论文标题
线性编程具有统一的约束
Linear programming with unitary-equivariant constraints
论文作者
论文摘要
统一性的均衡性是一种自然对称性，在物理和数学中的许多情况下发生。这种对称性的优化问题通常可以作为半趋化程序进行表述，以适用于$ d^{p+q} $ - 尺寸矩阵变量，以$ u^{\ otimes p} \ otimes \ otimes \ bar {u}^u}^{\ otimes q} $}（对于所有$ u \ \ in \ in \ in \ n \ = u}（u）即使$ p+q $很小，但是本地尺寸$ d $很大，解决此类问题也可能很昂贵。我们表明，在其他对称性假设下，此问题还原为一个线性程序，该程序可以在不扩展$ d $的时间内解决，我们提供了一个通用框架来在不同类型的对称性下执行此减少。我们方法的关键成分是通过围墙brauer代数图的线性组合对解决方案空间进行紧凑的参数化。此参数化需要Gelfand-Tsetlin的基础，我们通过适应一般方法Arxiv：1606.08900受Okounkov-vershik方法的启发。为了说明潜在的应用，我们使用了量子信息中的几个示例：确定量子状态的主要特征值，量子多数投票，非对称克隆和黑盒统一的不对称克隆和转换。我们还概述了将我们的方法扩展到一般统一的半决赛计划的可能途径。
Unitary equivariance is a natural symmetry that occurs in many contexts in physics and mathematics. Optimization problems with such symmetry can often be formulated as semidefinite programs for a $d^{p+q}$-dimensional matrix variable that commutes with $U^{\otimes p} \otimes \bar{U}^{\otimes q}$, for all $U \in \mathrm{U}(d)$. Solving such problems naively can be prohibitively expensive even if $p+q$ is small but the local dimension $d$ is large. We show that, under additional symmetry assumptions, this problem reduces to a linear program that can be solved in time that does not scale in $d$, and we provide a general framework to execute this reduction under different types of symmetries. The key ingredient of our method is a compact parametrization of the solution space by linear combinations of walled Brauer algebra diagrams. This parametrization requires the idempotents of a Gelfand-Tsetlin basis, which we obtain by adapting a general method arXiv:1606.08900 inspired by the Okounkov-Vershik approach. To illustrate potential applications, we use several examples from quantum information: deciding the principal eigenvalue of a quantum state, quantum majority vote, asymmetric cloning and transformation of a black-box unitary. We also outline a possible route for extending our method to general unitary-equivariant semidefinite programs.
