The goal of this paper is to elucidate the hermitian structure of the algebra of geometric quaternions \({\textbf {H}}\) (that is, the even algebra of the geometric algebra of the euclidean 3d space), use it to couch a geometric and dynamical presentation of the q-bit, and to see its bearing on the geometric structure of q-registers (arrangements of finite number of q-bits) and with that to pay a brief revisit to the formal structure of q-computations, with emphasis on the algebra structure of \(\textbf{H}^{\otimes n}\). The main underlying theme is the unraveling of the subtle geometric relations between \(\textbf{H}\) and the sphere \(S^2\) in the 3d euclidean space.

本文的目标是阐明几何四元数代数\({\textbf {H}}\)（即欧几里得 3d 空间几何代数的偶代数）的 Hermitian 结构，用它来表达q位的几何和动态表示，并了解其对q寄存器的几何结构（有限数量q位的排列）的影响，并由此简要回顾q计算的形式结构，重点是\(\textbf{H}^{\otimes n}\)的代数结构。主要的主题是揭示3d 欧几里德空间中\(\textbf{H}\)和球体\(S^2\)之间微妙的几何关系。