Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Clifford group dipoles in quantum computing

Michel R. P. Planat

Published Mar 31, 2009• 0 citations
OpenAlex

Abstract

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.

Related papers

Clifford group dipoles and the enactment of Weyl/Coxeter group W(E8) 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 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.

arXiv:0904.3691•Apr 23, 2009•0 citations

Clifford quantum computing

Michel R. P. Planat

One reports about the embedding of the Pauli group into the Clifford group, and one explicits the concept of Clifford quantum computation, following Gottesman-Knill approach. Normal subgroups of the two-qubit Clifford group are investigated and a link to the generalized quadrangle of other two and hexads in the Steiner system S(3,6,22) is found.

•Nov 21, 2007•0 citations

Generalized Clifford groups and simulation of associated quantum circuits

Stephen R. L. Clark, Richard Jozsa, Noah Linden

Quantum computations starting with computational basis states and involving only Clifford operations, are classically simulable despite the fact that they generate highly entangled states; this is the content of the Gottesman-Knill theorem. Here we isolate the ingredients of the theorem and provide generalisations of some of them with the aim of identifying new classes of simulable quantum computations. In the usual construction, Clifford operations arise as projective normalisers of the first and second tensor powers of the Pauli group. We consider replacing the Pauli group by an arbitrary finite subgroup $G$ of $U(d)$. In particular we seek $G$ such that $G\otimes G$ has an entangling normaliser. Via a generalisation of the Gottesman-Knill theorem the resulting normalisers lead to classes of quantum circuits that can be classically efficiently simulated. For the qubit case $d = 2$ we exhaustively treat all finite irreducible subgroups of $U(2)$ and find that the only ones (up to unitary equivalence and trivial phase extensions) with entangling normalisers are the groups generated by X and the $n^{\rm th}$ root of $Z$ for $n \in \N$.

Quantum Information and Computation•Jan 1, 2008•22 citations

Clifford groups of quantum gates, BN-pairs and smooth cubic surfaces

Michel R. P. Planat, Patrick Solé

Abstract. The recent proposal (M Planat and M Kibler, Preprint 0807.3650 [quantph]) of representing Clifford quantum gates in terms of unitary reflections is revisited. In this essay, the geometry of a Clifford group G is expressed as a BN-pair, i.e. a pair of subgroups B and N that generate G, is such that intersection H = B ∩N is normal

•Jan 1, 2008•9 citations

Clifford Group and Unitary Designs under Symmetry

Yosuke Mitsuhashi, Nobuyuki Yoshioka

We have generalized the well-known statement that the Clifford group is a unitary 3-design into symmetric cases by extending the notion of unitary design.Concretely, we have proven that a symmetric Clifford group is a symmetric unitary 3-design if and only if the symmetry constraint is described by some Pauli subgroup.We have also found a complete and unique construction method of symmetric Clifford groups with simple quantum gates for Pauli symmetries.For the overall understanding, we have also considered physically relevant U(1) and SU(2) symmetry constraints, which cannot be described by a Pauli subgroup, and have proven that the symmetric Clifford group is a symmetric unitary 1-design but not a 2-design under those symmetries.Our findings are numerically verified by computing the frame potentials, which measure the difference in randomness between the uniform ensemble on the symmetric group of interest and the symmetric unitary group.This work will open a new perspective into quantum information processing, such as randomized benchmarking, and give a deep understanding to many-body systems, such as monitored random circuits.

arXiv:2306.17559•PRX Quantum•Nov 27, 2023•12 citations

Clifford groups of quantum gates, BN-pairs and smooth cubic surfaces

Michel R. P. Planat, Patrick Solé

The recent proposal (M Planat and M Kibler, Preprint 0807.3650 [quantph]) of representing Clifford quantum gates in terms of unitary reflections is revisited. In this essay, the geometry of a Clifford group G is expressed as a BN-pair, i.e. a pair of subgroups B and N that generate G, is such that intersection H = B \cap N is normal in G, the group W = N/H is a Coxeter group and two extra axioms are satisfied by the double cosets acting on B. The BN-pair used in this decomposition relies on the swap and match gates already introduced for classically simulating quantum circuits (R Jozsa and A Miyake, Preprint arXiv:0804.4050 [quant-ph]). The two- and three-qubit steps are related to the configuration with 27 lines on a smooth cubic surface.

