Speaker
Description
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. 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 present our main result that, by assuming that the parametrised unitaries form a 2-design with respect to the full unitary group, the mean of the Fourier coefficients is concentrated at zero, while the variance scales at an exponentially decaying order with respect to the multi-dimensional frequency magnitude via Weingarten Calculus. When a noise channel with unitary Kraus operators and probabilities $\{p_k\}$ is taken into account, we show that the variance is further suppressed by a factor $\left(\sum_kp_k^2\right)^{Q}<1$, where $Q$ is the number of channel instances applied. On top of this, we demonstrate and validate the analytical results through simulations, both noiseless and noisy. Beyond the length of this extended abstract, this work also investigates how differences in the domain of amplitudes can lead to a distinct expressivity of the zero-frequency Fourier coefficient, as well as demonstrating its utility in fitting target functions with non-integer frequencies that are forbidden in vanilla angle-embedded models. 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.
| Topic | Quantum variational circuits and their applications |
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| Additional Topic | Encoding and processing of data in quantum systems |