Quantum Fourier integration of decay rates at next-to-leading order

We discuss the use of quantum neural networks (QNNs) as a tool for integrating decay rates at next-to-leading order (NLO) in perturbation theory. Data-encoding quantum circuits admit a truncated Fourier-series representation, enabling a trained QNN to approximate a target integrand and provide the coefficients subsequently integrated with Quantum Fourier Iterative Amplitude Estimation (QFIAE). The causal-unitary formulation of the Loop-Tree Duality (LTD) combines virtual and real contributions into locally finite integrands suitable for this representation. We consider $H\to q\bar q(g)$, $\gamma^*\to q\bar q(g)$ and $\Phi\to\phi\phi(\phi)$, comparing quantum-simulator and quantum-hardware results with analytic NLO predictions. The study illustrates the potential of Fourier-based QNN integration for high-energy physics.