发表机构
The University of Western Australia; University Paris City and University of Reunion（西澳大利亚大学; 巴黎城市大学和留尼汪大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文从第一性原理出发，将块编码、量子化、QSP、QSVT和GQSP统一为构建量子算法的框架，提出决策工作流和端到端流程，用于实现算子的多项式变换。
AI 中文摘要
现代量子算法日益被表述为用于实现算子和奇异值的多项式变换的相干过程。这一视角为量子算法设计提供了一种强大且统一的语言，通过五个紧密相关的关键工具将各种不同的问题联系起来：块编码、量子化、QSP、QSVT和GQSP。块编码将非酉矩阵嵌入到更大的酉矩阵中；量子化将块编码转换为结构化算子；QSP、QSVT和GQSP实现了具有近最优查询复杂度的多项式变换。这些技术共同构成了将矩阵函数转换为可实现的量子电路的一般性工具包。本文从第一性原理出发，将这些技术发展为一个用于构建量子算法的统一框架。我们将该框架应用于代表性应用，以突出设计原理，并展示如何从统一的算子变换序列中构建不同的算法。一个核心贡献是提供了一个系统的决策工作流，用于根据算子结构和所需变换多项式选择合适的方法。这一视角阐明了何时直接使用GQSP、通过量子化、使用洛朗展开或QSVT最为合适。我们将算法设计组织成一个端到端的流程：识别目标矩阵函数、构建合适的块编码、确定相关的谱域、设计多项式或洛朗多项式近似、合成相位因子，并将变换转换为可执行的量子电路。通过将该统一框架应用于示例应用，我们展示了一种基于多项式变换的量子算法推理、设计和实现的实用方法论。
英文摘要
Modern quantum algorithms are increasingly formulated as coherent procedures for implementing polynomial transformations of operators and singular values. This perspective provides a powerful and unifying language for quantum algorithm design, connecting a wide range of distinct problems through five closely related key tools: block-encoding, qubitization, QSP, QSVT and GQSP. Block-encoding embeds non-unitary matrices into larger unitaries; qubitization converts block-encodings into structured operators; QSP, QSVT and GQSP enable polynomial transformations with near-optimal query complexity. Together, these techniques form a general toolkit for transforming matrix functions into implementable quantum circuits. This paper develops these techniques from first principles as a unified framework for constructing quantum algorithms. We apply this framework to representative applications to highlight design principles and demonstrate how distinct algorithms can be constructed from a unified sequence of operator transformations. A central contribution is a systematic decision workflow for selecting the appropriate approach according to the operator structure and the desired transformation polynomial. This perspective clarifies when direct GQSP or through qubitization, or Laurent expansion, or QSVT is most appropriate. We organize algorithmic design into an end-to-end pipeline: identifying the target matrix function, constructing an appropriate block-encoding, determining the relevant spectral domain, designing a polynomial or Laurent-polynomial approximation, synthesizing the phase factors, and translating the transformation into an executable quantum circuit. By applying this unified framework to example applications, we showcase a practical methodology for reasoning, designing, and implementing quantum algorithms based on polynomial transformations.
