51,199 research outputs found
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
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
The invariants of the Clifford groups
The automorphism group of the Barnes-Wall lattice L_m in dimension 2^m (m not
3) is a subgroup of index 2 in a certain ``Clifford group'' C_m (an
extraspecial group of order 2^(1+2m) extended by an orthogonal group). This
group and its complex analogue CC_m have arisen in recent years in connection
with the construction of orthogonal spreads, Kerdock sets, packings in
Grassmannian spaces, quantum codes, Siegel modular forms and spherical designs.
In this paper we give a simpler proof of Runge's 1996 result that the space
of invariants for C_m of degree 2k is spanned by the complete weight
enumerators of the codes obtained by tensoring binary self-dual codes of length
2k with the field GF(2^m); these are a basis if m >= k-1. We also give new
constructions for L_m and C_m: let M be the Z[sqrt(2)]-lattice with Gram matrix
[2, sqrt(2); sqrt(2), 2]. Then L_m is the rational part of the mth tensor power
of M, and C_m is the automorphism group of this tensor power. Also, if C is a
binary self-dual code not generated by vectors of weight 2, then C_m is
precisely the automorphism group of the complete weight enumerator of the
tensor product of C and GF(2^m). There are analogues of all these results for
the complex group CC_m, with ``doubly-even self-dual code'' instead of
``self-dual code''.Comment: Latex, 24 pages. Many small improvement
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
- …