This paper assembles a conservative mathematical foundation for the use of quaternions in single-qubit quantum computing. The goal is not to replace standard quantum mechanics, but to organize familiar structures in a way that makes their geometry more explicit. We distinguish carefully between normalized state vectors in C^2 ~ R^4, the unit sphere S^3 inside that space, pure states modulo global phase, the projective space CP^1 ~ S^2, the single-qubit gate group U(2), and SU(2) representatives obtained after removing global phase. Within that framework, we exhibit an explicit isomorphism between the unit quaternions and SU(2), explain why unit quaternions are therefore a natural model for single-qubit gates modulo global phase, and show how quaternion multiplication exposes the axis-angle and composition structure of rotations. Standard facts are separated from the organizational choices adopted in the RQM Technologies series. The paper is restricted to single-qubit structure and makes no claim about new physics, multi-qubit quaternionic state evolution, or computational advantage.
Single-qubit quantum mechanics is standardly presented in matrix language, but quaternionic notation can make its rotation geometry more explicit without changing the theory. The purpose of this paper is to organize a conservative foundation for that statement and to distinguish carefully between state spaces, projective state spaces, gate groups, and quaternionic models of those gate groups.
The paper is restricted to single-qubit structure. It does not propose new physics and does not claim a replacement of ordinary quantum mechanics.
A normalized single-qubit state vector has the form (alpha, beta)^T in C^2 with |alpha|^2 + |beta|^2 = 1. Under C^2 ~ R^4, the normalized vectors form S^3.
Pure states are rays in C^2, or equivalently points of CP^1. The Bloch-sphere map identifies CP^1 with S^2, so the Bloch sphere is a quotient of S^3 by global phase rather than another name for S^3 itself.
Single-qubit gates live in U(2). After removing overall phase, each such gate admits a representative in SU(2), unique up to sign.
The quaternion algebra H has basis 1, i, j, k with i^2 = j^2 = k^2 = ijk = -1. The unit quaternions form Sp(1), a Lie group whose underlying manifold is S^3.
The paper fixes an explicit standard identification Phi from H to 2 x 2 complex matrices so that the quaternion basis units align with the Pauli x, y, and z generators.
Under this identification, the unit quaternions are isomorphic to SU(2), and sequential single-qubit gate composition corresponds to quaternion multiplication.
The standard axis-angle expression for SU(2) maps directly to the standard axis-angle form of a unit quaternion. This makes coordinate-axis rotations especially transparent.
Writing a quaternion as scalar part plus pure part exposes dot and cross products directly in the multiplication law. The cross-product term is the geometric source of the noncommutativity of sequential rotations.
Conjugation of pure quaternions defines a surjective homomorphism from Sp(1) to SO(3) with kernel {+1, -1}. This is the familiar double cover SU(2) -> SO(3).
The paper marks as standard the facts about normalized spinors, CP^1, S^2, U(2), SU(2), unit quaternions, and the double cover of SO(3). The project-specific contribution is only an organizational one: to use the quaternion model as a primary expository language for single-qubit gate geometry and to state the distinctions among these spaces explicitly.
This paper does not claim new physics, hardware advantage, or a multi-qubit quaternionic state formalism. It treats only the representation of single-qubit structure.
Unit quaternions belong naturally in single-qubit quantum computing because they provide an exact model of SU(2), the determinant-one representative group of single-qubit gates after overall phase has been removed. Their value in this context is geometric and organizational rather than revisionary.