Abstract
We consider the problem of measuring the similarity between two graphs using continuous-time quantum walks and comparing their time-evolution by means of the quantum Jensen-Shannon divergence. Contrary to previous works that focused solely on undirected graphs, here we consider the case of both directed and undirected graphs. We also consider the use of alternative Hamiltonians as well as the possibility of integrating additional node-level topological information into the proposed framework. We set up a graph classification task and we provide empirical evidence that: (1) our similarity measure can effectively incorporate the edge directionality information, leading to a significant improvement in classification accuracy; (2) the choice of the quantum walk Hamiltonian does not have a significant effect on the classification accuracy; (3) the addition of node-level topological information improves the classification accuracy in some but not all cases. We also theoretically prove that under certain constraints, the proposed similarity measure is positive definite and thus a valid kernel measure. Finally, we describe a fully quantum procedure to compute the kernel.
Highlights
In recent years, we have observed rapid advancements in the fields of machine learning and quantum computation
An increasing number of researchers is looking at challenges emerging at the intersection of these two fields, from quantum annealing as an alternative to classical simulated annealing [1,2] to quantum parallelism as a source of algorithmic speedup [3,4]
The rising popularity of quantum walks can be further understood by looking at the work of Childs [7], who showed that any quantum computation can be efficiently simulated by a quantum walk on a sparse and unweighted graph, elevating quantum walks to the status of universal computational primitives
Summary
We have observed rapid advancements in the fields of machine learning and quantum computation. An increasing number of researchers is looking at challenges emerging at the intersection of these two fields, from quantum annealing as an alternative to classical simulated annealing [1,2] to quantum parallelism as a source of algorithmic speedup [3,4]. The rich expressiveness of these representations comes with several issues when applying standard machine learning and pattern recognition techniques to them. These techniques usually require graphs to be mapped to corresponding vectorial representations, which in turn needs a canonical node order to be established first. Graphs with different number of nodes and edges present yet another challenge, as the dimension
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