Constrained and Vanishing Expressivity of Quantum Fourier Models
1Sorbonne Université, CNRS, LIP6, 75005 Paris, France
2ENSTA Paris, Institut Polytechnique de Paris, France
3CEMIS, Direction Technique, Naval Group, 83190 Ollioules, France
4Quantum Software Lab, School of Informatics, University of Edinburgh, United Kingdom
5PASQAL SAS, 7 avenue Léonard de Vinci, 91300 Massy, France
6QC Ware, Palo Alto, USA and Paris, France
| Published: | 2025-09-03, volume 9, page 1847 |
| Editor: | Kunal Sharma |
| Eprint: | arXiv:2403.09417v3 |
| Doi: | https://doi.org/10.22331/q-2025-09-03-1847 |
| Citation: | Quantum 9, 1847 (2025). |
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Abstract
In this work, we highlight an unforeseen behavior of the expressivity of Parameterized Quantum Circuits (PQCs) for machine learning. A large class of these models, seen as Fourier series whose frequencies are derived from the encoding gates, were thought to have their Fourier coefficients mostly determined by the trainable gates. Here, we demonstrate a new correlation between the Fourier coefficients of the quantum model and its encoding gates. In addition, we display a phenomenon of vanishing expressivity in certain settings, where some Fourier coefficients vanish exponentially as the number of qubits grows. These two behaviors imply novel forms of constraints which limit the expressivity of PQCs, and therefore imply a new inductive bias for quantum models. The key concept in this work is the notion of a frequency redundancy in the Fourier series spectrum, which determines its importance. Those theoretical behaviors are observed in numerical simulations.
Featured image: Variance of Fourier coefficients, under trainable unitaries forming a 2-design, is shown to be proportional to their respective redundancy for a spectrum generated by tensor product of a) the same single qubit Pauli rotation and b) rescaled single qubit Pauli rotations with different prefactors.
Popular summary
However, we emphasize in this work that the frequency spectrum alone does not fully capture the model's expressivity as the achievable values of the Fourier coefficients are subject to some constraints. We introduce the notion of frequency redundancy (i.e. the number of times that frequency appears in the theoretical construction of the quantum spectrum) and show that the variance of a Fourier coefficient is proportional to its frequency's redundancy. In other words, the ability to vary a coefficient through parameter tuning is limited by its frequency redundancy, leading to what we refer to as “constrained expressivity”.
Additionally, we identify a “vanishing expressivity” phenomenon, where the variance of some or all Fourier coefficients decays exponentially with the number of qubits. Notably, this effect can occur independently of the well-known barren plateau phenomenon, indicating a fundamental limitation on model expressivity rather than an obstacle to trainability.
Together, these findings imply novel forms of constraints which limit the expressivity of Quantum Fourier models, and therefore imply a new inductive bias for Quantum machine learning models.
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