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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
Coding Theory and Algebraic Combinatorics
This chapter introduces and elaborates on the fruitful interplay of coding
theory and algebraic combinatorics, with most of the focus on the interaction
of codes with combinatorial designs, finite geometries, simple groups, sphere
packings, kissing numbers, lattices, and association schemes. In particular,
special interest is devoted to the relationship between codes and combinatorial
designs. We describe and recapitulate important results in the development of
the state of the art. In addition, we give illustrative examples and
constructions, and highlight recent advances. Finally, we provide a collection
of significant open problems and challenges concerning future research.Comment: 33 pages; handbook chapter, to appear in: "Selected Topics in
Information and Coding Theory", ed. by I. Woungang et al., World Scientific,
Singapore, 201
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