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

• URL: http://arxiv.org/abs/2605.05321v1
• Date: Wed, 06 May 2026 18:00:26 GMT
• Title: Analytical Angle-Finding and Series Expansions for Quantum Signal Processing via Orthogonal Polynomial Theory
• Abstract summary: Quantum signal processing is a powerful framework in quantum algorithms, playing a central role in Hamiltonian simulation and related applications.<n>We characterize the achievable bases in terms of their orthogonality or biorthogonality with respect to a functional admitting an integral representation.<n>Explicit expressions for the quantum signal processing angles are derived for families of sequences.
• Score: 0.5156484100374059
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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-Szegő 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 $ε$-approximation of a smooth function $f$ can be block-encoded using $O(\log(1/ε))$ 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.

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