Asymptotics for Frequency Redundancies in Quantum Machine Learning Models

The redundancy distribution of the frequency spectrum has been shown in the literature to impact the expressivity and trainability of Quantum Fourier Models (QFMs). In this work, we address the question of how this redundancy spectrum is shaped by the choice of eigenvalues of the data re-uploading Hamiltonians, using a simple mathematical formalism based on generating functions. We derive exact and asymptotic redundancy profiles for several structured eigenvalue choices identical for every layer, including arithmetic progressions and single-qubit Pauli encodings, and show that both approach a Gaussian profile as the number of layers grows. We then show that this Gaussian limit is not specific to these constructions but is a generic feature of QFMs built from integer eigenvalues that are identical in every layer. These results can be tied to the central limit theorem for random walks. These results clarify why generic or unstructured encoding choices give rise to a redundancy bias that favours low frequencies, highlighting the importance of well-thought-out encoding strategies when constructing a QFM.