AI 中文总结
研究利用具有简单切比雪夫系数的多项式逼近执行量子算法，推导严格误差界，证明其近最优性，进而得出相关近最优多项式逼近，可用于量子相位估计等多种量子算法，基于量子奇异值变换实现。
AI 中文摘要
我们引入了在区间[-1,1]上具有简单切比雪夫系数的偶数和奇数阶跃函数的多项式逼近，使其数值实现变得直接。我们推导了严格的误差界，并证明这些多项式在其误差与理论最优误差的偏差由随多项式阶数对数增长的乘法因子决定的意义上是近最优的。从这些多项式中，我们推导出了相关的近最优多项式逼近，可用于通过量子奇异值变换执行量子相位估计、线性幅度放大、特征值阈值处理和其他量子算法。
英文摘要
We introduce polynomial approximations of the even and odd step functions on the interval $[-1,1]$ with simple Chebyshev coefficients, making their numerical implementation straightforward. We derive rigorous error bounds and demonstrate that these polynomials are nearly optimal in the sense that their error deviates from the theoretically optimal error by a multiplicative factor that grows logarithmically with the polynomial order. From these polynomials, we derive related nearly optimal polynomial approximations that can be used to perform quantum phase estimation, linear amplitude amplification, eigenvalue thresholding, and other quantum algorithms using the quantum singular value transform.
Comments13+10 pages, 6 figures
