Generalized Quantum Signal Processing

• Generalized quantum signal processing (GQSP) is a framework that synthesizes arbitrary operator functions using Fourier-based, operator-valued polynomial transformations.
• It addresses traditional limitations by employing a universal sequence of single-qubit rotations interleaved with fixed-time evolution oracles, requiring only one ancilla qubit.
• GQSP enables efficient implementation of quantum simulations, matrix inversions, and thermal state preparations, aligning with hybrid digital-analog hardware.

Generalized quantum signal processing (GQSP) is a framework for synthesizing arbitrary operator functions on quantum computers, extending traditional quantum signal processing (QSP) by implementing operator-valued Fourier (or Laurent) polynomial transformations without the algebraic and ancillary restrictions previously found in QSP methods. GQSP formulations enable the efficient realization of quantum algorithms for simulating operator functions—such as Hamiltonian powers, exponentials, and inverses—using only a single ancilla regardless of approximation degree, employing a universal sequence of single-qubit rotations. This approach is particularly relevant when the physical oracle provides access to fixed-time Hamiltonian evolution rather than block-encoded operators, broadening compatibility with a range of simulation and hybrid digital-analog implementations.

1. Fourier-Based Synthesis of Operator Functions

GQSP is rooted in approximating a target operator function $f(H)$, where $H$ is a Hermitian operator, by truncating its Fourier series: $$f(x) \approx \tilde{g}_q(x) = \sum_{m=-q/2}^{q/2} c_m e^{imx},$$ subject to $|\tilde{g}_q(x)| \leq 1$ for $x \in [-\pi, \pi]$. Phase arguments $x = \lambda t$ correspond to the eigenvalues $\lambda$ of $H$ evolved for fixed time $t$. The synthesis promotes $\tilde{g}_q(x)$ from a classical trigonometric polynomial to a matrix function via a sequence of single-qubit SU(2) operations (pulses) interleaved with controlled calls to the time-evolution oracle: $$O = \mathbb{I} \otimes |0\rangle\langle 0| + e^{-itH} \otimes |1\rangle\langle 1|.$$ This construction enables the embedding of $\tilde{g}_q(x)$ into a single-qubit unitary

$$U_{\tilde{g},\tilde{h}}(x) = \begin{bmatrix} \tilde{g}_q(x) & \tilde{h}_q(x) \ -\tilde{h}_q^*(x) & \tilde{g}_q^*(x) \end{bmatrix},$$

where the off-diagonal component $\tilde{h}_q(x)$ is a complementary Fourier series ensuring overall unitarity.

2. Universality and Expressibility

A central theoretical achievement is the proof of expressibility (Theorem 1 and Lemma 1 in the primary reference): for any normalized Fourier series $\tilde{g}_q(x)$ (with $|\tilde{g}_q(x)| \leq 1$), there exists a complementary polynomial $\tilde{h}_q(x)$ such that $U_{\tilde{g},\tilde{h}}(x)$ is in SU(2) for all $x$ and the operation is fully characterized by a constructive decomposition. This is achieved by representing $1-|\tilde{g}_q(x)|^2$ as a non-negative Laurent polynomial, factoring it as $G(z)G^*(z^{-1})$, and iteratively stripping high-degree components by solving linear systems that guarantee vanishing unwanted Fourier terms at each recursion step. The proposed method is constructive, with classical computational complexity polynomial in $q$ (the Fourier truncation degree).

Table: GQSP Versus Traditional QSP

Property	Traditional QSP	GQSP (Fourier-based)
Ancilla scaling with degree	Grows (log/linear)	Constant (1 qubit)
Oracle type	Block encoding	Fixed-time evolution
Polynomial symmetry/parity req.	Yes	No
Circuit expressibility	Parity-restricted	Unconstrained ($|P|\leq1$)

3. Efficient Parameter Calculation

The classical parameter extraction is efficient and robust: given the Fourier coefficients $\{c_m\}$ parametrizing the operator function, the algorithm first finds the complementary series $\tilde{h}_q(x)$ by solving for Laurent polynomial coefficients that satisfy the SU(2) unitarity constraint. The process removes the need for high-dimensional classical optimization or root-finding, as previously required in QSP approaches, by allowing recursive, linear system solutions at each layer (corresponding to the pulse sequence). The computational overhead scales as poly$(q)$, and the algorithm produces the required rotation angles for each step in the pulse sequence that defines the quantum circuit implementing $f[H]$.

4. Ancilla Efficiency and Circuit Architecture

A salient feature of this GQSP construction is the use of only a single ancilla qubit, regardless of the degree of the approximating Fourier series. Prior block-encoding-based approaches required an ancillary register whose size scales with the expansion degree, impacting both the experimental qubit count and circuit depth as well as the postselection success probability. The fixed ancilla architecture notably reduces circuit complexity, making the protocol amenable to short-depth, intermediate-scale, and hybrid digital-analog quantum hardware.

5. Compatibility with Existing Simulation Paradigms

The GQSP method is designed to integrate directly with Trotterized Hamiltonian simulation and hybrid digital-analog quantum platforms, as it requires as an oracle only a fixed time evolution operator $e^{-itH}$—exactly the output of most product formula or analog simulation routines. Unlike prior Fourier-based frameworks that demanded evolving the system at variable times (increasing query cost), this approach leverages a fixed $t$, thereby aligning with current hardware capabilities and minimizing control complexity.

6. Algorithmic Applications and Impact

Practical applications of GQSP include:

• Hamiltonian simulation: Direct synthesis of $e^{-iHt}$ and, via suitable Fourier decompositions, polynomial or spectral functions of $H$ with complexity resources that scale favorably with both simulation time and error.
• Matrix inversion and linear system solvers: Synthesis of $f(H)=H^{-1}$ or regularized functions (e.g., for HHL-type algorithms), exploiting Fourier approximants to achieve robust numerical behavior.
• Imaginary-time evolution (thermal Gibbs state preparation): Exact synthesis of $f(H)=e^{-\beta H}$, crucial for quantum thermodynamics and quantum chemistry models.
• Hybrid digital-analog workflows: The method’s fixed-time evolution oracle matches naturally with analog simulation primitives and digital, few-qubit control, broadening the range of compatible experimental platforms.

7. Conclusion and Significance

Fourier-based GQSP establishes a paradigm in which arbitrary (bounded) operator functions of Hermitian matrices can be efficiently and universally synthesized on quantum hardware with minimal ancilla overhead and without the polynomial symmetry or oracle constraints of previous approaches. The algorithmic design—iterated single-qubit rotations interleaved with fixed-time oracles—underpins a range of quantum computational primitives, lowers the barriers for implementation on near-term devices, and offers a flexible, scalable tool for both algorithm development and practical quantum simulation.