Quantum kernel method for learning graph data via quantum walk

Graph kernels are essential for learning the properties of graph data, a fundamental task in the analysis of such data. However, a major limitation of current graph kernel methods is their high computational cost, which becomes prohibitive when dealing with large-scale graphs. In order to address this bottleneck, we propose a quantum algorithm designed to analyze the intrinsic structural information within graph data through an efficient quantum kernel method. The core idea of our kernel method involves encoding pertinent information from graphs into quantum states, utilizing quantum kernel functions to represent the similarities among various graphs. This process is facilitated through the application of quantum walks and quantum signal processing techniques. Building upon this framework, we propose a quantum kernel ridge regression algorithm that consists of two key steps. First, we employ our quantum kernel method to prepare a quantum state, in which the amplitudes encode the kernel functions. Second, we implement quantum feature mapping in parallel to encode the corresponding kernel matrix into the quantum circuit. Crucially, we demonstrate that the computational complexity of our regression algorithm scales primarily as $O({M}^{3/2})$ with respect to the size of the graph set $M$. This suggests that our quantum kernel method could process graph data with resource requirements scaling approximately as $O(\text{poly}log(N))$, where $N$ denotes the number of nodes in the graph. Accordingly, this underscores the advantage of our approach over classical methods in the context of large-scale graph analysis. To validate our approach, we conduct numerical experiments on chemical molecular datasets, demonstrating the method's effectiveness and advantages. In summary, this work represents a significant advancement in quantum graph learning, enhancing its potential for analyzing large-scale graph data.