Unveiling the nature of graphs through quantum graphon learning

Graphons are continuous functions hidden behind large-scale graphs. Traditional approaches to graphon learning either suffer from the resolution limitations of piecewise-constant functions or struggle to capture the high-frequency structures intrinsic to graphons. In this paper, we propose quantum graphon neural representation (QGNR), a quantum-enhanced machine learning method that enables arbitrarily high-resolution and high-frequency graphon modeling through Fourier series expressivity of data re-uploading quantum circuits. We further extend QGNR to conditional graphon learning, thereby allowing the handling of complex graph-related tasks. Numerical results demonstrate that QGNR outperforms classical state-of-the-art methods in graphon learning and excels in graph classification tasks across domains like molecular property prediction and social network analysis. Notably, QGNR attains superior performance while requiring up to 81.65% fewer variational parameters than the most advanced neural network-based baselines. This work pioneers the application of quantum machine learning to graphons, demonstrating promising quantum advantages in representational power and learning efficiency.