By Bjorn Berntson and Christoph Sünderhauf
A new paper, published in Communications in Mathematical Physics, addresses a key challenge in Quantum Signal Processing (QSP)—the construction of "complementary polynomials."
Quantum Signal Processing (QSP) is a cornerstone of modern quantum algorithms, finding applications in everything from solving differential equations to computational fluid dynamics. A key challenge in QSP lies in constructing "complementary polynomials," a task that has traditionally relied on inefficient and inaccurate numerical methods.
Our paper, Complementary Polynomials in Quantum Signal Processing, tackles this problem head-on, providing an efficient and accurate method for calculating these essential polynomials. The breakthrough leverages sophisticated mathematical tools, including complex analysis, Fourier analysis, and the Fast Fourier Transform (FFT).
Why is this important? Complementary polynomials are crucial for building quantum circuits used in Quantum Singular Value Transformation (QSVT). QSVT, in turn, has broad applications, including matrix inversion, solving differential equations, and even computational fluid dynamics.
QSVT has had a tremendous impact on quantum computing already because polynomials may be used to approximate a wide variety of functions. This family of algorithms encompasses many prior quantum algorithms, including those for Hamiltonian simulation, Eigenvalue filtering (useful for finding fixed points, for example), and amplitude amplification.
The ability to compute these polynomials efficiently unlocks the full potential of QSVT and related quantum algorithms.
The impact of this research is already being felt. Companies, including Google and Mitsubishi, are utilizing this new method, demonstrating its practical significance.
By providing a faster and more reliable way to compute complementary polynomials, this work paves the way for more powerful and versatile quantum computations.
You can read the full paper here: Complementary Polynomials in Quantum Signal Processing. You can also find more about our recent work on unlocking future quantum applications here.