Generalised Quantum Signal Processing (GQSP)

• Generalised Quantum Signal Processing is an extension of conventional QSP that replaces fixed-axis phase rotations with arbitrary SU(2) operations to target complex polynomial transformations.
• It enables efficient implementation of unitary, Hermitian, and multivariate matrix functions, supporting applications such as Hamiltonian simulation and quantum algorithm design.
• State-of-the-art synthesis methods like inverse NLFT, layer stripping, and Riemann–Hilbert factorization address the challenges of constructing complementary polynomials and numerical stability.

Generalised Quantum Signal Processing (GQSP) denotes a family of extensions of standard quantum signal processing in which the single-qubit modulation layer is broadened from fixed-axis phase rotations to more general (SU(2)) operations, and the target transformation is broadened from the original real-polynomial setting on ([-1,1]) to complex polynomials, Laurent polynomials, matrix-valued polynomial matrices, and several multivariate or continuous-variable variants on the unit circle [2510.00443, 2402.03016]. In current usage, the term covers at least three tightly connected viewpoints: an (SU(2)) polynomial-of-a-unitary formalism with arbitrary complex (P,Q) subject to (|P|^2+|Q|^2=1) [2402.03016]; a nonlinear-Fourier-theoretic description in which GQSP phase synthesis is equivalent to inverse (\mathrm{SU}(2)) nonlinear Fourier transform (NLFT) [2505.12615, 2510.00443]; and algorithmic constructions that apply this machinery to Hamiltonian simulation, Hermitian matrix functions, multivariate matrix polynomials, and hybrid oscillator–qubit processing [2401.10321, 2512.18249, 2408.01439, 2510.10495].

1. Standard QSP and the point of generalisation

Standard QSP is usually written as an alternating product of a signal-dependent (SU(2)) matrix and single-qubit phase rotations. In the survey formulation,
[
U_d(x,\Psi):=e^{i\psi_0 Z}\prod_{j=1}^{d}\left[W(x)e^{i\psi_j Z}\right],\qquad

W(x)=e^{i\arccos(x)X}

\begin{pmatrix}
x & i\sqrt{1-x^2}\
i\sqrt{1-x^2} & x
\end{pmatrix},
]
and the achievable scalar data are constrained by degree, parity, and the unitarity identity (|P(x)|^2+(1-x^2)|Q(x)|^2=1) [2510.00443]. In the Laurent-polynomial picture, standard QSP equivalently acts on (w=e^{i\theta}\in U(1)) and realizes real Laurent polynomials with parity constraints through interleavings of (W_z(\theta)=\mathrm{diag}(w,w^{-1})) and fixed-axis rotations [2402.03016].

GQSP enlarges this framework by replacing the one-parameter phase rotations of standard QSP with more general (SU(2)) gates. In the survey’s (SU(2)) formulation, the elementary modulation is
[
R(\psi,\phi):=
\begin{pmatrix}
\cos\psi & e^{i\phi}\sin\psi\
-\,e^{-i\phi}\sin\psi & \cos\psi
\end{pmatrix},
]
and the target can be a general complex polynomial (b(z)\in\mathbb{C}[z]) on the unit circle, rather than a real parity-constrained polynomial on ([-1,1]) [2510.00443]. The practical consequence is that complex-valued transformations that would otherwise require decomposition into several real even/odd sectors in ordinary QSP can be represented in a single generalized sequence [2402.03016].

A second broadening, used prominently for Hamiltonian simulation, replaces control between identity and (U) by control between (U) and (U^\dagger). This produces Laurent polynomials in both positive and negative powers and is the source of the directional or bidirectional variants of GQSP [2401.10321].

2. Canonical (SU(2)) formulations on the unit circle

In the polynomial-of-a-unitary formulation adopted by Motlagh–Wiebe and subsequent work, the central theorem states that for a unitary (U) there exist generalized rotation parameters such that
[

\left(\prod_{j=1}^{d} R(\theta_j,\phi_j,0)\,CU_0\right)R(\theta_0,\phi_0,\lambda)

\begin{bmatrix}
P(U) & *\
Q(U) & *
\end{bmatrix}
]
if and only if (P,Q\in\mathbb{C}[z]) have degree at most (d) and satisfy
[
|P(z)|^2+|Q(z)|^2=1,\qquad |z|=1
]
[2507.11142, 2401.10321]. In this sense, GQSP is a unitary-valued completion theory: once a bounded target polynomial (P) is chosen, the central structural task is to find a complementary polynomial (Q) that restores unitarity on the unit circle.

