QuOp: A Quantum Operator Representation for Nodes

TL;DR

This paper introduces QuOp, a quantum operator-based node representation method that leverages local graph topology and quantum mechanics principles, achieving competitive similarity scoring without parameter training.

Contribution

It presents a novel quantum operator representation for graph nodes derived from adjacency matrices, enabling quantum-inspired embeddings without prior operator assumptions.

Findings

Outperforms classical methods GloVe and FastRP in node similarity tasks.

Uses local topology and Hamiltonian derivation for quantum node embeddings.

Creates a sub-vector space of the Lie algebra of special unitary operators.

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. 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.