发表机构
QuSoft; Institute for Logic, Language and Computation, University of Amsterdam; Korteweg-de Vries Institute for Mathematics, University of Amsterdam（QuSoft; 阿姆斯特丹大学逻辑、语言与计算研究所; 阿姆斯特丹大学科特韦格-德弗里斯数学研究所）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文重新研究对称群的量子傅里叶变换，修正其门复杂度与电路深度，针对横截元素选择的非最优性提出更简单的实现方案。
AI 中文摘要
一般群的量子傅里叶变换（QFT）在该领域早期就被认为是基础内容。非阿贝尔QFT的典型例子——对称群的QFT由Beals于1997年概述。后来Kawano和Sekigawa在2016年对该算法进行了更详细的分析。本文重新审视该构造，经仔细分析，将其门复杂度修正为$\boldsymbol{\tilde{\boldsymbol{O}}}(n^{3.5})$，电路深度修正为$\boldsymbol{\tilde{\boldsymbol{O}}}(n^3)$。此外，研究发现其构造在横截元素的选择上并非最优，因此提出了对称群QFT的更简单实现方案。
英文摘要
Quantum Fourier transforms (QFT) for general groups were recognized to be fundamental already early in the field. A canonical example of non-abelian QFT for the symmetric group was outlined by Beals (1997). Later, a more detailed analysis of this algorithm was carried out by Kawano and Sekigawa (2016). In this paper, we revisit that construction. After a careful analysis, we revise their gate complexity to $\widetilde{\mathcal{O}}(n^{3.5})$ and circuit depth to $\widetilde{\mathcal{O}}(n^3)$. Moreover, we observe that their construction is not optimal in the choice of transversal elements, so we propose simpler realization of the symmetric group QFT.
