# Continuous-Time Quantum Walks based Graph Neural Network

arXiv: https://arxiv.org/abs/2608.20738 · HTML: https://mlsanity.com/paper/2608.20738/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 27 of its 60 references that are in the map.
- First posted on arXiv 21 Aug 2026.

## Builds on (most cited first, 27 of 27)

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [Semi-Supervised Classification with Graph Convolutional Networks](https://mlsanity.com/paper/1609.02907/) (2016)
- [Graph Attention Networks](https://mlsanity.com/paper/1710.10903/) (2017)
- [Inductive Representation Learning on Large Graphs](https://mlsanity.com/paper/1706.02216/) (2017)
- [Simplifying Graph Convolutional Networks](https://mlsanity.com/paper/1902.07153/) (2019)
- [Deeper Insights into Graph Convolutional Networks for Semi-Supervised Learning](https://mlsanity.com/paper/1801.07606/) (2018)
- [Geom-GCN: Geometric Graph Convolutional Networks](https://mlsanity.com/paper/2002.05287/) (2020)
- [Adaptive Universal Generalized PageRank Graph Neural Network](https://mlsanity.com/paper/2006.07988/) (2020)
- [Masked Label Prediction: Unified Message Passing Model for Semi-Supervised Classification](https://mlsanity.com/paper/2009.03509/) (2020)
- [Beyond Low-frequency Information in Graph Convolutional Networks](https://mlsanity.com/paper/2101.00797/) (2021)
- [Large Scale Learning on Non-Homophilous Graphs: New Benchmarks and Strong Simple Methods](https://mlsanity.com/paper/2110.14446/) (2021)
- [Finding Global Homophily in Graph Neural Networks When Meeting Heterophily](https://mlsanity.com/paper/2205.07308/) (2022)
- [Towards Deeper Graph Neural Networks with Differentiable Group Normalization](https://mlsanity.com/paper/2006.06972/) (2020)
- [Understanding and Resolving Performance Degradation in Graph Convolutional Networks](https://mlsanity.com/paper/2006.07107/) (2020)
- [Label Efficient Semi-Supervised Learning via Graph Filtering](https://mlsanity.com/paper/1901.09993/) (2019)
- [Efficient and Equivariant Graph Networks for Predicting Quantum Hamiltonian](https://mlsanity.com/paper/2306.04922/) (2023)
- [Graph Neural Networks with Learnable and Optimal Polynomial Bases](https://mlsanity.com/paper/2302.12432/) (2023)
- [PC-Conv: Unifying Homophily and Heterophily with Two-fold Filtering](https://mlsanity.com/paper/2312.14438/) (2023)
- [Graph Neural Convection-Diffusion with Heterophily](https://mlsanity.com/paper/2305.16780/) (2023)
- [Finding the Missing-half: Graph Complementary Learning for Homophily-prone and Heterophily-prone Graphs](https://mlsanity.com/paper/2306.07608/) (2023)
- [Adaptive Message Passing: A General Framework to Mitigate Oversmoothing, Oversquashing, and Underreaching](https://mlsanity.com/paper/2312.16560/) (2023)
- [How Universal Polynomial Bases Enhance Spectral Graph Neural Networks: Heterophily, Over-smoothing, and Over-squashing](https://mlsanity.com/paper/2405.12474/) (2024)
- [Unleashing the Potential of Fractional Calculus in Graph Neural Networks with FROND](https://mlsanity.com/paper/2404.17099/) (2024)
- [Self-supervised Consensus Representation Learning for Attributed Graph](https://mlsanity.com/paper/2108.04822/) (2021)
- [Robust Preference-Guided Denoising for Graph based Social Recommendation](https://mlsanity.com/paper/2303.08346/) (2023)
- [Feature Transportation Improves Graph Neural Networks](https://mlsanity.com/paper/2307.16092/) (2023)
- [Distributed-Order Fractional Graph Operating Network](https://mlsanity.com/paper/2411.05274/) (2024)

## Abstract (arXiv)

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 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-iHt. 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. textitCTQW-based Aggregation evolves node features through the unitary propagator, preserving mid- and high-frequency signals for heterophilic graphs while preventing Dirichlet-energy collapse. textitCTQW-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. textitLF 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.