Analytical Angle-Finding and Series Expansions for Quantum Signal Processing via Orthogonal Polynomial Theory

TL;DR

This paper develops a new analytical framework for quantum signal processing using orthogonal polynomial theory, providing explicit formulas for angles and characterizing achievable polynomial bases.

Contribution

It introduces a novel method leveraging orthogonal and biorthogonal polynomial theory to explicitly determine quantum signal processing angles and characterize achievable polynomial bases.

Findings

Explicit formulas for quantum signal processing angles for Hermite, Jacobi, and Rogers-Szeg ext{"o} polynomial families.

Achieves $O( ext{log}(1/\epsilon))$ gate complexity for approximating smooth functions.

Characterizes polynomials achievable by $ ext{SU}(1,1)$-QSP in terms of roots.

Abstract

Quantum signal processing is a powerful framework in quantum algorithms, playing a central role in Hamiltonian simulation and related applications. The sequence of polynomials implemented at each step of this protocol provides a polynomial basis for block-encoding any polynomial of a unitary. We characterize the achievable polynomial bases in terms of their orthogonality or biorthogonality with respect to a linear functional admitting an integral representation. Explicit expressions for the quantum signal processing angles are derived for families of polynomial sequences, including Hermite, Jacobi, and Rogers-Szeg\H{o} polynomials. We show that $2n+2$ rotation angles are required to encode a sequence of polynomials in these classes up to degree $n$. We use this result to show that an $\epsilon$-approximation of a smooth function $f$ can be block-encoded using $O(\log(1/\epsilon))$ gates via its Hermite series expansion. The connections established with the theory of orthogonal and biorthogonal polynomials lead to a new method for solving the quantum signal processing angle-finding problem, yielding explicit expressions for the angles. They also provide a complete characterization of the polynomials achievable by $\mathrm{SU}(1,1)$-QSP in terms of their roots. Biorthogonality properties are shown to hold in the bivariate QSP setting, yielding a set of necessary conditions for achievable polynomials.