AI 中文总结
本论文研究量子计算中调和紧框架的实现，运用表示理论、POVM等工具，得到其可分性等条件，设计出实现循环群调和框架的量子电路并给出应用实例与未来方向。
AI 中文摘要
本论文旨在探索一类特殊量子测量——所谓的调和紧框架的实现。为实现这一目标，本文运用表示理论，重点阐述其由不可约表示的构造及满足的关系。这些工具因对称性的存在简化了研究，引出群框架的概念，其定义为群酉作用下的轨道，以对称群为特征。在各类群中，最简单的是阿贝尔群，由此产生调和框架，本文通过阿贝尔群的分类定理及其不可约表示的特征标对其进行分析。在量子力学语境下，框架元素可被解释为系统的纯态，因此本文探究了可分性的充要条件，因为可分态因局域特性更易实现。此外，当将框架视为测量时，本文研究了POVM（正算子值测度）和奈马克定理，将其作为量子计算机中测量设计的关键工具。利用这些知识，本文从阿贝尔群结构定理出发，给出了调和框架的特征刻画，由此得到其可分性的充要条件。本文还研究了这些态在 bipartite（二分）系统中的最大纠缠条件，得到了子系统维度的必要条件。最后，本文得到了一个量子电路，其可利用傅里叶矩阵和置换矩阵，将与循环群相关的调和框架实现为POVM。本文总结了将结果应用于量子计算机的简单实例，并提出了未来的研究方向，这些方向可用于改进电路设计并将其扩展到所有调和框架的情形。
英文摘要
The aim of this thesis is to explore the implementation of a special kind of quantum measurements, so called harmonic tight frames. To achieve this goal, representation theory is addressed, emphasizing its construction from irreducible representations and the relations they satisfy. These tools simplify the study due to the existence of symmetries, leading to the concept of group frames, defined as orbits under the unitary action of a group and characterized by their symmetry groups. Among the various groups, the simplest are the abelian ones, giving rise to harmonic frames, which are analyzed through the classification theorem of abelian groups and the characters of their irreducible representations. In the quantum mechanics context, the frame elements can be interpreted as pure states of a system. Therefore, the necessary and sufficient conditions for separability are explored, as separable states are easier to implement due to their local nature. Alternatively, when viewing frames as measurements, the concept of POVMs and Naimark's theorem are studied as key tools for designing measurements in quantum computers. Using this knowledge, a characterization of the harmonic frames is given from the abelian group structure theorem, which allows obtaining a necessary and sufficient condition for their separability. The conditions of maximum entanglement of these states for bipartite systems are also studied, obtaining a necessary condition on the dimensions of the subsystems. Finally, a quantum circuit is obtained that allows implementing the harmonic frames associated to cyclic groups as POVMs using a Fourier matrix and a permutation matrix. We conclude by giving simple examples where the results are applied to quantum computers and proposing future avenues of research that could serve to improve the circuit design and extend it to the case of all harmonic frames.
CommentsLicenciatura thesis, Pontificia Universidad Católica de Chile, 2024. Advisor: Dardo Goyeneche. 65 pages
