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

Abstract

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. By assuming that the ensemble of unitaries generated from the parameter space forms at least a 2-design with respect to the unitary group, we derive, via Weingarten calculus, that the mean of the Fourier coefficients is concentrated at zero, and the variance scales at an exponentially decaying order with respect to the multi-dimensional frequency magnitude. When a noise channel with unitary Kraus operators and probabilities $\{p_k\}$ is taken into account, the variance is further suppressed by a factor $\left(\sum_k p_k^2\right)^{Q}<1$, where $Q$ is the number of channel instances applied. Furthermore, we demonstrate and validate the analytical results through simulations, both noiseless and noisy, including a case 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.