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Physica Scripta

Multipartite entanglement arising from dense Euclidean lattices in dimensions 4–24

Michel R. P. Planat

Published Feb 1, 2012• 4 citations
Publisher

Abstract

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).

DOI
10.1088/0031-8949/2012/t147/014025

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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).
The group of automorphisms of Euclidean (embedded in Rn\mathbb{R}^nRn) dense lattices such as the root lattices D4D_4D4​ and E8E_8E8​, the Barnes-Wall lattice BW16BW_{16}BW16​, the unimodular lattice D12+D_{12}^+D12+​ and the Leech lattice Λ24Λ_{24}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 D4D_4D4​ and BW16BW_{16}BW16​, respectively, and the three-qubit real Clifford group is maximal in the Weyl group W(E8)W(E_8)W(E8​). Technically, the automorphism group Aut(Λ)Aut(Λ)Aut(Λ) of the lattice ΛΛΛ is the set of orthogonal matrices BBB such that, following the conjugation action by the generating matrix of the lattice, the output matrix is unimodular (of determinant ±1\pm 1±1, with integer entries). When the degree nnn is equal to the number of basis elements of ΛΛΛ, then Aut(Λ)Aut(Λ)Aut(Λ) also acts on basis vectors and is generated with matrices BBB such that the sum of squared entries in a row is one, i.e. BBB 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 E8E_8E8​ (the two- and three- tangles have equal magnitude 1/4) and a GHZ type entanglement in BW16_{16}16​. In this paper, we also investigate the entangled gates from D12+D_{12}^+D12+​ and Λ24Λ_{24}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}.
Abstract. We discuss three important classes of three-qubit entangled states and their encoding into quantum gates, finite groups and Lie algebras. States of the GHZ and W-type correspond to pure tripartite and bipartite entanglement, respectively. We introduce another generic class B of three-qubit states, that have balanced entanglement over two and three parties. We show how to realize the largest cristallographic group W(E8) in terms of three-qubit gates (with real entries) encoding states of type GHZ or W [M. Planat, Clifford group dipoles and the enactment of Weyl/Coxeter group W(E8) by entangling gates, Preprint 0904.3691 (quant-ph)]. Then, we describe a peculiar “condensation ” of W(E8) into the four-letter alternating group A4, obtained from a chain of maximal subgroups. Group A4 is realized from two B-type generators and found to correspond to the Lie algebra sl(3, C) ⊕ u(1). Possible applications of our findings to particle physics and the structure of genetic code are also mentioned. 1.
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.
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.
Universal quantum computers are expected to generate arbitrary complex quantum states of logical qubits encoded in many physical qubits. This capability hinges on a fault-tolerantly implemented universal gate set, which no single quantum error-correction code admits transversally but which becomes accessible by joining complementary codes via lattice surgery. Here we report on the experimental generation and certification of logical genuine multipartite entanglement in a trapped-ion quantum processor using a transversally implemented universal logical gate set. The gate set is accessed via lattice surgery across two different codes and comprises a Hadamard gate on a four-qubit surface code and a doubly controlled Pauli-ZZZ (CCZ‾\overline{\mathrm{CCZ}}CCZ) gate on an eight-qubit 3D colour code. To showcase this lattice-surgery toolbox, we generate both stabiliser (Greenberger-Horne-Zeilinger) and non-stabiliser (∣CCZ‾⟩|\overline{\mathrm{CCZ}}\rangle∣CCZ⟩) states of three logical qubits and verify their genuine multipartite entanglement--a form of correlation beyond statistical mixtures of bipartite entanglement across any bipartition. We further use these cross-code primitives to demonstrate arbitrary rotations of single logical qubits via a CCZ‾\overline{\mathrm{CCZ}}CCZ-based resource gadget accessing the full universal gate set through lattice surgery. Together, these demonstrations showcase the core building blocks of an architecture for fault-tolerant quantum computation and its ability to generate complex logical quantum states.