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arXiv:1002.4287

Entangling gates in even Euclidean lattices such as the Leech lattice

Michel R. P. Planat

Published Feb 23, 2010• 1 citation
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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 $Λ_{24}$ 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}.

DOI
10.48550/arxiv.1002.4287

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The group of automorphisms of Euclidean (embedded in ) 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 ±1, with integer entries). When the degree n is equal to the number of basis elements of Λ, Aut(Λ) also acts on basis vectors and is generated with matrices B such that the sum of squared entries in a row is 1, 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 the same magnitude 1/4) and a Greenberger–Horne–Zeilinger-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. In addition to quantum computing, the work may be related to particle physics in the spirit of Planat et al (2011 Rep. Math. Phys. 66 39–51).
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 B3B_3B3​ and G2G_2G2​ (for single qubits), D5D_5D5​ and A4A_4A4​ (for two qubits), E7E_7E7​ and E6E_6E6​ (for three qubits), the complex reflection groups G(2l,2,5)G(2^l,2,5)G(2l,2,5) 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 π/4π/4π/4 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.
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 C±\pm±n . 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.
Balanced tripartite entanglement, the alternating group A4A_4A4​\\ and the Lie algebra sl(3,C)⊕u(1)sl(3,\mathbb{C})\oplus u(1)sl(3,C)⊕u(1)
I feature three important classes of three-qubit entangled states and their encoding into quantum gates, discrete groups and Lie algebras. States of the GHZ and W-type correspond to pure tripartite and bipartite entanglement, respectively. I introduce another generic class B of three-qubit states, that are equally balanced over two and three parties. I show how to realize the largest cristallographic group W(E8)W(E_8)W(E8​) in terms of three-qubit gates (with real entries) encoding states of type GHZ or W [M. Planat, {\it Clifford group dipoles and the enactment of Weyl/Coxeter group W(E8) by entangling gates}, Preprint 0904.3691 (quant-ph)]. Then, I describe a peculiar condensation of W(E8)W(E_8)W(E8​) to the four-letter alternating group A4A_4A4​, obtained from a chain of maximal subgroups. Group A4A_4A4​ is realized from two B-type generators and also corresponds to the Lie algebra sl(3,C)⊕u(1)sl(3,\mathbb{C})\oplus u(1)sl(3,C)⊕u(1). Applications are to particle and gravitation physics, and to the genetic code.
In quantum information context, the groups generated by Pauli spin matrices, and Dirac gamma matrices, are known as the single qubit Pauli group P, and two-qubit Pauli group P2, respectively. It has been found [M. Socolovsky, Int. J. Theor. Phys. 43, 1941 (2004)] that the CPT group of the Dirac equation is isomorphic to P. One introduces a two-qubit entangling orthogonal matrix S basically related to the CPT symmetry. With the aid of the two-qubit swap gate, the S matrix allows the generation of the three-qubit real Clifford group and, with the aid of the Toffoli gate, the Weyl group W(E8) is generated (M. Planat, Preprint 0904.3691). In this paper, one derives three-qubit entangling groups ˜ P and ˜ P2, isomorphic to the CPT group P and to the Dirac group P2, that are embedded into W(E8). One discovers a new class of pure theequbit quantum states with no-vanishing concurrence and three-tangle that we name CPT states. States of the GHZ and CPT families encode the new representation of the Dirac group and its CPT subgroup.
One learned from Gottesman-Knill theorem that the Clifford model of quantum computing \cite{Clark07} may be generated from a few quantum gates, the Hadamard, Phase and Controlled-Z gates, and efficiently simulated on a classical computer. We employ the group theoretical package GAP\cite{GAP} for simulating the two qubit Clifford group C2\mathcal{C}_2C2​. We already found that the symmetric group S(6), aka the automorphism group of the generalized quadrangle W(2), controls the geometry of the two-qubit Pauli graph \cite{Pauligraphs}. Now we find that the {\it inner} group Inn(C2)=C2/Center(C2){Inn}(\mathcal{C}_2)=\mathcal{C}_2/{Center}(\mathcal{C}_2)Inn(C2​)=C2​/Center(C2​) exactly contains two normal subgroups, one isomorphic to Z2×4\mathcal{Z}_2^{\times 4}Z2×4​ (of order 16), and the second isomorphic to the parent A′(6)A'(6)A′(6) (of order 5760) of the alternating group A(6). The group A′(6)A'(6)A′(6) stabilizes an {\it hexad} in the Steiner system S(3,6,22)S(3,6,22)S(3,6,22) attached to the Mathieu group M(22). Both groups A(6) and A′(6)A'(6)A′(6) have an {\it outer} automorphism group Z2×Z2\mathcal{Z}_2\times \mathcal{Z}_2Z2​×Z2​, a feature we associate to two-qubit quantum entanglement.