发表机构
QuSoft; Institute for Logic, Language and Computation, University of Amsterdam; Department of Mathematical Sciences, University of Copenhagen; Mathematical Institute, Leiden University; 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; 哥本哈根大学数学科学系; 莱顿大学数学研究所; 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
该研究基于Mackey理论与Clifford理论开发两种QFT电路构造方法，分别实现GL₂(F_q)和圈积F≀Sₙ的QFT指数级电路开销优化，为广泛有限群族的QFT构造提供了新系统工具。
AI 中文摘要
量子傅里叶变换（QFTs）是量子算法中的核心基本组件。阿贝尔群可实现高效的QFT电路，其电路规模随群阶的对数呈多项式增长，但仅少数非阿贝尔群族存在已知的高效构造。我们分别基于Mackey理论和Clifford理论开发了两种新的QFT电路构造方法，利用这些方法证明了特定群族的电路开销可实现指数级提升。采用Mackey理论方法，我们得到了有限域上一般线性群GL₂(F_q)的QFT显式量子电路，其规模随log q呈多项式增长，而非随q呈多项式增长。采用Clifford理论方法，我们得到了圈积F≀Sₙ的QFT电路，其开销取决于F上QFT的开销及其表示寄存器的规模。这消除了此前通用构造要求|F|=poly(n)的限制，当F本身具有高效QFT时可实现指数级改进。这些方法共同为广泛有限群族的QFT构造提供了新的系统工具。
英文摘要
Quantum Fourier transforms (QFTs) are essential primitives in quantum algorithms. While abelian groups admit efficient QFT circuits, with circuit size polynomial in the logarithm of the group order, efficient constructions are known for relatively few non-abelian families. We develop two new approaches to QFT circuit construction, based on Mackey theory and Clifford theory, respectively, and use them to show exponential improvement in circuit cost for specific group families. Using the Mackey-theoretic approach, we obtain explicit quantum circuits for the QFT over $\mathrm{GL}_2(F_q)$ that scale polynomially in $\log q$, rather than polynomially in $q$. Using the Clifford-theoretic approach, we obtain QFT circuits for wreath products $F\wr S_n$, whose cost depends on the cost of a QFT over $F$ and the size of its representation registers. This removes the restriction $|F|=\operatorname{poly}(n)$ required by previous generic constructions and can yield exponential improvements when $F$ itself has an efficient QFT. Together, these methods provide new systematic tools to construct QFTs for broad classes of finite groups.
