偶数欧几里德格（如Leech格）中的纠缠门
Entangling gates in even Euclidean lattices such as the Leech lattice
摘要：
欧几里德（嵌入在$\mathbb{R}^n$中）稠密晶格的自同构群，诸如根晶格$D_4$和$E_8$、巴恩斯-沃尔晶格$BW_{16}$、幺模晶格$D_{12}^+$和利奇晶格$Λ_{24}$，可以由相应维度的纠缠量子门生成。这些（实）门/格对于量子纠错是有用的：例如，两个和四个量子位的实克利福德群分别是格$D_4$和$BW_{16}$的自同构群，并且三个量子位的实克利福德群在Weyl群$W(E_8)$中是最大的。从技术上讲，格$Λ$的自同构群$Aut(Λ)$是正交矩阵$B$的集合，使得在格的生成矩阵的共轭作用之后，输出矩阵是幺模的（行列式$\pm 1$，具有整数项）。当次数$n$等于$Λ$的基元素的数量时，则$Aut(Λ)$也作用于基向量，并且用矩阵$B$生成，使得行中的平方项的和为1，即$B$可以被视为量子门。对于上面列出的稠密晶格，出现了最大的多体纠缠。特别地，在$E_8$中发现了平衡的三重纠缠（两个和三个纠缠具有相等的幅度1/4），在BW $_{16}$中发现了GHZ型纠缠。在本文中，我们还研究了$D_{12}^+$和$Λ_{24}$的纠缠门，通过将它们看作分别将一个qutrit耦合到两个和三个qubit的系统。除了量子计算，这项工作可能与\cite{PLS2010}精神下的粒子物理学有关。
英文摘要：
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 $Λ_{24}$ 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 $Aut(Λ)$ 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 $\pm 1$, 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 $E_8$ (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 $Λ_{24}$, 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}.
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