We present a simplified exposition of some pieces of [Gily\'en, Su, Low, and Wiebe, STOC'19, arXiv:1806.01838], who introduced a quantum singular value transformation (QSVT) framework for applying polynomial functions to block-encoded matrices. The QSVT framework has garnered substantial recent interest from the quantum algorithms community, as it was demonstrated by [GSLW19] to encapsulate many existing algorithms naturally phrased as an application of a matrix function. First, we posit that the lifting of quantum singular processing (QSP) to QSVT is better viewed not through Jordan's lemma (as was suggested by [GSLW19]) but as an application of the cosine-sine decomposition, which can be thought of as a more explicit and stronger version of Jordan's lemma. Second, we demonstrate that the constructions of bounded polynomial approximations given in [GSLW19], which use a variety of ad hoc approaches drawing from Fourier analysis, Chebyshev series, and Taylor series, can be unified under the framework of truncation of Chebyshev series, and indeed, can in large part be matched via a bounded variant of a standard meta-theorem from [Trefethen, 2013]. We hope this work finds use to the community as a companion guide for understanding and applying the powerful framework of [GSLW19].
翻译:我们提出了一个简化的解析[Gily\'en, Su, Low, and Wiebe, STOC'19, STOC'19, ARXiv:1806.018388]中的一些部分[Gily\'en, Su, Low和Wiebe, STOC'19, STOC'19, ARXIV:1806.018388],这些部分采用了一个量子单值转换框架,用于将多元函数应用到块码矩阵矩阵矩阵。QSVT框架最近从量子算算法界获得了相当大的兴趣,正如[GSLW19]所证明的那样,将许多现有的算法自然表述为矩阵功能。首先,我们认为,将量子单处理(QSP)提升到QSVT QSSVT, 不是通过约旦的 Lemmma (GSL19) 框架来(QSSVT) 引入一个量子超值转换(QSVT) 框架,而是将量值转换成一个可被理解的精度框架。</s>