QuOp: A Quantum Operator Representation for Nodes

Abstract

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. 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, $\mathfrak{s u}\left(2^{n}\right)$. This sub-vector space in turn generates a subgroup of the Lie group of special unitary operators, $\text{SU}\left(2^{n}\right)$. 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.