52 research outputs found
Some new results on the self-dual [120,60,24] code
The existence of an extremal self-dual binary linear code of length 120 is a
long-standing open problem. We continue the investigation of its automorphism
group, proving that automorphisms of order 30 and 57 cannot occur. Supposing
the involutions acting fixed point freely, we show that also automorphisms of
order 8 cannot occur and the automorphism group is of order at most 120, with
further restrictions. Finally, we present some necessary conditions for the
existence of the code, based on shadow and design theory.Comment: 23 pages, 6 tables, to appear in Finite Fields and Their Application
Permutation Symmetry Determines the Discrete Wigner Function
The Wigner function provides a useful quasiprobability representation of
quantum mechanics, with applications in various branches of physics. Many nice
properties of the Wigner function are intimately connected with the high
symmetry of the underlying operator basis composed of phase point operators:
any pair of phase point operators can be transformed to any other pair by a
unitary symmetry transformation. We prove that, in the discrete scenario, this
permutation symmetry is equivalent to the symmetry group being a unitary
2-design. Such a highly symmetric representation can only appear in odd prime
power dimensions besides dimensions 2 and 8. It suffices to single out a unique
discrete Wigner function among all possible quasiprobability representations.
In the course of our study, we show that this discrete Wigner function is
uniquely determined by Clifford covariance, while no Wigner function is
Clifford covariant in any even prime power dimension.Comment: 5+2 pages, connection with unitary 2-designs added, accepted by Phys.
Rev. Lett. as Editors' Suggestio
Clifford groups of quantum gates, BN-pairs and smooth cubic surfaces
The recent proposal (M Planat and M Kibler, Preprint 0807.3650 [quantph]) of
representing Clifford quantum gates in terms of unitary reflections is
revisited. In this essay, the geometry of a Clifford group G is expressed as a
BN-pair, i.e. a pair of subgroups B and N that generate G, is such that
intersection H = B \cap N is normal in G, the group W = N/H is a Coxeter group
and two extra axioms are satisfied by the double cosets acting on B. The
BN-pair used in this decomposition relies on the swap and match gates already
introduced for classically simulating quantum circuits (R Jozsa and A Miyake,
Preprint arXiv:0804.4050 [quant-ph]). The two- and three-qubit steps are
related to the configuration with 27 lines on a smooth cubic surface.Comment: 7 pages, version to appear in Journal of Physics A: Mathematical and
Theoretical (fast track communications
Linear groups and distance-transitive graphs
A detailed treatment is given of the graphs on which a group with simple socle PSL(n, q) acts primitively and distance-transitively
Linear groups and distance-transitive graphs
A detailed treatment is given of the graphs on which a group with simple socle PSL(n, q) acts primitively and distance-transitively
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