线性化量子信号处理
Linearised quantum signal processing
浏览论文内容
中文总结 AI 辅助
本研究建立了通用哈密顿量本征值变换与量子信号处理框架的关联，提出通用哈密顿量奇异值变换算法，可高效变换哈密顿量编码矩阵的奇异值，且仅要求变换函数在原点消失。
中文摘要 AI 辅助
量子函数编程在近年通过两种不同范式发展而来：基于量子信号处理（QSP）的方法，包括量子奇异值变换（QSVT）；以及基于高阶量子变换的方法，如通用哈密顿量本征值变换（UHET）。尽管UHET与QSP技术存在明显结构相似性，但UHET对哈密顿量动力学进行函数变换的特性，使其与QSP技术的关联始终未明确。本研究填补了这一空白，建立了UHET与基于QSP框架的联系，具体而言，证明UHET可被解释为广义量子信号处理（GQSP）的（随机）线性化。基于该结果，本文引入了（基于哈密顿量的）量子奇异值变换的线性化变体，称为通用哈密顿量奇异值变换（UHSVT）。该算法可对任意哈密顿量块中编码的任意矩阵A的奇异值进行高效变换，其中哈密顿量的动力学可作为黑箱访问，变换函数为任意足够可微的复值函数f。该算法仅要求f在原点处消失，而此前基于QSVT的方法要么假设矩阵A的奇异值存在下界，要么假设能在诱导的二维‘量子化’子空间上执行X旋转门。
英文摘要
Quantum functional programming has been developed through two distinct paradigms in the last few years: Quantum Signal Processing (QSP)-based methods, including the Quantum Singular Value Transformation (QSVT), and methods based on higher-order quantum transformations, such as the Universal Hamiltonian Eigenvalue Transformation (UHET). While UHET performs functional transformations of Hamiltonian dynamics, its relationship to QSP-based techniques has remained unclear despite evident structural similarities. In this work, we resolve this gap by establishing a connection between UHET and QSP-based frameworks; specifically, we show that UHET can be interpreted as a (randomised) linearisation of Generalised QSP (GQSP). Building on this result, we introduce a linearised variant of (Hamiltonian-based) QSVT, which we call Universal Hamiltonian Singular Value Transformation (UHSVT), that enables the efficient transformation of the singular values of any arbitrary matrix $A$ encoded in a block of a Hamiltonian, whose dynamics is accessible as a black box, by any sufficiently differentiable complex-valued function $f$. Our algorithm requires the sole condition that $f$ vanishes at the origin, in contrast to previous QSVT-based approaches that assumed either a lower bound on the singular values of $A$ or the ability to perform $X$-rotation gates on the induced two-dimensional 'qubitised' subspace.
