Fourier analysis of quantum neural network with non-linear data embedding

Fourier analysis has become a crucial tool for understanding the expressivity of Variational Quantum Circuit (VQC) models, as well as an important indicator of barren plateaus (BP). While existing literature has only studied angle-embedded VQCs in a noiseless environment, here we develop the Fourier analysis of VQCs with non-linear data embedding, with particular focus on amplitude embedding, which provides a naturally compact encoding scheme. We first investigate a subtle difference in the domain of input features within amplitude embedding that leads to a distinct expressivity of the zero-frequency Fourier coefficient. Assuming that the ensemble of unitaries generated from the parameter space forms at least a unitary 2-design, we derive that, in integer-valued frequencies, the mean of the Fourier coefficients is concentrated at zero, and the variance is upper bounded by an inverse power-law function with respect to the infinity norm of the multi-dimensional frequency. A numerical evidence of exponentially decaying pattern with respect to the frequency magnitude is also established, providing a frequency-perspective of viewing VQC model expressivity and its trainability. When a noise channel with unitary Kraus operators and probabilities p k is taken into account, the variance is further suppressed by a factor (∑ k p k 2 ) Q < 1, where Q is the number of channel instances applied. Furthermore, we demonstrate and validate the analytical results through simulations, both noiseless and noisy, and contrast its expressivity against the angle-embedded model Fourier features. We also examine the model's performance where target functions are decomposed into non-integer frequencies, highlighting the practical utility of the approach. Our results establish a rigorous Fourier framework for amplitude-encoded VQCs, offering both theoretical guarantees on expressivity, hence trainability scaling in the frequency domain, as well as practical simulations for deployment on noisy quantum devices.