Variational quantum machine learning methods for graph structured data with edge and multi-node features

Abstract

This study aims to build equivariant variational quantum models for learning tasks on graph data with edge features and multi-node features. The existing equivariantly diagonalizable unitary quantum graph circuit framework was extended in two aspects. First, the equivariantly diagonalizable unitary gates were parameterized with variables representing multi-node features and variables that depend on the layer number. Second, the equivariantly diagonalizable unitary gates were generalized to act on more than two nodes. These models were tested on the QM9 dataset for proof-of-concept experiments that showed similar performance to classical models with comparable numbers of variables. Quantum models with and without multi-node equivariantly diagonalizable unitary gates were tested on a molecular dynamics dataset, and the models with multi-node equivariantly diagonalizable unitary gates showed better performance. This showed that the multi-node features were incorporated into the model. In order to check the trainability of the proposed model, the variance of the gradient values was numerically calculated while the number of qubits, number of layers, and the average degree of the input graph nodes were changed. The gradient vanished exponentially as a function of the number of qubits and layers while the degree of the input graph did not affect the gradient size. Thus, this model may have difficulty applying to graph data with high numbers of nodes.