Quantum Models as Fourier Series

Quantum Models as Fourier Series

This PennyLane tutorial explores a deep and useful result about quantum machine-learning models: a parameterized quantum circuit that encodes data through rotation gates computes, as a function of that data, a Fourier series, a sum of sinusoids. This perspective, developed by Schuld, Sweke, and Meyer, demystifies what quantum models can and cannot represent: the frequencies available in the Fourier series are determined by the data-encoding gates (how often and with what eigenvalues the data is uploaded), while the trainable gates control the Fourier coefficients. The tutorial builds simple encoding circuits in PennyLane and shows empirically that the functions they realize are indeed truncated Fourier series, demonstrating how adding more data-encoding repetitions enriches the accessible frequency spectrum and thus the model's expressivity. It connects this to the data-reuploading idea and explains the practical implication: to fit a target function, the circuit's encoding must support the frequencies that function contains. By grounding model expressivity in the concrete and familiar language of Fourier analysis, the tutorial gives clear, principled insight into the representational power of quantum machine-learning models in PennyLane.

Run it

pip install -r requirements.txt
python demo.py

Source and license

Imported from demonstrations_v2/tutorial_expressivity_fourier_series/demo.py in PennyLaneAI/demos at c52c0abeb5122218aa96b38eea848864cce7323f, under the Apache License 2.0. Original authors: Xanadu and the PennyLane community. The upstream LICENSE is included alongside this example.