We introduce Quantum Graph Neural Networks (QGNN), a new class of quantum neural network ansatze which are tailored to represent quantum processes which have a graph structure, and are particularly suitable to be executed on distributed quantum systems over a quantum network. Along with this general class of ansatze, we introduce further specialized architectures, namely, Quantum Graph Recurrent Neural Networks (QGRNN) and Quantum Graph Convolutional Neural Networks (QGCNN). We provide four example applications of QGNN's: learning Hamiltonian dynamics of quantum systems, learning how to create multipartite entanglement in a quantum network, unsupervised learning for spectral clustering, and supervised learning for graph isomorphism classification. Variational Quantum Algorithms are a promising class of algorithms are rapidly emerging as a central subfield of Quantum Computing (McClean et al., 2016; Farhi et al., 2014; Farhi & Neven, 2018) . Similar to parameterized transformations encountered in deep learning, these parameterized quantum circuits are often referred to as Quantum Neural Networks (QNNs). Recently, it was shown that QNNs that have no prior on their structure suffer from a quantum version of the no-free lunch theorem (McClean et al., 2018) and are exponentially difficult to train via gradient descent. Thus, there is a need for better QNN ansatze. One popular class of QNNs has been Trotter-based ansatze (Farhi et al., 2014; Hadfield et al., 2019) . The optimization of these ansatze has been extensively studied in recent works, and efficient optimization methods have been found (Verdon et al., 2019b; Li et al., 2019) . On the classical side, graph-based neural networks leveraging data geometry have seen some recent successes in deep learning, finding applications in biophysics and chemistry (Kearnes et al., 2016) . Inspired from this success, we propose a new class of Quantum Neural Network ansatz which allows for both quantum inference and classical probabilistic inference for data with a graph-geometric structure. In the sections below, we introduce the general framework of the QGNN ansatz as well as several more specialized variants and showcase four potential applications via numerical implementation. Results featured in this paper should be viewed as a promising set of first explorations of the potential applications of QGNNs. Through our numerical experiments, we have shown the use of these QGNN ansatze in the context of quantum dynamics learning, quantum sensor network optimization, unsupervised graph clustering, and supervised graph isomorphism classification. Given that there is a vast set of literature on the use of Graph Neural Networks and their variants to quantum chemistry, future works should explore hybrid methods where one can learn a graph-based hidden quantum representation (via a QGNN) of a quantum chemical process. As the true underlying process is quantum in nature and has a natural molecular graph geometry, the QGNN could serve as a more accurate model for the hidden processes which lead to perceived emergent chemical properties. We seek to explore this in future work. Other future work could include generalizing the QGNN to include quantum degrees of freedom on the edges, include quantum-optimization-based training of the graph parameters via quantum phase backpropagation (Verdon et al., 2018) , and extending the QSGCNN to multiple features per node.
Introducing a new class of quantum neural networks for learning graph-based representations on quantum computers.
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