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

Advances in Applied Clifford Algebras

A Clifford Algebra of Signature (n,3n) and the Density Operators of Quantum Information Theory

Nolmar Melo, Carlile Lavor

Published Jun 5, 2012• 3 citations
Publisher
DOI
10.1007/s00006-012-0341-9

Related papers

Clifford Algebra, its Generalizations, and Applications to Quantum Information and Computing

Thalanayar S. Santhanam

Advances in Applied Clifford Algebras•May 8, 2010•1 citation

Hybrid Clifford Codes via Operator Algebra Quantum Error Correction and Projective Representation Theory

Jonas Eidesen, David W. Kribs, Andrew Nemec

Clifford codes are a natural generalization of quantum stabilizer codes based primarily on representation theory. This class of codes has previously been extended to the setting of quantum subsystem codes. We formulate a two-fold generalization of Clifford codes, for both the hybrid classical and quantum information and projective representation theory settings. This leads to new classes of hybrid subspace and subsystem Clifford codes. We extend the fundamental representation theoretic quantum error correction theorem to include these codes, based on the operator algebra quantum error correction framework. We also discuss several examples throughout the presentation, of both stabilizer and non-stabilizer type.

quant-pharXiv:2606.02531•Jun 2, 2026•0 citations

The Clifford Algebra of the Density Matrix: An Elementary Approach

Pedro Amao, Hernán Castillo

Advances in Applied Clifford Algebras•Jun 29, 2024•0 citations

Uma álgebra de Clifford de assinatura (n,3n) e os operadores densidade da teoria da informação quântica

Nolmar Melo

This work presents an algebraic language for two basic elements of quantum information theory (the quantum bits and density operators), based in the properties of a Clifford algebra of signature (n,3n).We prove that the new description of these elements preserves the same mathematical properties obtained with the classical description.We also extend some results presented in the literature that relate Clifford algebra and quantum information.v Dedico esse trabalho àqueles que me apoiaram incondicionalmente: meus pais, João Ferreira de Souza e Maria de Fátima Mello de Souza.vi Agradecimentos É difícil escrever a parte de agradecimentos desse trabalho e contemplar todos aqueles que merecem ser agradecidos, isso pelo fato de muitas pessoas terem contribuído de alguma forma para a conclusão desse trabalho, seja com dicas de como resolver um problema ou mesmo apoiando-me em momentos difíceis ou ainda gerando momentos alegres durante esse período de reclusão e pesquisa.

•Jan 19, 2011•0 citations

A Unified Clifford Algebra Framework for Quantum Computation: From Specific Spinor Realizations to General Operator Representations

Luciano Silva

This paper presents a unified algebraic framework for $n$-qubit quantum computation rooted in the complex Clifford algebra $Cl(2n, \mathbb{C})$. We leverage the fundamental isomorphism $Cl(2n, \mathbb{C}) \cong M(2^n, \mathbb{C})$ to directly identify the algebra of all $n$-qubit quantum operators (including states, gates, and observables) with elements of $Cl(2n, \mathbb{C})$. This operator-centric perspective treats quantum states, whether pure or mixed, as density multivectors $\rho \in Cl(2n, \mathbb{C})$ satisfying standard physical constraints ($\rho^\dagger = \rho$, $2^n \langle \rho \rangle_0 = 1$, positivity), and unitary gates $U$ as multivectors obeying $U^\dagger U = I$. Quantum evolution $\rho' = U\rho U^\dagger$ and measurements $p(m) = 2^n \langle P_m \rho \rangle_0$ are thus described by intrinsic Clifford algebraic operations. This contrasts with, yet encompasses, specific realizations like spinor representations based on Witt ideals ($S = Cl(2n, \mathbb{C})\prod_j f_j f_j^\dagger$), which are shown to be particular constructions within this universal algebra. By analyzing Deutsch's algorithm through these different lenses, we demonstrate the generality and unifying power of the operator-centric multivector approach. This formulation not only provides a comprehensive algebraic language that naturally handles mixed states and general quantum processes but also opens avenues for applying the rich geometric and structural tools of Clifford algebras—such as grade decomposition and invariant theory—to gain deeper insights into quantum information, entanglement, and dynamics.

