Berntson Bjorn K, Sünderhauf Christoph
Riverlane Research, Cambridge, Massachusetts, USA.
Riverlane, Cambridge, UK.
Commun Math Phys. 2025;406(7):161. doi: 10.1007/s00220-025-05302-9. Epub 2025 Jun 4.
Quantum signal processing is a framework for implementing polynomial functions on quantum computers. To implement a given polynomial , one must first construct a corresponding . 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.
量子信号处理是一种在量子计算机上实现多项式函数的框架。为了实现给定的多项式，首先必须构造一个相应的 。解决这个问题的现有方法采用的数值方法不适合进行显式误差分析。我们提出了一种使用复分析来处理互补多项式的新方法。我们的主要数学结果是一个规范互补多项式的围道积分表示。在单位圆上，这种表示具有特别简单且有效的傅里叶分析解释，我们利用它来开发一种基于快速傅里叶变换的算法，以便在具有显式误差保证的单项式基中高效计算 。提供了数值证据表明我们的算法在计算互补多项式方面优于基于优化的现有方法。
