Quantum Generating Functions: Formulation and Applications in Quantum Machine Learning

In this paper, we study quantum generating functions (QGFs) as algorithmic primitives for quantum computation with significant focus in machine learning. We develop a systematic formulation for encoding sequences and functions into quantum states, enabling efficient quantum algorithms for evaluation, transformation, and analysis. 
We demonstrate the utility of QGFs across three key areas of machine learning: i) Quantum activation functions with efficient, explicit circuit implementations. ii) Quantum kernel methods via generating function representations of feature maps. iii) Quantum Kolmogorov–Arnold networks (KANs) for high-dimensional function approximation.

We analyze circuit complexities and applications offering a foundation for quantum machine learning. The quantum implementations of generating functions for structured function classes consist of simple quantum rotations. Therefore, it can provide efficient algorithmic primitives to design machine learning methods.