Quregisters, symmetry groups and Clifford algebras

TL;DR

This paper explores the mathematical structure of qubits and entanglement through Clifford algebras and symmetry groups, providing a geometric and algebraic framework for quantum information.

Contribution

It introduces a novel approach linking qubits, quregisters, and qugates to symmetry groups and Clifford algebras, extending these concepts to tensor powers.

Findings

Representation of qubits within SU(2) via Clifford algebra

Extension of symmetry group framework to tensor products

Realization of entanglement as group-theoretic constructs

Abstract

The Clifford algebra over the three-dimensional real linear space includes its linear structure and its exterior algebra, the subspaces spanned by multivectors of the same degree determine a gradation of the Clifford algebra. Through these geometric notions, natural one-to-one and two-to-one homomorphisms from $\mbox{SO}(3)$ into $\mbox{SU}(2)$ are built conventionally, and the set of qubits, is identified with a subgroup of $\mbox{SU}(2)$. These constructions are suitable to be extended to corresponding tensor powers. The notions of qubits, quregisters and qugates are translated into the language of symmetry groups. The corresponding elements to entangled states in the tensor product of Hilber spaces. realise a notion of entanglement in the tensor product of symmetry groups.