Analytical Angle-Finding and Series Expansions for Quantum Signal Processing via Orthogonal Polynomial Theory
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Quantum Physics arXiv:2605.05321 (quant-ph) [Submitted on 6 May 2026] Title:Analytical Angle-Finding and Series Expansions for Quantum Signal Processing via Orthogonal Polynomial Theory Authors:Pierre-Antoine Bernard, Nathan Wiebe View a PDF of the paper titled Analytical Angle-Finding and Series Expansions for Quantum Signal Processing via Orthogonal Polynomial Theory, by Pierre-Antoine Bernard and Nathan Wiebe View PDF 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 $\epsilon$-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. Comments: Subjects: Quantum Physics (quant-ph); Mathematical Physics (math-ph) MSC classes: 68Q12, 33C45 Cite as: arXiv:2605.05321 [quant-ph] (or arXiv:2605.05321v1 [quant-ph] for this version) https://doi.org/10.48550/arXiv.2605.05321 Focus to learn more arXiv-issued DOI via DataCite (pending registration) Submission history From: Pierre-Antoine Bernard [view email] [v1] Wed, 6 May 2026 18:00:26 UTC (44 KB) Full-text links: Access Paper: View a PDF of the paper titled Analytical Angle-Finding and Series Expansions for Quantum Signal Processing via Orthogonal Polynomial Theory, by Pierre-Antoine Bernard and Nathan WiebeView PDFTeX Source view license Current browse context: quant-ph new | recent | 2026-05 Change to browse by: math math-ph math.MP References & Citations INSPIRE HEP NASA ADSGoogle Scholar Semantic Scholar export BibTeX citation Loading... BibTeX formatted citation × loading... Data provided by: Bookmark Bibliographic Tools Bibliographic and Citation Tools Bibliographic Explorer Toggle Bibliographic Explorer (What is the Explorer?) Connected Papers Toggle Connected Papers (What is Connected Papers?) Litmaps Toggle Litmaps (What is Litmaps?) scite.ai Toggle scite Smart Citations (What are Smart Citations?) Code, Data, Media Code, Data and Media Associated with this Article alphaXiv Toggle alphaXiv (What is alphaXiv?) Links to Code Toggle CatalyzeX Code Finder for Papers (What is CatalyzeX?) DagsHub Toggle DagsHub (What is DagsHub?) GotitPub Toggle Gotit.pub (What is GotitPub?) Huggingface Toggle Hugging Face (What is Huggingface?) ScienceCast Toggle ScienceCast (What is ScienceCast?) Demos Demos Replicate Toggle Replicate (What is Replicate?) Spaces Toggle Hugging Face Spaces (What is Spaces?) Spaces Toggle TXYZ.AI (What is TXYZ.AI?) Related Papers Recommenders and Search Tools Link to Influence Flower Influence Flower (What are Influence Flowers?) Core recommender toggle CORE Recommender (What is CORE?) Author Venue Institution Topic About arXivLabs arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs. Which authors of this paper are endorsers? | Disable MathJax (What is MathJax?)