Analytical Angle-Finding and Series Expansions for Quantum Signal Processing via Orthogonal Polynomial Theory
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arXiv
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| Natura: | Preprint |
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2026
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| _version_ | 1866913096826617856 |
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| author | Bernard, Pierre-Antoine Wiebe, Nathan |
| author_facet | Bernard, Pierre-Antoine Wiebe, Nathan |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_05321 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Analytical Angle-Finding and Series Expansions for Quantum Signal Processing via Orthogonal Polynomial Theory Bernard, Pierre-Antoine Wiebe, Nathan Quantum Physics Mathematical Physics 68Q12, 33C45 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. |
| title | Analytical Angle-Finding and Series Expansions for Quantum Signal Processing via Orthogonal Polynomial Theory |
| topic | Quantum Physics Mathematical Physics 68Q12, 33C45 |
| url | https://arxiv.org/abs/2605.05321 |