AI 中文总结
针对量子信号处理（QSP）用于哈密顿量模拟时的常数因子差距，本文通过精细处理贝塞尔尾项，将模拟开销降低约e/2倍，几乎完全缩小了最优常数因子的差距。
AI 中文摘要
量子信号处理（QSP）是量子计算机上哈密顿量模拟的渐近最优技术。通过雅可比-安格尔展开近似时间演化算子，哈密顿量模拟问题可简化为多项式逼近理论问题：寻找足够次数为d的多项式序列，以误差ε在[-1,1]上逼近e^{-iτx}。已知d∈Õ(τ)是渐近最优的，但当前最优常数乘子（约等于1）的最新界与实际值之间存在差距。本文几乎完全缩小了该差距，使未来可能的改进不具备实际意义。我们的改进在于利用卡普坦不等式和沃森不等式对雅可比-安格尔级数中的贝塞尔尾项进行精细处理，从而将量子计算机上所有哈密顿量模拟任务的开销估计降低了约e/2倍。
英文摘要
Quantum signal processing (QSP) serves as the asymptotically optimal technique for Hamiltonian simulation on a quantum computer. By approximating the time evolution operator via the Jacobi-Anger expansion, the Hamiltonian simulation problem reduces to a problem in polynomial approximation theory: find a sufficient degree-$d$ polynomial series to approximate $e^{-iτx}$ on $[-1,1]$ within error $ε$. While $d\in\tilde{\mathcal{O}}(τ)$ is known to be asymptotically optimal, there exists a gap between state-of-the-art bounds and the optimal constant multiplicative factor, which is approximately equal to 1. Here, we close this gap almost entirely, to the point where possible future improvements will not be of practical significance. Our improvement resides in a careful treatment of the Bessel tail in the Jacobi-Anger series using Kapteyn's and Watson's inequalities, thereby reducing the overhead estimates for all Hamiltonian simulation tasks on quantum computers by a factor of $\approx e/2$.
Comments8 pages, 1 figure
