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 a significant focus on 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 function approximation. For each area we provide proof-of-concept numerical simulations of the proposed circuits. 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, they can provide efficient algorithmic primitives to design machine learning methods. At the same time, we are explicit about the scope of this efficiency: the simplest primitives are product-state circuits with classically computable angles, which are classically simulable, so a quantum advantage can only materialize when the constructions are embedded in coherent pipelines with entangling processing; we state these limitations alongside the corresponding results. Throughout the paper, complexity statements concern the core quantum computation; the costs of initial data loading and final classical readout are stated separately where relevant.