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

Luciano Silva

Published May 23, 2025• 0 citations
Publisher

Abstract

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.

DOI
10.20944/preprints202505.1877.v1

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We introduce Natural Qubit Algebra (NQA), a compact real operator calculus for qubit systems based on a 2×22\times22×2 block alphabet {I,X,Z,W}⊂Mat(2,R)\{I,X,Z,W\}\subset\mathrm{Mat}(2,\mathbb{R}){I,X,Z,W}⊂Mat(2,R) and tensor-word representations. The resulting multiplication law induces a canonical (Z2)2m(\mathbb{Z}_2)^{2m}(Z2​)2m-grading with a bicharacter that controls commutation signs, placing the framework naturally within the theory of color-graded and Clifford-type algebras. Within this language, we provide: (i) an explicit real Clifford normal form for two-qubit operators via the identification Mat(4,R)≅Cl(2,2;R)\mathrm{Mat}(4,\mathbb{R})\cong\mathrm{Cl}(2,2;\mathbb{R})Mat(4,R)≅Cl(2,2;R); (ii) a purely algebraic reformulation of the Bell--CHSH scenario, where the quantum violation is expressed as a spectral non-embeddability of a noncommutative spinor algebra into any commutative Kolmogorov algebra; and (iii) compact factored representations of the Bernstein--Vazirani and Grover phase oracles, showing that both Clifford and non-Clifford examples can admit similarly structured symbolic descriptions. We clarify that Grover's iterate remains outside the Clifford group due to its continuous spectral rotation, consistent with the Gottesman--Knill theorem, while retaining a compact tensor-block form in NQA. The framework isolates spectral, algebraic, and syntactic aspects of operator structure, providing a graded operator language compatible with standard quantum mechanics.
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 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.
The Clifford hierarchy of unitary operators is a foundational concept for universal quantum computation. It was introduced to show that universal quantum computation can be realized via quantum teleportation, given access to certain standard resources. While the full structure of the hierarchy is still not understood, Cui et al. [S. X. Cui et al., Phys. Rev. A 95, 012329 (2017)] recently described the structure of diagonal unitaries in the hierarchy. They considered diagonal unitaries whose action on a computational basis qudit state is described by a 2kth{2}^{k}\mathrm{th}2kth root of unity raised to some polynomial function of the state, and they established the level of such unitaries in the hierarchy as a function of kkk and the degree of the polynomial. For qubit systems, we consider kthk\mathrm{th}kth-level diagonal unitaries that can be described just by quadratic forms of the state over the ring Z2k{\mathbb{Z}}_{{2}^{k}}Z2k​ of integers modulo 2k{2}^{k}2k. The quadratic forms involve symmetric matrices over Z2k{\mathbb{Z}}_{{2}^{k}}Z2k​ that can be used to efficiently describe all two-local and certain higher locality diagonal gates in the hierarchy. We also provide explicit algebraic descriptions of their action on Pauli matrices, which establishes a natural recursion to diagonal unitaries from lower levels. The result involves symplectic matrices over Z2k{\mathbb{Z}}_{{2}^{k}}Z2k​ and hence our perspective unifies a subgroup of diagonal gates in the Clifford hierarchy with the binary symplectic framework for gates in the Clifford group. We augment our description with simple examples for certain standard gates. In addition to demonstrating structure, these formulas might prove useful in applications such as (i) classical simulation of quantum circuits, especially via the stabilizer rank approach, (ii) synthesis of logical non-Clifford unitaries, specifically alternatives to expensive magic state distillation, and (iii) decomposition of arbitrary unitaries beyond the Clifford+T\mathrm{Clifford}+TClifford+T set of gates, perhaps leading to shorter depth circuits. Our results suggest that some nondiagonal gates in the hierarchy might also be understood by generalizing other binary symplectic matrices to integer rings.