A GNN based on Continuous-Time Quantum Walks (CTQW) and exploiting two properties of the CTQW propagator, preserving mid- and high-frequency signals for heterophilic graphs while preventing Dirichlet-energy collapse.
Abstract
Graph Neural Networks (GNNs) are widely used on graph-structured data, but most suffer from two key weaknesses. First, message passing behaves as a low-pass filter under the homophily assumption, leading to poor performance on heterophilic graphs. Second, stacking layers drives node features toward constants, causing over-smoothing. Existing methods usually address these issues separately, while the few joint solutions rely largely on empirical heuristics, and many over-smoothing remedies sacrifice model expressiveness. We propose \textbf{CTQW-GNN}, a GNN based on Continuous-Time Quantum Walks (CTQW), to address both issues with theoretical justification. Its design exploits two properties of the CTQW propagator $e^{-\mathrm{i}Ht}$. First, it is unitary and has eigenvalues on the unit circle, so no frequency component is damped, counteracting the low-pass bias. Second, unitarity preserves feature norms and prevents the Dirichlet energy from decaying exponentially with depth, thereby mitigating over-smoothing. CTQW-GNN combines three complementary aggregation modules. \textit{CTQW-based Aggregation} evolves node features through the unitary propagator, preserving mid- and high-frequency signals for heterophilic graphs while preventing Dirichlet-energy collapse. \textit{CTQW-Attention Aggregation} constructs a multi-hop neighbor graph from CTQW amplitudes and applies attention over it, enabling access to distant homophilic nodes missed by single-hop aggregation. \textit{LF Aggregation} uses a standard low-pass GAT branch to retain strong performance on homophilic graphs, where pure CTQW aggregation can be suboptimal. We further provide a spectral-gap analysis explaining energy preservation and a Lieb--Robinson-type bound that gives a principled rule for selecting the walk time $t$.
We investigate message-passing graph neural networks with random node features. Random node features are known to enhance the expressiveness of graph neural networks (GNNs) both theoretically and empirically. Here, we establish a novel universality result focusing on permutation-equivariant neural networks (PENNs), a class of GNNs built from feedforward neural network components that subsumes many prominent GNN architectures. We show that PENNs, combined with partially random node features, can approximate arbitrarily well in probability any measurable permutation-invariant or permutation-equivariant function on directed graphs of fixed size with multidimensional node and edge features. For $k$-times continuously differentiable functions, $k\geq 2$, we also derive upper bounds on the approximation rates, relating the complexity of the feedforward components of a PENN in terms of layer depth and number of nonzero weights to the desired approximation accuracy.
Lukas Gonon, T. Meyer‐Brandis, Niklas H. Weber· arXiv.org· 0 citations
Graph neural networks (GNNs) have demonstrated strong capabilities in graph representation learning but still face limitations in efficiency and scalability. Quantum GNNs (QGNNs) offer a promising alternative. However, existing approaches often fail to fully exploit edge information, require substantial quantum resources, and insufficiently account for permutation invariance in graph learning. To address these challenges, this article proposes a permutation-invariant quantum GNN (PIQGNN). The proposed model introduces a low-qubit-cost quantum encoding strategy that jointly embeds node features, edge features, and graph topology into entangled quantum states using only $n$ qubits, where $n$ denotes the number of nodes, while explicitly enforcing permutation invariance. Furthermore, a symmetry-aware variational quantum neural network (QNN) is designed to enable end-to-end permutation-invariant learning. Its hyperparameters are optimized via Bayesian optimization to alleviate barren plateau (BP) effects and enhance training stability. Experimental results on multiple graph binary classification benchmark datasets demonstrate that, compared with classical GNNs, PIQGNN achieves competitive performance with a significantly reduced number of trainable parameters. Compared with existing QGNNs, PIQGNN attains higher accuracy with lower quantum resource requirements and exhibits stronger robustness under noisy conditions. These results indicate that PIQGNN provides an efficient, scalable, and noise-resilient quantum framework for graph learning, highlighting its practical potential in the noisy intermediate-scale quantum (NISQ) era.
Maoduo Li, Wen Liu, Lei Shi et al.· IEEE Transactions on Neural...· 0 citations
The Learnable Graph Network Model (LGNM) is introduced, a heterogeneous ENM in which per-edge spring constants θij = fi · fj · (dc/rij)2 are parameterised by per-residue flexibility coefficients {fi} predicted by a physics-constrained Graph Neural Network (GNN).
CoRe-GNN is proposed, which performs both propagations in parallel at each layer: a coarsened inter-cluster term capturing long-range structure, and a local intra-cluster term preserving per-node discriminability.
Antonin Joly, Nicolas Keriven, Aline Roumy· 0 citations
DIAL, a message-passing layer that gives nodes access to graph structure through the diagonal of graph-derived operators, is introduced, which uses randomized probing to provide nodes with learnable, permutation equivariant access to diagonal entries.
Saku Peltonen, H. Bilgi, Kubilay Atasu· 0 citations
QDAGer is introduced, a quantum-inspired graph-pair Transformer that injects quantum-dynamical features from time series of node occupations and connected two-point correlators directly into the attention mechanism, and applies it to learning Graph Edit Distance, an NP-hard similarity measure.
Mehdi Djellabi, L. Henry· 0 citations
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