标题： 一种生成复空间中最大正交框架的遗传算法 显示英文标题

标题： A genetic algorithm to generate maximally orthogonal frames in complex space

摘要： 框架是向量空间基底概念的推广，它是一组冗余覆盖的向量集，这些向量是线性相关的。框架在信号处理和量子信息理论中找到应用。 我们提出了一种遗传算法，可以生成任意大小为$n$的$d$维复数空间中的最大正交框架。 首先，我们形式化了最大正交框架的概念，并证明它依赖于能量函数的选择来衡量向量之间的不同成对重叠。 然后，我们讨论了不同的能量函数与著名的框架变体（如紧框架和Grassmannian框架以及复射影$p$设计）之间的关系。 获得最大正交框架构成了一个全局非凸最小化问题。 我们讨论了与已建立的数值问题（如汤姆森问题和在复射影空间中寻找最优填充的问题）的关系。 为了处理最小化问题，我们设计了一种混合遗传算法，该算法具有父代的局部优化特性。 为了评估算法的性能，我们提出了两种可视化技术，使我们可以分析高维框架的一致性和均匀性。 对于配置到$d=6$和$n=36$的情况，该遗传算法能够生成高度对称的通用框架，如等角紧框架、对称信息完整、正算子值测量（SIC-POVMs）以及最大互无基组，对于最大的配置，运行时间约为几分钟的普通台式计算机时间。 显示英文摘要

摘要： A frame is a generalization of a basis of a vector space to a redundant overspanning set whose vectors are linearly dependent. Frames find applications in signal processing and quantum information theory. We present a genetic algorithm that can generate maximally orthogonal frames of arbitrary size $n$ in $d$-dimensional complex space. First, we formalize the concept of maximally orthogonal frame and demonstrate that it depends on the choice of an energy function to weigh the different pairwise overlaps between vectors. Then, we discuss the relation between different energy functions and well-known frame varieties such as tight and Grassmannian frames and complex projective $p$-designs. Obtaining maximally orthogonal frames poses a global non-convex minimization problem. We discuss the relation with established numerical problems such as the Thomson problem and the problem of finding optimal packings in complex projective space. To tackle the minimization, we design a hybrid genetic algorithm that features local optimization of the parents. To assess the performance of the algorithm, we propose two visualization techniques that allow us to analyze the coherence and uniformity of high-dimensional frames. The genetic algorithm is able to produce highly-symmetric universal frames, such as equiangular tight frames, symmetric, informationally complete, positive operator-valued measurements (SIC-POVMs) and maximal sets of mutually unbiased bases, for configurations of up to $d=6$ and $n=36$, with runtimes of the order of several minutes on a regular desktop computer for the largest configurations.