The group of automorphisms of Euclidean (embedded in Rn) dense
lattices such as the root lattices D4 and E8, the Barnes-Wall lattice
BW16, the unimodular lattice D12+ and the Leech lattice
Λ24 may be generated by entangled quantum gates of the corresponding
dimension. These (real) gates/lattices are useful for quantum error correction:
for instance, the two and four-qubit real Clifford groups are the automorphism
groups of the lattices D4 and BW16, respectively, and the three-qubit
real Clifford group is maximal in the Weyl group W(E8). Technically, the
automorphism group Aut(Λ) of the lattice Λ is the set of
orthogonal matrices B such that, following the conjugation action by the
generating matrix of the lattice, the output matrix is unimodular (of
determinant ±1, with integer entries). When the degree n is equal to the
number of basis elements of Λ, then Aut(Λ) also acts on basis
vectors and is generated with matrices B such that the sum of squared entries
in a row is one, i.e. B may be seen as a quantum gate. For the dense lattices
listed above, maximal multipartite entanglement arises. In particular, one
finds a balanced tripartite entanglement in E8 (the two- and three- tangles
have equal magnitude 1/4) and a GHZ type entanglement in BW16. In this
paper, we also investigate the entangled gates from D12+ and
Λ24, by seeing them as systems coupling a qutrit to two- and
three-qubits, respectively. Apart from quantum computing, the work may be
related to particle physics in the spirit of \cite{PLS2010}.Comment: 11 pages, second updated versio