Towards Quantum Graph Neural Networks: An Ego-Graph Learning Approach

TL;DR

This work introduces the Ego-graph based Quantum Graph Neural Network (egoQGNN), a hybrid quantum–classical model for graph classification that operates on a fixed-size quantum device by decomposing graphs into ego-graphs. It replaces classical weight matrices with unitary gates and uses tensor product encodings to injectively embed neighborhood information, while a trainable mapping preserves Euclidean distances in Hilbert space. The framework leverages a three-part Ulayer circuit and a decomposition strategy to scale to real-world graphs, with von Neumann entropy used as a graph representation for classification. Empirical results show egoQGNN often outperforms state-of-the-art methods using only a fraction of the parameters, and the trainable mapping further reduces information loss, underscoring the method’s potential for near-term quantum hardware.

Abstract

Quantum machine learning is a fast-emerging field that aims to tackle machine learning using quantum algorithms and quantum computing. Due to the lack of physical qubits and an effective means to map real-world data from Euclidean space to Hilbert space, most of these methods focus on quantum analogies or process simulations rather than devising concrete architectures based on qubits. In this paper, we propose a novel hybrid quantum-classical algorithm for graph-structured data, which we refer to as the Ego-graph based Quantum Graph Neural Network (egoQGNN). egoQGNN implements the GNN theoretical framework using the tensor product and unity matrix representation, which greatly reduces the number of model parameters required. When controlled by a classical computer, egoQGNN can accommodate arbitrarily sized graphs by processing ego-graphs from the input graph using a modestly-sized quantum device. The architecture is based on a novel mapping from real-world data to Hilbert space. This mapping maintains the distance relations present in the data and reduces information loss. Experimental results show that the proposed method outperforms competitive state-of-the-art models with only 1.68\% parameters compared to those models.

Figures (5)

• Figure 1: Ulayer with Uinit, Ucov, and Uent. Uinit contains $n$ quantum gates to map $n$-dimensional data $X$ into quantum states. In Ucov, RX, RY, RZ quantum gates are applied to each qubit. Uent utilizes CNOT gates to entangle all qubits.
• Figure 2: Following the three-bit error correction code proposed by raussendorf2012key, the error correction of egoQGNN can be achieved by applying $U_c$ and $U_p$ modules after and before Ulayer respectively.
• Figure 3: In Euclidean space, the distance between the green point and the orange one is smaller than that between the green and the red ones. After mapping the data points to Hilbert space, points close to each other in Euclidean space become more distant quantum states (the green and orange lines), while the distance between the points which are farther away in Euclidean space becomes closer (the red and green lines).
• Figure 4: An instance of the egoQGNN. Step 1 and Step 4 are implemented on a classical computer. The quantum circuits of Step 2 and Step 3 can run on a quantum circuit or simulator of a classical computer.
• Figure 5: Running time comparison on MUTAG. The blue and orange bars indicate the per-epoch training time and test time respectively.

Theorems & Definitions (5)

• Lemma 1
• Lemma 2
• Lemma 3
• Theorem 1
• Theorem 2