发表机构
University of Michigan; University of California, Berkeley（密歇根大学; 加州大学伯克利分校）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
针对量子信号处理经典设计阶段的约束极小极大近似问题，研究人员提出结合有效集约束的Remez交换法与非线性傅里叶回退两种方法，在保持逼近精度的同时解决可行性问题，相关工作已在qsppack软件中实现。
AI 中文摘要
量子信号处理（QSP）提供了一种使用量子电路实现多项式变换的简单高效框架。其经典设计阶段会产生一个约束极小极大近似问题：寻找一个指定奇偶性的多项式，使其在拟合集上一致逼近目标函数，同时在区间[0,1]上的幅值不超过1，这可视为半无限约束。离散化将该问题转化为线性规划，但有限采样点上的可行性无法保证整个定义域上的可行性，尤其是当最优逼近函数达到可行集边界时。我们研究了两种解决该问题的方法：结合有效集约束执行的Remez交换法在多数测试实例中高效，但其稳定性取决于目标函数和问题几何；随后我们引入非线性傅里叶回退，利用QSP完备性和相位合成将近可行多项式转化为可行QSP多项式的相位因子，且不增加次数。在代表性问题中，回退方法在很大程度上保持了逼近精度，且在Remez启发式方法不稳定的实例中仍有效。该工作流连接了经典极小极大近似、半无限优化与非线性傅里叶分析，并在qsppack软件包中实现。
英文摘要
Quantum signal processing (QSP) provides a simple and efficient framework for implementing polynomial transformations using quantum circuits. Its classical design stage leads to a constrained minimax approximation problem: find a polynomial of prescribed parity that approximates a target function uniformly on a fitting set while remaining bounded in magnitude by one on the domain $[0,1]$, which can be viewed as a semi-infinite constraint. Discretization converts the problem into a linear program, but feasibility at a set of finitely many sampled points does not ensure feasibility on the whole domain, especially when an optimal approximant reaches the boundary of the feasible set. We investigate two approaches to address this difficulty. A Remez exchange method combined with active-set constraint enforcement is efficient on many tested instances, but its stability depends on the target and problem geometry. We then introduce nonlinear Fourier retraction, which uses QSP completion and phase synthesis to turn a nearly feasible polynomial into phase factors for a feasible QSP polynomial without increasing the degree. Across representative problems, retraction largely preserves approximation accuracy and remains effective on instances where the Remez heuristic is unstable. The resulting workflow connects classical minimax approximation and semi-infinite optimization with nonlinear Fourier analysis, and is implemented in the qsppack software package.
Comments27 pages, 13 figures
