Understanding the Qubit Trajectory in Parameterized Quantum Circuits

Seungcheol Oh, Chaemoon Im, Soohyun Park, Joongheon Kim

Keywords: Quantum Neural Networks, Parameterized Quantum Circuits

Abstract:

Quantum neural networks (QNNs) employ parameterized quantum circuits (PQCs) to map data inputs to desired predictions. Similar to classical supervised learning, PQCs are often treated as black-box function mappers. Recent analytical research demonstrates that PQCs consisting of data reuploading structure are naturally expressed as partial Fourier series and that a single qubit circuit can serve as a universal approximator for univariate functions. However, this prior work largely focuses on representational capacity and does not provide intuitive or structural explanations of how the individual circuit components govern the resulting Fourier coefficients. In this paper, we peel back the black-box and analyze the intrinsic structure of PQCs. In particular, we investigate how data encoding, repeated reuploading, and trainable unitary operators combine to represent function classes characterized by a Fourier expansion with specific accessible frequency components. Our key contribution is to show, both mathematically and empirically, that the configuration of trainable parameters creates an output qubit trajectory that is critical for the representation of Fourier coefficients.

(We want to have the paper as extended abstract, if accepted)

Submission Number: 17