发表机构
Université libre de Bruxelles（布鲁塞尔自由大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文利用对称群表示论简化量子换能器的构造，证明最优催化剂可协变且换能器可块对角化，从而为无结构搜索、振幅放大和估计等原语导出最优换能器。
AI 中文摘要
换能器（Belovs、Jeffery 和 Yolcu，2024）是一个量子计算框架，它将量子算法描述为一个酉变换，该变换利用催化剂（一个保持不变的辅助向量）将输入态转换为目标态。它们是量子算法设计中的强大工具，尤其是在量子查询复杂度领域：对偶对手半定规划（adversary SDP）的可行点直接转化为换能器，且最优换能复杂度等于对手界（即拉斯维加斯复杂度），已知该复杂度可刻画有界误差量子查询复杂度。此外，与有界误差算法相反，换能器可精确复合，这限制了由复合构建的算法中因控制误差而产生的开销。然而，就量子查询复杂度而言，构建高效的甚至最优的换能器仍然是一项艰巨的任务，因为这仍需求解对手SDP并构造酉变换以获得显式算法。在本文中，我们展示了如何利用状态转换问题的对称群来简化这两个步骤。首先，利用对称化论证，我们证明最优催化剂总可以被选择为在对称群的某个表示下协变。其次，我们证明换能器交织群的两种不同表示，因此可以在希尔伯特空间的等型分解下选择为块对角形式。利用这些方法，我们随后为多种广泛使用的量子算法原语（如无结构搜索、振幅放大和振幅估计）推导出具有最优常数的最优换能器。我们的方法将先前利用表示论计算对手下界的工作（Høyer、Lee 和 Špalek，2007；Ambainis、Magnin、Roetteler 和 Roland，2011）扩展到了最优算法的系统构造。
英文摘要
Transducers (Belovs, Jeffery and Yolcu, 2024) are a quantum computing framework describing a quantum algorithm as a unitary converting an input state into a target state using a catalyst, an auxiliary vector that is left unchanged. They are a powerful tool in quantum algorithm design, especially in the context of quantum query complexity: feasible points of the (dual) adversary semidefinite program directly translate into transducers and the optimal transduction complexity is equal to the adversary bound, i.e. the Las Vegas complexity, which is known to characterize bounded-error quantum query complexity. Moreover, contrary to bounded-error algorithms, transducers compose exactly, which limits overheads due to controlling errors in algorithms constructed by composition. Constructing efficient, let alone optimal, transducers in terms of quantum query complexity nevertheless remains a hard task since it still requires solving the adversary SDP and constructing the unitary to obtain an explicit algorithm. In this paper, we show how using the symmetry group of state-conversion problems simplifies both steps. First, using a symmetrization argument, we prove an optimal catalyst can always be chosen covariant under a representation of the symmetry group. Second, we prove that the transducer intertwines two different representations of the group and can thus be chosen block diagonal in the isotypic decomposition of the Hilbert space. Using those methods, we then derive optimal transducers, with optimal constants, for different widely used quantum algorithmic primitives, such as unstructured search, amplitude amplification and amplitude estimation. Our approach extends previous work on the use of representation theory to compute adversary lower bounds (Høyer, Lee, and {\v S}palek, 2007; Ambainis, Magnin, Roetteler and Roland, 2011) to the systematic construction of optimal algorithms.
Comments39 pages, 6 figures
