Nearly optimal polynomial approximations for the quantum singular value transform
arXiv:2607.12190
The paper presents simple Chebyshev‑based polynomial approximations of even and odd step functions on [-1,1] that achieve nearly optimal error, and shows how these can be employed in quantum singular value transform applications such as phase estimation and amplitude amplification.
Abstract
We introduce polynomial approximations of the even and odd step functions on the interval 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.
13+10 pages, 6 figures