AI 中文总结
介绍部分可交换多项式优化框架，该框架包含交换和非交换多项式优化，允许变量间有任意交换关系，以部分可交换幺半群为基础代数结构，用于构建半定规划松弛解决相关问题。
AI 中文摘要
半定规划层次结构用于交换和非交换多项式优化，是量子信息中强大的计算工具。在应用中，变量通常并非全交换或全非交换，而是部分交换。虽可通过半定规划松弛中的线性约束纳入部分交换关系，但从一开始利用其代数性质可得到更紧凑的松弛。为此引入部分可交换多项式优化框架，其基础代数结构是部分可交换幺半群。文中介绍并回顾了此类幺半群的关键方面，展示如何用于构建半定规划松弛。
英文摘要
Semidefinite programming hierarchies for commutative and non-commutative polynomial optimization represent a powerful computational tool with many applications in quantum information. In such applications, a given variable is typically not either commuting or non-commuting with all other variables, but instead commutes with some variables and does not commute with others, i.e., the variables satisfy some partial commutation relations. While such partial commutation relations can always be incorporated in a fully non-commutative setting through suitable linear constraints in the semidefinite programming relaxations, exploiting their algebraic properties from the onset can result in more compact relaxations. This leads us to introduce partially-commutative polynomial optimization, a framework that encompasses commutative and non-commutative polynomial optimization, allowing for arbitrary commutation relations among the variables. We point out that the underlying algebraic structure is that of a partially-commutative monoid. We present and review several key aspects of such monoids and show how they can be used to build SDP relaxations for partially-commutative polynomial optimization problems in which the partial commutations are natively implemented in the monomial structure, without the need of additional linear constraints.
Comments52 pages
