[Paper Review] Quantum Walk Inspired Neural Networks for Graph-Structured Data

Stefan Dernbach, Arman Mohseni-Kabir+3 more|arXiv (Cornell University)|Jan 16, 2018

Quantum Computing Algorithms and Architecture24 references3 citations

TL;DR

This paper proposes Quantum Walk Neural Networks (QWNNs), a novel graph neural network architecture that leverages quantum random walks to learn diffusion operators on graph-structured data. By modeling information propagation through unitary evolution instead of classical transition matrices, QWNNs achieve state-of-the-art or competitive performance on node and graph regression, and classification tasks across molecular, biological, and temperature datasets, demonstrating the potential of quantum-inspired methods in graph learning.

ABSTRACT

In recent years, new neural network architectures designed to operate on graph-structured data have pushed the state-of-the-art in the field. A large set of these architectures utilize a form of classical random walks to diffuse information. We propose quantum walk neural networks (QWNN), a novel graph neural network architecture based on quantum random walks, the quantum parallel to classical random walks. A QWNN learns a quantum walk on a graph to construct a diffusion operator which can then be applied to graph-structured data. We demonstrate the use of QWNNs on a variety of prediction tasks on graphs involving temperature, biological, and molecular datasets.

Motivation & Objective

• To develop a new graph neural network architecture inspired by quantum random walks to improve information diffusion on graphs.
• To explore whether quantum-inspired techniques can offer advantages over classical random walk-based GNNs in learning graph-structured representations.
• To evaluate the performance of QWNNs on diverse graph prediction tasks, including node regression, graph regression, and graph classification.
• To analyze the limitations of QWNNs, particularly in terms of space complexity and sensitivity to graph structure ordering.
• To lay the foundation for future quantum machine learning models that can be implemented on quantum hardware.

Proposed method

• QWNNs model information diffusion on graphs using discrete-time quantum walks governed by unitary evolution, replacing classical transition matrices with quantum operators.
• The architecture employs a unitary coin operator and shift operator to evolve the quantum state across graph nodes over multiple time steps.
• The quantum walk is parameterized with learnable unitary operations, allowing the network to adapt the diffusion process to the learning task.
• The final quantum walk state is used to construct a diffusion matrix that serves as a representation of the graph for downstream prediction tasks.
• The model uses a spatial coin implementation where each node's coin is independently tuned to its edge ordering, improving invariance to graph representation.
• The QWNN is trained via backpropagation with a differentiable loss function, and the learned diffusion matrix can be cached for inference.

Experimental results

Research questions

• RQ1 Can quantum walk-based diffusion mechanisms outperform classical random walk-based methods in graph representation learning?
• RQ2 How does the quantum walk's unitary evolution affect the expressiveness and generalization of graph neural networks?
• RQ3 What are the performance gains of QWNNs on node-level, graph-level regression, and classification tasks compared to state-of-the-art GNNs?
• RQ4 How do QWNNs handle structural variations such as edge ordering and graph isomorphism?
• RQ5 What are the computational and memory limitations of implementing quantum walk layers in classical settings?

Key findings

• On the QM7 molecular dataset, QWNNs achieved an RMSE of 12.52±0.91 on atomization energy prediction, outperforming GCNN (19.26±0.47) and DCNN (15.51±1.12).
• For the PTC and PROTEINS datasets, QWNNs achieved test F1-scores of 0.71±0.01 and 0.85±0.06, respectively, matching or exceeding those of GCNN and DCNN.
• The QWNN with similarity-based learning achieved an F1-score of 0.92±0.03 on the PROTEINS dataset, significantly outperforming GCNN (0.83±0.09) and DCNN (0.67).
• The model demonstrated improved performance on graph regression tasks, with QWNN (centrality) achieving a lower MAE of 8.93±0.14 compared to GCNN's 14.77±0.26.
• Despite better performance, QWNNs incurred higher computational cost due to the need for independent quantum walk simulations per node, leading to increased training time.
• The space complexity of O(N²dT) for full diffusion matrices limits scalability to large graphs, especially on GPU hardware.