几何代数及其在量子系统控制中的应用

摘要

Based on the concepts of geometric algebra, the relations between inner product, outer product and geometric product are introduced. Then from the plane geometry point of view, the relations between geometric product and parallel vector as well as vertical vector are described, especially for the relation between the pseudo-scalar constructed by the geometric product of three orthogonal vectors and the Pauli matrix rotation algebra used in Spin 1/2 quantum system. Based on these, combining with another representation of quantum algebra motion trajectory, Bloch sphere, an arbitrary unitary operator in single qubit can be written as a combination of rotation with phase shift. The manipulation of any arbitrary qubit can be completed by the unitary operator. The concepts elaborated in this paper arebasicandabstractbutimportantandpractical.%　　首先从几何代数最基本的概念出发，介绍内积、外积与几何积之间的关系；然后从平面几何代数的层面，介绍平行向量、垂直向量与几何积之间的关系。重点介绍由3个正交向量的几何积所构成的伪标量(虚数)与自旋1/2粒子的量子系统中所使用的泡利旋转矩阵代数之间的关系。基于所描述的关系，并结合另一种描述单量子几何运行轨迹的表达方式Bloch球，将单个量子位上的任意幺正算符写成旋转与总相位移动的结合，任何一个量子位的操控可以通过幺正算符来完成。所阐述的概念虽然很基础，很抽象，但很重要，也很实用。