Complementary polynomials in quantum signal processing
arXiv:2406.04246 · doi:10.1007/s00220-025-05302-9
Abstract
Quantum signal processing is a framework for implementing polynomial functions on quantum computers. To implement a given polynomial , one must first construct a corresponding complementary polynomial . Existing approaches to this problem employ numerical methods that are not amenable to explicit error analysis. We present a new approach to complementary polynomials using complex analysis. Our main mathematical result is a contour integral representation for a canonical complementary polynomial. On the unit circle, this representation has a particularly simple and efficacious Fourier analytic interpretation, which we use to develop a Fast Fourier Transform-based algorithm for the efficient calculation of in the monomial basis with explicit error guarantees. Numerical evidence that our algorithm outperforms the state-of-the-art optimization-based method for computing complementary polynomials is provided.
29 pages, 6 figures
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