AI 中文总结
从范畴张量图语义角度研究量子计算协议，对多种算法进行拓扑重新解释，超越标准量子比特模型，形式化新算法并给出图形表示，还实现相关门、追溯量子纠缠起源及简化协议，为量子硬件自动电路优化提供图形工具包。
AI 中文摘要
本文从范畴张量图语义的直观角度，在FHilb范畴内系统研究量子计算协议。传统希尔伯特空间形式将量子算法结构本质隐藏于高维矩阵运算之后，而本文开发的拓扑框架直接将算法功能编码到图形框架中。对伯恩斯坦 - 瓦齐拉尼算法和西蒙算法进行了全面拓扑重新解释，展示拓扑变换如何提炼其核心数学本质并阐明预言机操作机制。超越标准量子比特模型，形式化了三量子比特适应的广义德乌施 - 约扎算法和广义单次格罗弗算法并给出图形表示。通过互补弗罗贝尼乌斯结构实现CNOT门，追溯包括贝尔态和GHZ态在内的量子纠缠的图形起源，给出W态制备协议的图形简化。通过将张量范畴理论与实际量子算法设计相结合，为跨不断发展的量子硬件生态系统的自动电路优化提供了可组合、可扩展的图形工具包。
英文摘要
This paper investigates foundational quantum computing protocols from the intuitive perspective of categorical tensor-graph semantics within the category \textbf{FHilb}. While conventional Hilbert-space formalisms often conceal the structural nature of quantum algorithms behind high-dimensional matrix operations, the topological framework directly encodes algorithmic functionalities into their graphical skeletons. We provide a comprehensive topological reinterpretation of the Bernstein--Vazirani and Simon algorithms, demonstrating how topological transformations distill their core mathematical essence and clarify the operational mechanisms of oracles. Going beyond the standard qubit model, we construct explicit representations for the qutrit-adapted topological Deutsch--Jozsa and single-shot Grover algorithms. In particular, we establish a necessary and sufficient condition for the single-shot Grover search. We further implement CNOT gates via complementary Frobenius structures and investigate a diagrammatic decomposition scheme for the W-state preparation protocol. By bridging tensor category theory with practical quantum algorithmic design, this work furnishes a composable, scalable diagrammatic toolkit essential for automated circuit optimization across the evolving quantum hardware ecosystem.
Comments20 pages, new template with updated references and simplified the W-state preparation
