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<p>Quantum signal processing is a framework for implementing polynomial functions on quantum computers. To implement a given polynomial <i>P</i>, one must first construct a corresponding <i>complementary polynomial</i> <i>Q</i>. 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 <i>Q</i> 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.</p>

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Complementary Polynomials in Quantum Signal Processing

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2025年6月4日

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  • Bjorn K. Berntson,
  • Christoph Sünderhauf

摘要

Quantum signal processing is a framework for implementing polynomial functions on quantum computers. To implement a given polynomial P, one must first construct a corresponding complementary polynomial Q. 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 Q 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.