arXiv:0811.2109•Journal of Physics A Mathematical and Theoretical•Dec 19, 2008•8 citations
Clifford group {\it dipoles} Cn±\mathcal{C}_n^{\pm}Cn±​ are subgroups of the nnn-qubit Clifford group Cn\mathcal{C}_nCn​ defined as (non-split) extensions of the extraspecial subgroups E2n+1±E_{2n+1}^{\pm}E2n+1±​ of the Pauli error group Pn\mathcal{P}_nPn​. The subgroups Cn±:=E2n+1±.Ω±(2n,2)\mathcal{C}_n^{\pm}:=E_{2n+1}^{\pm}.\Omega^{\pm}(2n,2)Cn±​:=E2n+1±​.Ω±(2n,2) involve pairs of orthogonal subgroups Ω±(2n,2)\Omega^{\pm}(2n,2)Ω±(2n,2) of the symplectic group Sp(2n,2)Sp(2n,2)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.
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.
Quantum computations starting with computational basis states and involving only Clifford operations, are classically simulable despite the fact that they generate highly entangled states; this is the content of the Gottesman-Knill theorem. Here we isolate the ingredients of the theorem and provide generalisations of some of them with the aim of identifying new classes of simulable quantum computations. In the usual construction, Clifford operations arise as projective normalisers of the first and second tensor powers of the Pauli group. We consider replacing the Pauli group by an arbitrary finite subgroup GGG of U(d)U(d)U(d). In particular we seek GGG such that G⊗GG\otimes GG⊗G has an entangling normaliser. Via a generalisation of the Gottesman-Knill theorem the resulting normalisers lead to classes of quantum circuits that can be classically efficiently simulated. For the qubit case d=2d = 2d=2 we exhaustively treat all finite irreducible subgroups of U(2)U(2)U(2) and find that the only ones (up to unitary equivalence and trivial phase extensions) with entangling normalisers are the groups generated by X and the nthn^{\rm th}nth root of ZZZ for n∈Nn \in \Nn∈N.
We have generalized the well-known statement that the Clifford group is a unitary 3-design into symmetric cases by extending the notion of unitary design.Concretely, we have proven that a symmetric Clifford group is a symmetric unitary 3-design if and only if the symmetry constraint is described by some Pauli subgroup.We have also found a complete and unique construction method of symmetric Clifford groups with simple quantum gates for Pauli symmetries.For the overall understanding, we have also considered physically relevant U(1) and SU(2) symmetry constraints, which cannot be described by a Pauli subgroup, and have proven that the symmetric Clifford group is a symmetric unitary 1-design but not a 2-design under those symmetries.Our findings are numerically verified by computing the frame potentials, which measure the difference in randomness between the uniform ensemble on the symmetric group of interest and the symmetric unitary group.This work will open a new perspective into quantum information processing, such as randomized benchmarking, and give a deep understanding to many-body systems, such as monitored random circuits.
The recent proposal (M Planat and M Kibler, Preprint 0807.3650 [quantph]) of representing Clifford quantum gates in terms of unitary reflections is revisited. In this essay, the geometry of a Clifford group G is expressed as a BN-pair, i.e. a pair of subgroups B and N that generate G, is such that intersection H = B \cap N is normal in G, the group W = N/H is a Coxeter group and two extra axioms are satisfied by the double cosets acting on B. The BN-pair used in this decomposition relies on the swap and match gates already introduced for classically simulating quantum circuits (R Jozsa and A Miyake, Preprint arXiv:0804.4050 [quant-ph]). The two- and three-qubit steps are related to the configuration with 27 lines on a smooth cubic surface.