The survey presents the same idea in a slightly different normal form. Given a target polynomial (b(z)\in\mathbb{C}[z]) of degree (d) with (\sup_{z\in\mathbb{T}}|b(z)|\le 1), GQSP seeks parameters such that
[
\begin{pmatrix}
\cdot & b(z)\
\cdot & \cdot

\end{pmatrix}

R(\psi_0,\phi_0)\prod_{k=1}^{d}
\left(
\begin{pmatrix}
z&0\
0&1
\end{pmatrix}
R(\psi_k,\phi_k)
\right),
]
so the target appears directly as an entry of an (SU(2))-valued polynomial matrix [2510.00443]. This formulation removes the standard QSP parity restriction at the level of the target polynomial.

Directional GQSP extends the theorem further to Laurent polynomials. Using the signal operator (\mathrm{diag}(U,U^\dagger)), one obtains
[
\begin{bmatrix}
P(U) & -Q(U)^\dagger\
Q(U) & P(U)^\dagger
\end{bmatrix}
]
for (P,Q\in\mathbb{C}[z^{-1},z]), with degree bounds, a parity constraint (\mathrm{Parity}(P),\mathrm{Parity}(Q)=d\bmod 2), and the same unitarity condition (|P(z)|^2+|Q(z)|^2=1) on (|z|=1) [2401.10321]. This bidirectional form is especially natural when Fourier expansions contain both positive and negative harmonics.

3. Complementary polynomials, inverse NLFT, and angle synthesis

A persistent theme in GQSP is that the main synthesis difficulty is shifted from direct phase solving to the construction of a complementary polynomial. In the Hermitian-matrix-function formulation, this shift is stated explicitly: GQSP “shifts the main difficulty from angle synthesis to constructing a complementary polynomial” [2512.18249]. In the survey language, this is the problem of completing a bounded target (b(z)) to a pair ((a,b)) satisfying
[
aa^+bb^=1
]
on the unit circle, with (a^*) outer and (a^*(0)>0) [2510.00443].

The deeper mathematical explanation is that GQSP is equivalent to the (\mathrm{SU}(2)) nonlinear Fourier transform. For a compactly supported sequence (\gamma:\mathbb{Z}\to\mathbb{C}),
[
\overbrace{\gamma}(z):=\prod_{k=m}^{n}
\left[
\frac{1}{\sqrt{1+|\gamma_k|^2}}
\begin{pmatrix}
1 & \gamma_k z^k\
-\overline{\gamma_k}z^{-k} & 1
\end{pmatrix}

\right]

\begin{pmatrix}
a(z) & b(z)\
-b^*(z) & a^*(z)
\end{pmatrix},
]
and the GQSP parameters are simply the polar coordinates of the NLFT coefficients:
[
\gamma_k=\tan\psi_k\,e^{i\phi_k},\qquad
\psi_k=\arctan|\gamma_k|,\qquad
\phi_k=\Arg(\gamma_k)
]
[2505.12615, 2510.00443]. Under this identification, GQSP phase synthesis becomes inverse NLFT.

This viewpoint led to a sequence of increasingly robust classical synthesis algorithms. The survey records three principal inverse methods: layer stripping with (O(d^2)) complexity, a Riemann–Hilbert factorization method reducible to (O(d^2)) using displacement structure, and an inverse nonlinear fast Fourier transform (INLFFT) with (O(d\log^2 d)) complexity [2510.00443]. The paper devoted specifically to inverse NLFT proves numerical stability of layer stripping under an outer-complementary-polynomial condition and introduces a stable INLFFT on (\mathrm{SU}(2)) with near-linear complexity, explicitly applicable to both QSP and GQSP [2505.12615].

A complementary line of work addresses completion directly. The survey describes a Weiss algorithm in which, given (b(z)), one computes (R(z)=\log\sqrt{1-|b(z)|^2}), takes a periodic Hilbert transform, and sets (a(z)=e^{G(z)}), thereby constructing the outer complement [2510.00443]. For finite-degree GQSP, “Robust Angle Finding for Generalized Quantum Signal Processing” develops a Prony-based completion and carving pipeline. It reports angle sequences of precision (10^{-13}) up to polynomial degrees of hundreds within a second and identifies GQSP’s loss of standard QSP symmetries as a main reason why naive optimization-based phase finding performs poorly in the generalized setting [2402.03016].

