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.
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.
Alternative mathematical explorations in quantum computing can be of great scientific interest, especially if they come with penetrating physical insights. In this paper, we present a critical revisitation of our application of geometric (Clifford) algebras (GAs) in quantum computing as originally presented in [C. Cafaro and S. Mancini, Adv. Appl. Clifford Algebras 21, 493 (2011)]. Our focus is on testing the usefulness of geometric algebras (GAs) techniques in two quantum computing applications. First, making use of the geometric algebra of a relativistic configuration space (namely multiparticle spacetime algebra or MSTA), we offer an explicit algebraic characterization of one- and two-qubit quantum states together with a MSTA description of one- and two-qubit quantum computational gates. In this first application, we devote special attention to the concept of entanglement, focusing on entangled quantum states and two-qubit entangling quantum gates. Second, exploiting the previously mentioned MSTA characterization together with the GA depiction of the Lie algebras SO3;R and SU2;C depending on the rotor group Spin+3,0 formalism, we focus our attention to the concept of universality in quantum computing by reevaluating Boykin’s proof on the identification of a suitable set of universal quantum gates. At the end of our mathematical exploration, we arrive at two main conclusions. Firstly, the MSTA perspective leads to a powerful conceptual unification between quantum states and quantum operators. More specifically, the complex qubit space and the complex space of unitary operators acting on them merge in a single multivectorial real space. Secondly, the GA viewpoint on rotations based on the rotor group Spin+3,0 carries both conceptual and computational advantages compared to conventional vectorial and matricial methods.
This paper presents a unified algebraic framework for n-qubit quantum computation rooted in the complex Clifford algebra Cl(2n,C). We leverage the fundamental isomorphism Cl(2n,C)≅M(2n,C) to directly identify the algebra of all n-qubit quantum operators (including states, gates, and observables) with elements of Cl(2n,C). This operator-centric perspective treats quantum states, whether pure or mixed, as density multivectors ρ∈Cl(2n,C) satisfying standard physical constraints (ρ†=ρ, 2n⟨ρ⟩0=1, positivity), and unitary gates U as multivectors obeying U†U=I. Quantum evolution ρ′=UρU† and measurements p(m)=2n⟨Pmρ⟩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†), 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.
Algebras with given (anti-)commutativity structure are widespread in quantum mechanics. This structure is captured by quasi-Clifford algebras (QCA): a QCA generated by α1,…,αn is is given by the relations αi2=ki and αjαi=(−1)χijαiαj, where ki∈C and χij∈{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.
We revisit the Pauli-Clifford connection to introduce a real, grade-preserving algebraic framework for n-qubit quantum computation based on the tensor product Cℓ2,0(R)⊗n. In this setting, the bivector J=e12 satisfies J2=−1 and supplies the complex structure on the J-closure of a minimal left ideal via right multiplication, while Pauli operations arise as left actions of Clifford elements. The Peirce decomposition organizes the algebra into sector blocks determined by primitive idempotents, with nilpotent elements generating transitions between sectors. Quantum states are represented as equivalence classes modulo the left annihilator, exhibiting the quotient description underlying the minimal left ideal. Adopting a canonical stabilizer mapping, the n-qubit computational basis state ∣0⋯0⟩ is given natively by a tensor product of these idempotents. This structural choice leads to a compatibility law that is stable under the geometric product for n qubits and aligns symbolic Clifford multiplication with unitary evolution on the Hilbert space.
Quantum computation is, at bottom, a branch of applied algebra and functional analysis: the state of a quantum register is a unit vector in a finite-dimensional complex Hilbert space, computation is the action of the unitary group on that space, and the structure of the most important quantum algorithms is dictated by the representation theory of finite and continuous groups. This paper develops the mathematical scaffolding of the field from first principles and uses it to explain why quantum computers can outperform classical ones. We review complex inner-product spaces, self-adjoint and unitary operators, the spectral theorem, and the tensor-product construction that gives rise to entanglement and the exponential dimension of the state space. We then describe the qubit and its geometry, the group-theoretic classification of quantum gates (the special unitary groups, the Pauli and Clifford groups, and their Lie-algebraic generators), and the universality results that justify finite gate sets together with the Solovay–Kitaev theorem. With this apparatus we analyse the quantum Fourier transform, phase estimation, and the hidden subgroup problem, showing how Shor’s factoring algorithm, Grover’s search, and the quantum linear-systems algorithm arise as instances of a small number of algebraic templates. Finally we treat the stabilizer formalism of quantum error correction—where finite abelian groups over GF(2) play the central role—and the variational, near-term (NISQ) algorithms that dominate current practice. Throughout, the emphasis is on the algebraic invariants that make quantum speed-ups possible.