A Gradient-Descent Approach to Quantum Signal Processing Phase Angle Determination

THESSALONIKI, Greece - May 6, 2026 - PRLog -- A newly released open-source demonstration from independent quantum computing researcher Ross Peili showcases a practical and numerically stable method for training Quantum Signal Processing circuits, replacing complex analytic solvers with standard gradient-based optimization. The project, hosted on GitHub under the repository `rosspeili/qsp-pennylane-demo`, provides a working blueprint for practitioners seeking to implement high-degree polynomial transformations on quantum hardware without the traditional mathematical overhead.

The canonical method for finding these angles relies on analytic decomposition. Given a target polynomial, sophisticated algorithms compute the exact sequence of phases required. While theoretically sound, this analytic approach presents significant practical challenges. For high-degree polynomials, the solvers are notoriously susceptible to numerical instability, often failing to converge or producing inaccurate results due to the accumulation of floating-point errors. This instability places a hard ceiling on the complexity of polynomials one can reliably encode, hampering the application of QSP to real-world problems.

The community demo (https://pennylane.ai/qml/demos_community) introduced by Peili sidesteps the analytic bottleneck entirely by reformulating the problem as a machine learning task. Instead of computing the phase angles from a polynomial, the system initializes with random angles and treats the circuit's actual output as a parameterized function to be optimized. The objective is simple: minimize the mean squared error (MSE) between the circuit's expectation value and the target polynomial evaluated over a grid of signal inputs.

The implementation marries two powerful software libraries to achieve this. The quantum circuit is defined using PennyLane, a leading framework for differentiable quantum programming. Crucially, the circuit is constructed from elementary gates (`RZ` rotations and Hadamard gates) rather than a high-level QSVT template. This low-level construction renders the entire simulation traceable by JAX, a high-performance numerical computing library. JAX's automatic differentiation engine computes the gradient of the loss function with respect to every phase angle in the circuit. The Optax library then leverages these gradients to update the angles iteratively using the Adam optimizer.