4. Multivariate, higher-rank, and matrix-valued generalisations

The multivariate literature shows that “generalised” QSP is not a single construction but a hierarchy of signal models. In the commuting homogeneous bivariate setting, the sequence
[
F(a,b)=U_0
\begin{pmatrix}
a&0\
0&b
\end{pmatrix}
U_1\cdots
\begin{pmatrix}
a&0\
0&b
\end{pmatrix}
U_d
]
admits a complete Haah-type characterization: (F(a,b)) is achievable if and only if it is a homogeneous degree-(d) polynomial in commuting variables (a,b), unitary on (U(1)\times U(1)), and satisfies (\det F(a,b)=(ab)^d) [2312.09072]. For an alternative inhomogeneous bivariate scheme built from two QSP-type oracles, the same paper proves sufficiency only when the degree in one variable is at most (1), and gives an explicit degree-((2,2)) counterexample refuting the earlier Rossi–Chuang characterization conjecture [2312.09072].

A distinct three-dimensional analytic multivariate QSP formalism replaces classical choice of signal variable by a qutrit signal operator (\widetilde W=\mathrm{diag}(1,a,b)). In that framework, a polynomial state of degree (n) with non-zero corner coefficients at (1), (a^n), and (b^n) is fully decomposable, while simple span conditions on the edge coefficients give immediate non-implementability, and a determinant-based invariant yields an inapproximability radius around certain non-decomposable polynomials [2407.20823]. These results make explicit that the univariate “almost any bounded polynomial” intuition does not survive in the multivariate setting.

A different direction generalizes the ancilla group from (U(2)) to (U(N)). In the (U(N))-QSP/QSVT framework, one uses an (N)-dimensional signal ancilla together with projector-controlled applications of the input unitary and obtains an (N\times N) matrix of polynomial blocks (P_{jk}(U)), so multiple polynomials are realized simultaneously from one block-encoded input [2408.01439]. The same paper characterizes achievable matrix-valued polynomial targets by the condition that all singular values of (P(z)) lie in ([0,1]) on (|z|\le 1), gives a recursive construction, and uses the framework to realize bi-variate polynomial functions and to analyze quantum amplitude estimation with asymptotically optimal query complexity [2408.01439].

5. Functions of unitaries and Hermitians

The most visible algorithmic impact of GQSP so far lies in unitary and Hermitian matrix-function synthesis. In Hamiltonian simulation, directional GQSP replaces control between identity and the walk operator by control between forward and reverse walk steps. The resulting Laurent-polynomial construction packs the Jacobi–Anger harmonics into roughly half as many controlled walk applications as standard QSP, while preserving the optimal (O(\lambda t+\log(1/\epsilon))) asymptotic scaling [2401.10321]. The improvement is a constant-factor query reduction rather than a change in asymptotic order.

For Hermitian matrix functions, one 2025 construction uses GQSP as a polynomial-of-a-unitary engine without ever block-encoding the Hermitian matrix. For a Hermitian (A) with (|A|\le 1), it defines the Halmos-type unitary
[
U=A+i\sqrt{I-A^2},\qquad
A=\frac{1}{2}(U+U^\dagger),
]
proves that every power (A^n) can be written as a symmetric combination (R_n(U)+R_n(U^\dagger)) with closed-form (R_n), and then implements
[
P(A)=\widetilde P(U)+\widetilde P(U^\dagger)
]
as a linear combination of two GQSP circuits with only two ancillas and post-selection on (\ket{00}) [2512.18249]. The same work emphasizes that GQSP natively handles complex polynomials and that the main structural requirement is efficient implementation of (\sqrt{I-A^2}).

A related application domain is iterative linear algebra. “Quantum Power Iteration Unified Using Generalized Quantum Signal Processing” formulates quantum power iteration, power Lanczos, inverse iteration, and folded spectrum methods as different polynomial filters (P(H)) acting on a block-encoded Hamiltonian, all implemented by the same GQSP circuit template. The paper states that the number of queries is equal to the polynomial degree used in each method, reports numerical benchmarks on molecular Hamiltonians, and highlights that the framework can avoid Suzuki–Trotter decomposition [2507.11142].

The framework has also been exported to non-Hermitian problems through Hermitian embeddings. In a 2026 proof-of-principle for the two-dimensional Black–Scholes equation, the non-Hermitian backward-Euler matrix (\tilde M) is embedded into
[
H=
\begin{pmatrix}
0 & \tilde M\
\tilde M^T & 0
\end{pmatrix},
]
an odd polynomial approximates (1/x) on the spectrum of the rescaled Hermitian (H), and Hermitian-GQSP implements the resulting inverse transformation. Numerical simulations for two-asset European call options show close agreement with the classical backward-Euler finite-difference benchmark [2606.00458].

