Beyond Gates: Pulse Level Quantum Fourier Models

Abstract: In the domain of variational quantum algorithms, quantum Fourier models (QFMs) provide a mathematically well defined structure for quantum machine learning (QML). There has been a substantial amount of work on the scalability and trainability of such models showcasing the potential but also the limitations for the prospective application of QFMs. However, much less is known in the context of pulse-level quantum computing, where the microwave parameters that implement unitary operations on the hardware are used to perform computations directly instead of through the interface of quantum circuits. In this work, we evaluate QFMs through the lens of pulse parameters and link metrics such as expressibility and Fourier coefficient correlation (FCC) to this extended set of variational parameters. We show that while control over pulse shapes does not significantly alter the global expressibility or structural correlations of the Ansatz, it fundamentally alters the local optimisation landscape. For composite gates, independent pulse scalings replace a single logical angle by multiple independently tunable sub-angles. This relaxes the rigid monomial couplings induced by the gate-level parameterisation, and provides gradient descent with higher-dimensional escape routes, decoupling local parameter constraints and significantly boosting performance during training. Following an analytical proof, we show numerical results validating our theory on training a QFM with an exponential (ternary) feature map on a Fourier series with the same frequencies.