Geometric Algebra Speaks Quantum Esperanto

Abstract

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.