In this work, we develop a new Quantum-based Matching Kernel (QBMK) for un-attributed graphs, by computing the quantum Shannon entropies between the aligned vertices based on the Continuous-time Quantum Walk (CTQW). The theoretical analysis reveals that the proposed QBMK kernel cannot only address the shortcoming of neglecting the structural correspondence information between graphs arising in most existing R-convolution graph kernels, but also overcome the problem of neglecting the structural differences between pairs of aligned vertices arising in existing vertex-based matching kernels. Moreover, the proposed QBMK kernel can simultaneously capture both global and local structural information through the quantum Shannon entropies. The experimental evaluation on standard graph datasets demonstrates that the proposed QBMK kernel is able to outperform state-of-the-art graph kernels for graph classification tasks.