exactly contains two normal subgroups, one isomorphic to $\mathcal{Z}_2^{\times 4}$ (of order 16), and the second isomorphic to the parent $A'(6)$ (of order 5760) of the alternating group A(6). The group $A'(6)$ stabilizes an {\it hexad} in the Steiner system $S(3,6,22)$ attached to the Mathieu group M(22). Both groups A(6) and $A'(6)$ have an {\it outer} automorphism group $\mathcal{Z}_2\times \mathcal{Z}_2$, a feature we associate to two-qubit quantum entanglement.Comment: version for the journal Entrop

Resolving sets for Johnson and Kneser graphs

A set of vertices $S$ in a graph $G$ is a {\em resolving set} for $G$ if, for any two vertices $u,v$, there exists $x\in S$ such that the distances $d(u,x) \neq d(v,x)$. In this paper, we consider the Johnson graphs $J(n,k)$ and Kneser graphs $K(n,k)$, and obtain various constructions of resolving sets for these graphs. As well as general constructions, we show that various interesting combinatorial objects can be used to obtain resolving sets in these graphs, including (for Johnson graphs) projective planes and symmetric designs, as well as (for Kneser graphs) partial geometries, Hadamard matrices, Steiner systems and toroidal grids.Comment: 23 pages, 2 figures, 1 tabl

Low-Density Parity-Check Codes From Transversal Designs With Improved Stopping Set Distributions

This paper examines the construction of low-density parity-check (LDPC) codes from transversal designs based on sets of mutually orthogonal Latin squares (MOLS). By transferring the concept of configurations in combinatorial designs to the level of Latin squares, we thoroughly investigate the occurrence and avoidance of stopping sets for the arising codes. Stopping sets are known to determine the decoding performance over the binary erasure channel and should be avoided for small sizes. Based on large sets of simple-structured MOLS, we derive powerful constraints for the choice of suitable subsets, leading to improved stopping set distributions for the corresponding codes. We focus on LDPC codes with column weight 4, but the results are also applicable for the construction of codes with higher column weights. Finally, we show that a subclass of the presented codes has quasi-cyclic structure which allows low-complexity encoding.Comment: 11 pages; to appear in "IEEE Transactions on Communications

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

The Perfect Binary One-Error-Correcting Codes of Length 15: Part II--Properties

A complete classification of the perfect binary one-error-correcting codes of length 15 as well as their extensions of length 16 was recently carried out in [P. R. J. \"Osterg{\aa}rd and O. Pottonen, "The perfect binary one-error-correcting codes of length 15: Part I--Classification," IEEE Trans. Inform. Theory vol. 55, pp. 4657--4660, 2009]. In the current accompanying work, the classified codes are studied in great detail, and their main properties are tabulated. The results include the fact that 33 of the 80 Steiner triple systems of order 15 occur in such codes. Further understanding is gained on full-rank codes via switching, as it turns out that all but two full-rank codes can be obtained through a series of such transformations from the Hamming code. Other topics studied include (non)systematic codes, embedded one-error-correcting codes, and defining sets of codes. A classification of certain mixed perfect codes is also obtained.Comment: v2: fixed two errors (extension of nonsystematic codes, table of coordinates fixed by symmetries of codes), added and extended many other result