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

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.

• The paper introduces an analytical framework to solve the angle-finding problem in QSP using orthogonal polynomial theory.
• It derives explicit angle parametrizations for polynomial families such as Hermite, Jacobi, and Rogers–Szegő, facilitating precise resource estimation.
• The study delineates the expressibility limits of standard, SU(1,1), and bivariate QSP by establishing necessary polynomial conditions for block-encoding.

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

Introduction and Motivation

Quantum Signal Processing (QSP) is central to modern quantum algorithmics, especially as the foundational primitive enabling optimal Hamiltonian simulation, quantum eigenvalue transformations, and quantum block-encoding constructions. However, the precise characterization of which polynomial transformations of a unitary $U$ are achievable by various QSP protocols, and the explicit determination of the required single-qubit rotation angles ("angle-finding problem"), remain unresolved except in trivial cases (e.g., Chebyshev). Most practical instances rely on indirect numerical methods that impede analytical understanding and impede resource analysis.

This work introduces a framework leveraging orthogonal and biorthogonal polynomial theory to provide a comprehensive, analytic characterization of polynomial sequences implemented by both standard and generalized QSP protocols. The methodology establishes a rigorous correspondence between variants of QSP and specific classes of orthogonal polynomials (OPs), Laurent biorthogonal polynomials (LBPs), and orthogonal polynomials on the unit circle (OPUC). Using this connection, explicit angle parametrizations for a wide range of non-trivial polynomial families (including Hermite, Jacobi, and Rogers–Szegő) are derived, and the expressibility and limitations of both univariate and bivariate QSP protocols are sharply delineated.

QSP as Block-Encoding of Polynomial Bases

The analysis commences by formalizing QSP protocols as iterative block-encoding circuits. At each iteration $i$, a subcircuit involving controlled applications of $U$ and single-qubit rotations produces a block-encoding of a degree-$i$ monic polynomial $\hat{P}_i(U)$. The set $\{\hat{P}_i(U)\}_{i=0}^n$ provides a polynomial basis for constructing higher-degree polynomials. This enables an efficient hybrid LCU-QSP protocol wherein any target polynomial $P(U)$ admits an expansion:

$$P(U) = \sum_{i=0}^n v_i \frac{\hat{P}_i(U)}{\alpha_i}$$

where $\alpha_i$ denotes normalization factors.

Thus, the set of polynomials implementable by a QSP protocol is determined by the structural properties of the sequence $\{\hat{P}_i\}$, specifically the type of recurrence relation it satisfies. The question of angle-finding is thereby reduced to determining angle sequences such that the induced recurrence matches that of the desired polynomial family.

Orthogonal Polynomial-Based QSP (OP-QSP)

A novel QSP variant, OP-QSP, is constructed where each recursion step is engineered such that the sequence $i$0 satisfies a real three-term recurrence:

$i$1

This is precisely the recurrence type characterizing OPs relative to a positive-definite (integral or moment) form, within Favard’s theorem. The explicit mapping between $i$2 and protocol angles $i$3 is derived, so that, given any orthogonal polynomial family (e.g., Hermite, shifted Chebyshev, Jacobi), the unique sequence of QSP angles can be constructed analytically.

Concrete examples are provided:

• Hermite family: Allows block-encoding of $i$4, with $i$5 smooth and square-integrable with respect to a Gaussian, using only $i$6 gates for accuracy $i$7, and normalization scaling as $i$8 for a tuning parameter $i$9.
• Jacobi and Rogers–Szegő: Similar methodology is applied, with explicit angle recursions demonstrated.

The general complexity and normalization scaling properties are analyzed for these families.

Generalized QSP and Laurent Biorthogonal Polynomials

The standard (generalized) QSP recursion does not in general produce orthogonal polynomial sequences, but instead induces a coupled recurrence yielding a pair $U$0, which, as proven, are Laurent biorthogonal polynomials.

A key result is the identification of the necessary and sufficient conditions for a class of LBPs to be achievable by QSP: the recurrence coefficients must satisfy specific trigonometric relations determined by the circuit-level QSP parameters. For a given target pair $U$1, explicit expressions for the required angle sequence are derived in terms of moments of a suitable quasi-definite linear functional, with the dependence rationalized via determinantal formulae.

SU(1,1)-QSP and OPUC

A structurally distinct variant, "SU(1,1)-QSP" (continuous-variable QSP), is mapped directly to the theory of OPUC. The protocol naturally realizes the Szegő recurrence:

$U$2

with Verblunsky coefficients $U$3 determined by the SU(1,1) circuit angles. The Zeros Theorem of OPUC theory immediately yields a complete characterization: SU(1,1)-QSP can block-encode all polynomials whose zeros lie strictly inside the open unit disk. Conversely, only such polynomials are achievable.

Angle-finding for arbitrary polynomials with roots in the disk is reduced to the computation of OPUC moments (via the Bernstein–Szegő approach) and thus posed as a deterministic, explicit procedure.

Bivariate QSP and Biorthogonality Constraints

The extension of QSP to multivariate (especially bivariate) settings is highly nontrivial. This work rigorously establishes that while the bivariate recursion produces polynomial sequences with certain biorthogonality properties, not all polynomials satisfying boundedness constraints on the torus are block-encodable by bivariate QSP. Matrix-valued functional constraints, derived from the structure of the protocol, are established as necessary conditions.

A counterexample to the "FRT = QSP" conjecture is examined, and it is analytically confirmed to violate the new necessary conditions. This substantially clarifies the limitations of bivariate and multivariate QSP expressivity.

Implications and Future Directions

This analysis puts the study of QSP on a rigorous algebraic and analytic foundation, enabling:

• Analytical resource estimation: The complexity of block-encoding is now determined by the convergence rate and normalization in the corresponding orthogonal polynomial expansion.
• Angle-finding solubility: The recasting of angle-finding as orthogonalization in polynomial sequences yields determinantal closed forms for broad families.
• Expressibility demarcation: The full reach and limitations of current QSP protocols, especially for generalized, SU(1,1), and multivariate settings, are precisely characterized in terms of polynomial algebraic properties.

Immediate ramifications include improved design of quantum algorithms for simulation, control, and metrology where analytic control of the function class is required. The recognition of deep limitations in multivariate QSP points to the need for fundamentally new circuit constructions, possibly incorporating more general matrix-valued polynomial recursions or higher-dimensional ancilla schemes.

Conclusion

This work provides a comprehensive algebraic and analytic treatment of angle-finding and polynomial expressibility in quantum signal processing protocols. By establishing strong connections with the theory of orthogonal and biorthogonal polynomials and exploiting recurrence relations, a unified framework for analytic block-encoding of polynomials is developed. These results offer immediate, concrete tools for quantum algorithm designers and furnish formal boundaries for the power of QSP-based constructions, while also motivating further research into generalized architectures for multivariate quantum polynomial transformation.