ExMD: SU(2) matrix decomposition

In quantum information science, Berry’s geometric phase, corresponding to the evolution of the quantum system, has important information. Notably, the geometric phase is essential in developing fault-tolerant quantum computing. However, a general description of the geometric phase has not yet been reported, especially for non-commuting and non-anticommuting properties (we call these non-diabolical properties). We explain the cyclic evolution of arbitrary quantum states using multiple rotation operators with non-diabolical properties, and compare it with the quantum evolution by commuting and anticommuting properties. In addition, we analyze the conditions of such evolution in terms of geometric and dynamic phases using Uhlmann’s fidelity and visualization on the Bloch sphere, drawn in the ExMD Mathematica package.在量子信息科学（quantum information science）中，与量子系统演化相关联的贝里几何相位（Berry’s geometric phase）承载着关键信息。值得关注的是，该几何相位在容错量子计算（fault-tolerant quantum computing）的研发中具有核心地位。但目前尚未有针对该几何相位的普适性描述，尤其针对非对易与非反对易性质（我们称之为非恶魔性质（non-diabolical properties））的相关成果仍付阙如。本文采用具备非恶魔性质的多重旋转算符（rotation operators），阐释了任意量子态的循环演化过程，并将其与基于对易与反对易性质（commuting and anticommuting properties）的量子演化进行对比分析。此外，本文借助乌尔曼保真度（Uhlmann’s fidelity）以及通过ExMD Mathematica工具箱绘制的布洛赫球（Bloch sphere）可视化结果，从几何相位与动力学相位两个维度，剖析了此类演化的满足条件。