6. Continuous-variable and hybrid-architecture GQSP

Hybrid qubit–oscillator systems motivate another major branch of GQSP. In “Single-shot Quantum Signal Processing Interferometry,” a qubit–oscillator coupling
[

W_z=e^{-i h(\hat x,\hat p)\sigma_z t}

\begin{bmatrix}
\omega(\hat x,\hat p) & 0\
0 & \omega(\hat x,\hat p)^{-1}
\end{bmatrix}
]
is treated as a block encoding of a bosonic unitary, and ordinary QSP is lifted to operator-valued Laurent polynomials (F(\omega)), (G(\omega)) on an infinite-dimensional oscillator Hilbert space [2311.13703]. Conjugating a single-shot displacement signal (S_\beta) by a bosonic QSP sequence and its inverse produces a response probability that is itself a Laurent polynomial in (\nu=e^{i2\kappa\beta}). The paper uses this to implement binary threshold tests for displacement sensing and argues that the sensing accuracy, given a single-shot qubit measurement, scales inversely with the sensing time or circuit depth of the algorithm [2311.13703].

An even more explicitly generalized construction appears in oscillator–qubit GQSP for vibronic simulation. There the scalar signal (x\in U(1)) is promoted to the bosonic unitary
[
\hat U=e^{i\frac{\pi}{L}\hat Q},
]
and GQSP is used to synthesize arbitrary bosonic phase gates (e^{if(\hat Q)}) from Fourier/Laurent expansions in (\hat U). For analytic target phases, the Fourier truncation degree required for error (\varepsilon) is (d=\mathcal O(\ln(1/\varepsilon))), so the paper states that arbitrary bosonic phase gates can be synthesized with moderate circuit depth (O(\log(1/\varepsilon))), with approximation cost scaling by the Fourier bandwidth of the target bosonic phase rather than by the degree of nonlinearity [2510.10495]. In the uracil-cation case study, this compiler is used for state preparation and nonadiabatic time evolution on hybrid oscillator–qubit processors [2510.10495].

These continuous-variable examples make clear that GQSP is not restricted to qubitized finite-dimensional spectra. In the survey’s broader terminology, it also includes infinite QSP and Szegő-class generalizations, where infinitely many (SU(2)) factors represent non-polynomial functions under an outer-function condition and the relevant analysis is carried by NLFT and Riemann–Hilbert factorization [2510.00443].

7. Limitations, numerical bottlenecks, and open problems

Despite its broadened expressivity, GQSP retains the central classical bottleneck of phase synthesis. The 2024 angle-finding paper identifies this as “the largest bottleneck” of QSP and especially of GQSP once arbitrary single-qubit unitaries are allowed, because the symmetry structures that aid standard-QSP optimization are lost [2402.03016]. The NLFT-based literature sharpens this point: inverse synthesis is numerically stable only under suitable conditions, notably when the complementary polynomial is chosen outer and the target remains a fixed distance from the unit circle, typically (|b(z)|\le 1-\eta) [2505.12615, 2510.00443].

The multivariate theory is also incomplete. Homogeneous commuting bivariate QSP is characterized, and several sufficient and necessary conditions are known for inhomogeneous or higher-dimensional variants, but no complete characterization exists for general multivariate non-commuting constructions [2312.09072, 2407.20823]. The survey further notes that no general QSVT-like lifting exists yet for these multi-parameter generalizations, so their algorithmic consequences remain only partially developed [2510.00443].

Application-specific constraints remain substantial. The block-encoding-free Hermitian synthesis route requires efficient implementation of (\sqrt{I-A^2}) and accepts a post-selection overhead that is not removed within the construction itself [2512.18249]. GQSP-based quantum power methods can have very small success probabilities and would generally require amplitude amplification, which the paper leaves for future resource analysis [2507.11142]. In hybrid oscillator–qubit GQSP, one trades truncation overhead for heralded success probability, and the total success probability over many Trotter layers can become small unless the off-diagonal GQSP block is strongly suppressed [2510.10495].

Several open directions recur across the literature. These include extending Hermitian constructions to normal matrices and rational functions, developing better rational or polynomial approximations for inverse-like filters, completing the characterization of multivariate and non-commuting GQSP, integrating outer-complement construction and inverse NLFT into a single stable high-performance synthesis stack, and clarifying how the broader (SU(N)), multivariate, and continuous-variable generalizations should be lifted into full matrix-transformation frameworks analogous to QSVT [2512.18249, 2507.11142, 2505.12615, 2510.00443]. In that sense, GQSP is already a mature design language for polynomial transformations of unitaries, but still an active research program at the level of characterization, numerical synthesis, and fault-tolerant implementation.