•May 23, 2025•0 citations

On quantum mechanics and generalized Clifford algebras

A. K. Kwaśniewski, W. Bajguz, I. Jaroszewski

Advances in Applied Clifford Algebras•Dec 1, 1998•6 citations
Clifford codes are a natural generalization of quantum stabilizer codes based primarily on representation theory. This class of codes has previously been extended to the setting of quantum subsystem codes. We formulate a two-fold generalization of Clifford codes, for both the hybrid classical and quantum information and projective representation theory settings. This leads to new classes of hybrid subspace and subsystem Clifford codes. We extend the fundamental representation theoretic quantum error correction theorem to include these codes, based on the operator algebra quantum error correction framework. We also discuss several examples throughout the presentation, of both stabilizer and non-stabilizer type.
This work presents an algebraic language for two basic elements of quantum information theory (the quantum bits and density operators), based in the properties of a Clifford algebra of signature (n,3n).We prove that the new description of these elements preserves the same mathematical properties obtained with the classical description.We also extend some results presented in the literature that relate Clifford algebra and quantum information.v Dedico esse trabalho àqueles que me apoiaram incondicionalmente: meus pais, João Ferreira de Souza e Maria de Fátima Mello de Souza.vi Agradecimentos É difícil escrever a parte de agradecimentos desse trabalho e contemplar todos aqueles que merecem ser agradecidos, isso pelo fato de muitas pessoas terem contribuído de alguma forma para a conclusão desse trabalho, seja com dicas de como resolver um problema ou mesmo apoiando-me em momentos difíceis ou ainda gerando momentos alegres durante esse período de reclusão e pesquisa.
This paper presents a unified algebraic framework for nnn-qubit quantum computation rooted in the complex Clifford algebra Cl(2n,C)Cl(2n, \mathbb{C})Cl(2n,C). We leverage the fundamental isomorphism Cl(2n,C)≅M(2n,C)Cl(2n, \mathbb{C}) \cong M(2^n, \mathbb{C})Cl(2n,C)≅M(2n,C) to directly identify the algebra of all nnn-qubit quantum operators (including states, gates, and observables) with elements of Cl(2n,C)Cl(2n, \mathbb{C})Cl(2n,C). This operator-centric perspective treats quantum states, whether pure or mixed, as density multivectors ρ∈Cl(2n,C)\rho \in Cl(2n, \mathbb{C})ρ∈Cl(2n,C) satisfying standard physical constraints (ρ†=ρ\rho^\dagger = \rhoρ†=ρ, 2n⟨ρ⟩0=12^n \langle \rho \rangle_0 = 12n⟨ρ⟩0​=1, positivity), and unitary gates UUU as multivectors obeying U†U=IU^\dagger U = IU†U=I. Quantum evolution ρ′=UρU†\rho' = U\rho U^\daggerρ′=UρU† and measurements p(m)=2n⟨Pmρ⟩0p(m) = 2^n \langle P_m \rho \rangle_0p(m)=2n⟨Pm​ρ⟩0​ are thus described by intrinsic Clifford algebraic operations. This contrasts with, yet encompasses, specific realizations like spinor representations based on Witt ideals (S=Cl(2n,C)∏jfjfj†S = Cl(2n, \mathbb{C})\prod_j f_j f_j^\daggerS=Cl(2n,C)∏j​fj​fj†​), which are shown to be particular constructions within this universal algebra. By analyzing Deutsch's algorithm through these different lenses, we demonstrate the generality and unifying power of the operator-centric multivector approach. This formulation not only provides a comprehensive algebraic language that naturally handles mixed states and general quantum processes but also opens avenues for applying the rich geometric and structural tools of Clifford algebras—such as grade decomposition and invariant theory—to gain deeper insights into quantum information, entanglement, and dynamics.