Independent researcher Ross Peili has released an open-source demonstration detailing a numerically stable method for training Quantum Signal Processing (QSP) circuits using gradient-based optimization. The project, hosted on GitHub (rosspeili/qsp-pennylane-demo), provides a methodology for implementing high-degree polynomial transformations on quantum hardware by bypassing the traditional reliance on complex analytic solvers, which are often prone to numerical instability.
Addressing the Analytic Bottleneck
Quantum Signal Processing is a fundamental subroutine used to apply polynomial transformations to a signal encoded within a quantum circuit. The canonical approach involves interleaved applications of a signal oracle W(x) and a sequence of controlled phase rotations Rz(ϕ). Conventionally, the set of phase angles {ϕ0,ϕ1,…,ϕd} required to achieve a specific polynomial is found through analytic decomposition. However, as polynomial degrees increase, these solvers frequently suffer from floating-point errors and convergence failures, limiting the complexity of the transformations that can be reliably encoded.
Optimization-Based Framework
Peili’s approach reformulates phase angle determination as a variational machine learning task. Rather than decomposing a target polynomial, the system initializes with random angles and uses the Adam optimizer to minimize the mean squared error (MSE) between the circuit’s output and the target function. The implementation utilizes PennyLane for differentiable quantum programming and JAX for high-performance automatic differentiation. By constructing the circuit from elementary Hadamard and RZ gates, the entire workflow becomes traceable by JAX, allowing for the iterative update of phase angles.
Technical Results and Scaling
The demonstration includes a Jupyter notebook that successfully reproduces a degree-5 Chebyshev approximation of the sine function on the interval [−1,1] with an MSE below 10−3 after approximately 500 training steps. This methodology offers three distinct technical advantages:
- Numerical Stability: It avoids the precision-loss issues inherent in sequential analytic solvers.
- Implicit Specification: Researchers can define target transformations via custom loss functions rather than explicit mathematical formulas.
- Integration: QSP sequences can be embedded as trainable layers within broader Variational Quantum Algorithms (VQAs), enabling end-to-end optimization for specific objective functions.
This optimization-based paradigm is applicable to Hamiltonian simulations, quantum machine learning feature maps, and any domain utilizing the Quantum Singular Value Transform (QSVT). By decoupling circuit deployment from the mathematical overhead of phase decomposition, the project provides a recipe for building numerically stable quantum subroutines on contemporary hardware.
You can find the full technical demonstration on DEV Community here and access the open-source code repository on GitHub here.
May 6, 2026
