In this work, we propose a family of novel quantum kernels, namely the
Hierarchical Aligned Quantum Jensen-Shannon Kernels (HAQJSK), for un-attributed
graphs. Different from most existing classical graph kernels, the proposed
HAQJSK kernels can incorporate hierarchical aligned structure information
between graphs and transform graphs of random sizes into fixed-sized aligned
graph structures, i.e., the Hierarchical Transitive Aligned Adjacency Matrix of
vertices and the Hierarchical Transitive Aligned Density Matrix of the
Continuous-Time Quantum Walk (CTQW). For a pair of graphs to hand, the
resulting HAQJSK kernels are defined by measuring the Quantum Jensen-Shannon
Divergence (QJSD) between their transitive aligned graph structures. We show
that the proposed HAQJSK kernels not only reflect richer intrinsic global graph
characteristics in terms of the CTQW, but also address the drawback of
neglecting structural correspondence information arising in most existing
R-convolution kernels. Furthermore, unlike the previous Quantum Jensen-Shannon
Kernels associated with the QJSD and the CTQW, the proposed HAQJSK kernels can
simultaneously guarantee the properties of permutation invariant and positive
definiteness, explaining the theoretical advantages of the HAQJSK kernels.
Experiments indicate the effectiveness of the proposed kernels