  We derive an intuitive and novel method to represent nodes in a graph with
special unitary operators, or quantum operators, which does not require
parameter training and is competitive with classical methods on scoring
similarity between nodes. This method opens up future possibilities to apply
quantum algorithms for NLP or other applications that need to detect anomalies
within a network structure. Specifically, this technique leverages the
advantage of quantum computation, representing nodes in higher dimensional
Hilbert spaces. To create the representations, the local topology around each
node with a predetermined number of hops is calculated and the respective
adjacency matrix is used to derive the Hamiltonian. While using the local
topology of a node to derive a Hamiltonian is a natural extension of a graph
into a quantum circuit, our method differs by not assuming the quantum
operators in the representation a priori, but letting the adjacency matrix
dictate the representation. As a consequence of this simplicity, the set of
adjacency matrices of size $2^n \times 2^n$ generates a sub-vector space of the
Lie algebra of the special unitary operators, $\mathfrak{su}(2^n)$. This
sub-vector space in turn generates a subgroup of the Lie group of special
unitary operators, $\mathrm{SU}(2^n)$. Applications of our quantum embedding
method, in comparison with the classical algorithms GloVe (a natural language
processing embedding method) and FastRP (a general graph embedding method,
display superior performance in measuring similarity between nodes in graph
structures.
