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arXiv:2605.05321

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

Pierre-Antoine Bernard, Nathan Wiebe

quant-phmath-phSubmitted May 6, 2026• Announced May 8, 2026• 0 citations
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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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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+22n+22n+2 rotation angles are required to encode a sequence of polynomials in these classes up to degree nnn. We use this result to show that an εεε-approximation of a smooth function fff can be block-encoded using O(log⁡(1/ε))O(\log(1/ε))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 SU(1,1)\mathrm{SU}(1,1)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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Quantum Signal Processing (QSP), together with the quantum singular value transformation, is one of the central quantum algorithms due to its efficiency and generality in many fields including quantum simulation, quantum machine learning, and quantum cryptography. The largest bottleneck of QSP and its family is its difficulty in finding the phase angle sequence for signal processing. We find that this is in particular prominent when one employs the generalized formalism of the QSP, or the GQSP, to employ arbitrary single-qubit unitaries for signal processing operator. In this work, we extend the framework of GQSP and propose a robust angle finding algorithm. The proposed angle finding algorithm, based on Prony's method, successfully generates angle sequence of precision 10−1310^{-13}10−13 up to polynomial degrees of hundreds within a second. By applying our method to Hamiltonian simulation, we find that the number of calls, or queries, to signal operators are essentially halved compared to the ordinary framework of QSP.
Quantum signal processing (QSP) and its extensions are increasingly popular frameworks for developing quantum algorithms. Yet QSP implementations still struggle to complete a classical pre-processing step (‘QSP-processing’) that determines the set of SU (2) rotation matrices defining the QSP circuit. We introduce a method of QSP-processing for complex polynomials that identifies a solution without optimization or root-finding and verify the success of our methods with polynomials characterized by floating point precision coefficients. We demonstrate the success of our technique for relevant target polynomials and precision regimes, including the Jacobi-Anger expansion used in QSP Hamiltonian Simulation. For popular choices of sign and inverse function approximations, we characterize regimes where all known QSP-processing methods should be expected to struggle without arbitrary precision arithmetic.
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Modern quantum algorithms are increasingly formulated as coherent procedures for implementing polynomial transformations of operators and singular values. This perspective provides a powerful and unifying language for quantum algorithm design, connecting a wide range of distinct problems through five closely related key tools: block-encoding, qubitization, QSP, QSVT and GQSP. Block-encoding embeds non-unitary matrices into larger unitaries; qubitization converts block-encodings into structured operators; QSP, QSVT and GQSP enable polynomial transformations with near-optimal query complexity. Together, these techniques form a general toolkit for transforming matrix functions into implementable quantum circuits. This paper develops these techniques from first principles as a unified framework for constructing quantum algorithms. We apply this framework to representative applications to highlight design principles and demonstrate how distinct algorithms can be constructed from a unified sequence of operator transformations. A central contribution is a systematic decision workflow for selecting the appropriate approach according to the operator structure and the desired transformation polynomial. This perspective clarifies when direct GQSP or through qubitization, or Laurent expansion, or QSVT is most appropriate. We organize algorithmic design into an end-to-end pipeline: identifying the target matrix function, constructing an appropriate block-encoding, determining the relevant spectral domain, designing a polynomial or Laurent-polynomial approximation, synthesizing the phase factors, and translating the transformation into an executable quantum circuit. By applying this unified framework to example applications, we showcase a practical methodology for reasoning, designing, and implementing quantum algorithms based on polynomial transformations.