论文标题
有限基团的量子表示
Quantum representation of finite groups
论文作者
论文摘要
有限群体（QRFG）的量子表示的概念已经是量子计算的基本方面，在相当长的一段时间内，在每个角落都起着作用，从基本量子逻辑门到著名的Shor's和Grover的算法。在本文中，我们使用群体理论和差异几何形状提供了对该概念的正式定义。我们的工作证明了任何有限组的量子表示存在，并概述了两种方法，用于利用单位矩阵和变异量子算法的栅极分解将组的每个发电机转换为量子电路。此外，我们在开放访问平台上提供了明确示例的数值模拟。最后，我们通过在实施某些量子算法和量子有限自动机的实现中表现出其作用来证明QRFG的有用性和潜力。
The concept of quantum representation of finite groups (QRFG) has been a fundamental aspect of quantum computing for quite some time, playing a role in every corner, from elementary quantum logic gates to the famous Shor's and Grover's algorithms. In this article, we provide a formal definition of this concept using both group theory and differential geometry. Our work proves the existence of a quantum representation for any finite group and outlines two methods for translating each generator of the group into a quantum circuit, utilizing gate decomposition of unitary matrices and variational quantum algorithms. Additionally, we provide numerical simulations of an explicit example on an open-access platform. Finally, we demonstrate the usefulness and potential of QRFG by showing its role in the implementation of some quantum algorithms and quantum finite automata.
