Mathematical and numerical analysis of quantum signal processing

Abstract:Quantum signal processing (QSP) provides a representation of scalar polynomials of degree $d$ as products of matrices in $\mathrm{SU}(2)$, parameterized by $(d+1)$ real numbers known as phase factors. QSP is the mathematical foundation of quantum singular value transformation (QSVT), which is often regarded as one of the most important quantum algorithms of the past decade, with a wide range of applications in scientific computing, from Hamiltonian simulation to solving linear systems of equations and eigenvalue problems. In this article we survey recent advances in the mathematical and numerical analysis of QSP. In particular, we focus on its generalization beyond polynomials, the computational complexity of algorithms for phase factor evaluation, and the numerical stability of such algorithms. The resolution to some of these problems relies on an unexpected interplay between QSP, nonlinear Fourier analysis on $\mathrm{SU}(2)$, fast polynomial multiplications, and Gaussian elimination for matrices with displacement structure.

Quantum Physics (quant-ph); Classical Analysis and ODEs (math.CA); Numerical Analysis (math.NA)

arXiv:2510.00443 [quant-ph]

(or arXiv:2510.00443v2 [quant-ph] for this version)

https://doi.org/10.48550/arXiv.2510.00443 Focus to learn more arXiv-issued DOI via DataCite

https://doi.org/10.1137/25M1806120 Focus to learn more DOI(s) linking to related resources

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