arXiv:0904.3691

Clifford group dipoles and the enactment of Weyl/Coxeter group W(E8) by entangling gates

Michel R. P. Planat

Published Apr 23, 2009• 0 citations

Abstract

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.

DOI: 10.48550/arxiv.0904.3691

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arXiv:0904.3691

Clifford group dipoles and the enactment of Weyl/Coxeter group W(E8) by entangling gates

Michel R. P. Planat

Published Apr 23, 2009• 0 citations

arXiv HTML PDF

Abstract

\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.

DOI: 10.48550/arxiv.0904.3691

Related papers

Clifiord Group Dipoles and the Enactment of Weyl/Coxeter Group W(E 8 ) by Entangling Gates

Michel R. P. Planat

Peres/Mermin arguments about no-hidden variables in quantum mechanics are used for displaying a pair (R;S) of entangling Clifiord quantum gates, acting on two qubits. From them, a unitary realization of Coxeter/Weyl groups W(D5) and W(F4) emerges, which is also re∞ected into the splitting of the nqubit Clifiord group Cn into subgroup dipoles C § . Generators of the three-qubit real Clifiord group C + 3 with the Tofioli gate ensures a orthogonal realization of the Weyl/Coxeter group W(E8), and of its relatives. Other concepts involved are complex re∞ection groups, BN pairs, unitary group designs and entangled states of the GHZ, W and chain families.Peres/Mermin arguments about no-hidden variables in quantum mechanics are used for displaying a pair (R;S) of entangling Clifiord quantum gates, acting on two qubits. From them, a unitary realization of Coxeter/Weyl groups W(D5) and W(F4) emerges, which is also re∞ected into the splitting of the nqubit Clifiord group Cn into subgroup dipoles C § . Generators of the three-qubit real Clifiord group C + 3 with the Tofioli gate ensures a orthogonal realization of the Weyl/Coxeter group W(E8), and of its relatives. Other concepts involved are complex re∞ection groups, BN pairs, unitary group designs and entangled states of the GHZ, W and chain families.

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Three-qubit entangled embedding of CPT and Dirac groups within E8 Weyl group

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Dans un premier temps, je pr´esenterais bri`evement comment David Mermin rend compte des aspects paradoxaux de la r´ealit´e quantique, `a partir de la non-compatibilit´e alg´ebrique des r`egles pour les valeurs propres (mesures quantiques) et pour les vecteurs propres (´etats quantiques) d'ensembles d'observables qui ne commutent pas tous. Je montrerai alors comment on peut transf´erer un tel questionnement `a un groupe `a sym´etrie octahedrale et `a ses invariants. Ensuite, j'introduirais au calcul quantique, o`u les portes unitaires sont les ´el´ements d'un groupe ad´equat. On cherche `a contenir les erreurs du calcul, repr´esent´ees par les produits tensoriels des matrices de Pauli, dans un sous-groupe fini de portes quantiques, ´el´ements du groupe unitaire complexe U(2n) `a n qubits, et `a repr´esenter au mieux toute matrice unitaire (universalit´e du calcul), d'o`u le groupe de Clifford et sa relation au groupe symplectique Sp(2n, 2). La d´ecomposition de Sp(2n, 2) en ses sous-groupes orthogonaux (extrasp´eciaux) se refl`ete en une d´ecomposition du groupe de Clifford en deux dipoles, dont les ´el´ements rel`event de plusieurs concepts int´eressants: les groupes de r´eflections complexes, les designs unitaires, ainsi que l'intrication quantique. Je les introduirai en d´etail dans le s´eminaire. Puis je montrerai que le groupe de Coxeter/Weyl fini le plus grand W(E8), d'ordre 696 729 600, peux-ˆetre repr´esent´e par des portes quantiques r´eelles `a trois qubits, dont l'une au moins, la porte de Toffoli, est ´etrang`ere au groupe de Clifford. La structure de W(E8), ainsi repr´esent´ee, est intimement li´ee au ´etats GHZ, et sa d´ecomposition en sous-groupes maximaux est li´ee aux ´etats bipartites W ou aux ´etats `a intrication en chaˆıne. Quelques perspectives pour le calcul quantique et l'unification de la physique seront discut´ees. Bibliographie Clifford group dipoles and the enactment of Weyl/Coxeter group W(E8) by entangling gates. Preprint 0904.3691Dans un premier temps, je pr´esenterais bri`evement comment David Mermin rend compte des aspects paradoxaux de la r´ealit´e quantique, `a partir de la non-compatibilit´e alg´ebrique des r`egles pour les valeurs propres (mesures quantiques) et pour les vecteurs propres (´etats quantiques) d'ensembles d'observables qui ne commutent pas tous. Je montrerai alors comment on peut transf´erer un tel questionnement `a un groupe `a sym´etrie octahedrale et `a ses invariants. Ensuite, j'introduirais au calcul quantique, o`u les portes unitaires sont les ´el´ements d'un groupe ad´equat. On cherche `a contenir les erreurs du calcul, repr´esent´ees par les produits tensoriels des matrices de Pauli, dans un sous-groupe fini de portes quantiques, ´el´ements du groupe unitaire complexe U(2n) `a n qubits, et `a repr´esenter au mieux toute matrice unitaire (universalit´e du calcul), d'o`u le groupe de Clifford et sa relation au groupe symplectique Sp(2n, 2). La d´ecomposition de Sp(2n, 2) en ses sous-groupes orthogonaux (extrasp´eciaux) se refl`ete en une d´ecomposition du groupe de Clifford en deux dipoles, dont les ´el´ements rel`event de plusieurs concepts int´eressants: les groupes de r´eflections complexes, les designs unitaires, ainsi que l'intrication quantique. Je les introduirai en d´etail dans le s´eminaire. Puis je montrerai que le groupe de Coxeter/Weyl fini le plus grand W(E8), d'ordre 696 729 600, peux-ˆetre repr´esent´e par des portes quantiques r´eelles `a trois qubits, dont l'une au moins, la porte de Toffoli, est ´etrang`ere au groupe de Clifford. La structure de W(E8), ainsi repr´esent´ee, est intimement li´ee au ´etats GHZ, et sa d´ecomposition en sous-groupes maximaux est li´ee aux ´etats bipartites W ou aux ´etats `a intrication en chaˆıne. Quelques perspectives pour le calcul quantique et l'unification de la physique seront discut´ees. Bibliographie Clifford group dipoles and the enactment of Weyl/Coxeter group W(E8) by entangling gates. Preprint 0904.3691

Jun 11, 20090 citations

\pm

B_3

and

G_2

(for single qubits),

D_5

and

A_4

(for two qubits),

E_7

and

E_6

(for three qubits), the complex reflection groups

G(2^l,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/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.

Clifford group {\it dipoles}

\mathcal{C}_n^{\pm}

are subgroups of the

n

-qubit Clifford group

\mathcal{C}_n

defined as (non-split) extensions of the extraspecial subgroups

E_{2n+1}^{\pm}

of the Pauli error group

\mathcal{P}_n

. The subgroups

\mathcal{C}_n^{\pm}:=E_{2n+1}^{\pm}.\Omega^{\pm}(2n,2)

involve pairs of orthogonal subgroups

\Omega^{\pm}(2n,2)

of the symplectic group

Sp(2n,2)

. They are of special relevance to the geometry of quantum error correction, and encompass several concepts proposed so far, including complex reflection groups, BN-pairs in the Clifford group, and some unitary designs.