Algebras with given (anti-)commutativity structure are widespread in quantum mechanics. This structure is captured by quasi-Clifford algebras (QCA): a QCA generated by $α_1, \dots, α_n$ is is given by the relations $α_i^2 = k_i$ and $α_j α_i = (-1)^{χ_{ij}} α_i α_j$ , where $k_i \in \mathbb{C}$ and $χ_{ij} \in \{0, 1\}$ . We present a mapping from QCA to Pauli algebras and discuss its use in quantum information and computation. The mapping also provides a Wedderburn decomposition of matrix groups with quasi-Clifford structure. This provides a block-diagonalization for e.g. Pauli groups, while for Majorana operators the Jordan-Wigner transform is recovered. Applications to the symmetry reduction of semidefinite programs and for constructing maximal anti-commuting subsets are discussed.Algebras with given (anti-)commutativity structure are widespread in quantum mechanics. This structure is captured by quasi-Clifford algebras (QCA): a QCA generated by $α_1, \dots, α_n$ is is given by the relations $α_i^2 = k_i$ and $α_j α_i = (-1)^{χ_{ij}} α_i α_j$, where $k_i \in \mathbb{C}$ and $χ_{ij} \in \{0, 1\}$. We present a mapping from QCA to Pauli algebras and discuss its use in quantum information and computation. The mapping also provides a Wedderburn decomposition of matrix groups with quasi-Clifford structure. This provides a block-diagonalization for e.g. Pauli groups, while for Majorana operators the Jordan-Wigner transform is recovered. Applications to the symmetry reduction of semidefinite programs and for constructing maximal anti-commuting subsets are discussed.

arXiv:2508.01470Aug 2, 20250 citations

Geometric Algebra Quantum Gate Decomposition

Youssef Amraoui, Zeno Toffano

Quantum gates are usually described through matrix and tensor-product formalisms that often obscure their geometric structure. In this work, we formulate the Pauli and Clifford groups within the complex Geometric Algebra (GA) framework. We show that the Pauli group is naturally identified with the group of blades up to a global phase, thereby providing a geometric interpretation of Pauli operators and their commutation relations in terms of oriented subspaces. We further prove that Clifford operators are generated by products of π/4-Pauli rotors and introduce a greedy Pauli rotor decomposition algorithm whose empirical behavior suggests unexpectedly compact decompositions for Clifford operators. Finally, we show that Clifford+T universality admits a natural geometric interpretation through π/8-rotors within this framework.Quantum gates are usually described through matrix and tensor-product formalisms that often obscure their geometric structure. In this work, we formulate the Pauli and Clifford groups within the complex Geometric Algebra (GA) framework. We show that the Pauli group is naturally identified with the group of blades up to a global phase, thereby providing a geometric interpretation of Pauli operators and their commutation relations in terms of oriented subspaces. We further prove that Clifford operators are generated by products of π/4-Pauli rotors and introduce a greedy Pauli rotor decomposition algorithm whose empirical behavior suggests unexpectedly compact decompositions for Clifford operators. Finally, we show that Clifford+T universality admits a natural geometric interpretation through π/8-rotors within this framework.

quant-pharXiv:2606.12480Jun 11, 20260 citations

The Pauli Principle Revisited

Murat Altunbulak, Alexander Klyachko

arXiv:0802.0918Communications in Mathematical PhysicsJul 3, 2008156 citations

Geometric Algebra Quantum Gate Decomposition

Youssef Amraoui, Zeno Toffano

Quantum gates are traditionally described using matrix and tensor-product formalisms, representations that provide limited geometric intuition. In this work, we develop a formulation of the Pauli and Clifford groups within the complex Geometric Algebra (GA) framework in order to obtain useful quantum gate decompositions. We show that the Pauli group is naturally identified with the group of blades up to a global phase, providing an intuitive geometric interpretation of Pauli operators and their commutation relations in terms of oriented subspaces. We further prove that Clifford operators are generated by products of π/4-Pauli rotors and introduce a greedy Pauli rotor decomposition algorithm whose empirical performance reveals remarkably compact decompositions of Clifford operators. Finally, we show that Clifford+T universality also acquires a natural geometric interpretation through π/8-rotors within this framework. This work highlights Geometric Algebra as a potential geometric tool for quantum computation applications.Quantum gates are traditionally described using matrix and tensor-product formalisms, representations that provide limited geometric intuition. In this work, we develop a formulation of the Pauli and Clifford groups within the complex Geometric Algebra (GA) framework in order to obtain useful quantum gate decompositions. We show that the Pauli group is naturally identified with the group of blades up to a global phase, providing an intuitive geometric interpretation of Pauli operators and their commutation relations in terms of oriented subspaces. We further prove that Clifford operators are generated by products of π/4-Pauli rotors and introduce a greedy Pauli rotor decomposition algorithm whose empirical performance reveals remarkably compact decompositions of Clifford operators. Finally, we show that Clifford+T universality also acquires a natural geometric interpretation through π/8-rotors within this framework. This work highlights Geometric Algebra as a potential geometric tool for quantum computation applications.

AxiomsJul 13, 20260 citations

The Pauli Equation and the Dirac Operator

Peter Woit

Springer eBooksJan 1, 20171 citation

Algebras with given (anti-)commutativity structure are widespread in quantum mechanics. This structure is captured by quasi-Clifford algebras (QCA): a QCA generated by

α_1, \dots, α_n

is is given by the relations

α_i^2 = k_i

and

α_j α_i = (-1)^{χ_{ij}} α_i α_j

, where

k_i \in \mathbb{C}

and

χ_{ij} \in \{0, 1\}

. We present a mapping from QCA to Pauli algebras and discuss its use in quantum information and computation. The mapping also provides a Wedderburn decomposition of matrix groups with quasi-Clifford structure. This provides a block-diagonalization for e.g. Pauli groups, while for Majorana operators the Jordan-Wigner transform is recovered. Applications to the symmetry reduction of semidefinite programs and for constructing maximal anti-commuting subsets are discussed.