Entangling gates in even Euclidean lattices such as the Leech lattice

Abstract

The group of automorphisms of Euclidean (embedded in $\mathbb{R}^n$) dense lattices such as the root lattices $D_4$ and $E_8$, the Barnes-Wall lattice $BW_{16}$, the unimodular lattice $D_{12}^+$ and the Leech lattice $\Lambda_{24}$ 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 $D_4$ and $BW_{16}$, respectively, and the three-qubit real Clifford group is maximal in the Weyl group $W(E_8)$. Technically, the automorphism group $Aut(\Lambda)$ of the lattice $\Lambda$ 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 $\pm 1$, with integer entries). When the degree $n$ is equal to the number of basis elements of $\Lambda$, then $Aut(\Lambda)$ 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 $E_8$ (the two- and three- tangles have equal magnitude 1/4) and a GHZ type entanglement in BW$_{16}$. In this paper, we also investigate the entangled gates from $D_{12}^+$ and $\Lambda_{24}$, 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}.