Continuous time quantum walks built on graph Laplacians produce non monotonic graph propagation features through quantum interference during evolution, which classical diffusion cannot achieve. Such walks deliver a physically motivated propagation scheme for temporal graph signal modeling. We propose Quantum Walk Temporal Architecture (QWTA), which replaces classical graph propagation with a CTQW-type parameterized spectral propagator. QWTA embeds effective observation intervals into the spectral phase modulation of the propagation kernel. QWTA-Base preserves exact CTQW spectral evolution and serves as a physically faithful propagation reference. QWTA-GR further introduces phase soft clipping and gated residual fusion to stabilize propagation. The results show that explicit phase encoding of irregular time intervals, when combined with suitable stabilization, provides a physically motivated and competitive graph propagation design for temporal graph forecasting with missing historical observations.
Continuous time quantum walks built on graph Laplacians produce non monotonic graph propagation features through quantum interference during evolution, which classical diffusion cannot achieve. Such walks deliver a physically motivated propagation scheme for temporal graph signal modeling. We propose Quantum Walk Temporal Architecture (QWTA), which replaces classical graph propagation with a CTQW-type parameterized spectral propagator. QWTA embeds effective observation intervals into the spectral phase modulation of the propagation kernel. QWTA-Base preserves exact CTQW spectral evolution and serves as a physically faithful propagation reference. QWTA-GR further introduces phase soft clipping and gated residual fusion to stabilize propagation. The results show that explicit phase encoding of irregular time intervals, when combined with suitable stabilization, provides a physically motivated and competitive graph propagation design for temporal graph forecasting with missing historical observations.
Graph Neural Networks (GNN) and Transformer-based architectures have achieved remarkable progress in graph learning, yet they still struggle to capture both global structural dependencies and model the dynamic information propagation. In this paper, we propose CTQWformer, a hybrid graph learning framework that integrates continuous-time quantum walks (CTQW) with GNN. CTQWformer employs a trainable Hamiltonian that fuses graph topology and node features, enabling physically grounded modeling of quantum walk dynamics that captures rich and intricate graph structure information. The extracted CTQW-based representations are incorporated into two complementary modules:(i) a Graph Transformer module that embeds final-time propagation probabilities as structural biases in the self-attention mechanism, and (ii) a Graph Recurrent Module that captures temporal evolution patterns with bidirectional recurrent networks. Extensive experiments on benchmark graph classification datasets demonstrate that CTQWformer outperforms graph kernel and GNN-based methods, demonstrating the potential of integrating quantum dynamics into trainable deep learning frameworks for graph representation learning. To the best of our knowledge, CTQWformer is the first hybrid CTQW-based Transformer, integrating CTQW-derived structural bias with temporal evolution modeling to advance graph learning.
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−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. \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.
The continuous-time quantum walk denotes the evolution of a particle on a given graph governed by Schr\"odinger's equation. For different applications, the corresponding Hamiltonian is proportional to either the Laplacian or the adjacency matrix of the graph. The two quantum walks are equivalent on regular graphs, since each vertex has the same degree. However, for irregular graphs, the evolutions of the two quantum walks are generally different. In this paper, we report an experimental investigation of the two quantum walks on irregular graphs with single photons and interferometric networks. We demonstrate that it is possible to obtain equivalent probability distributions with the two quantum walks on specific irregular graphs, where the particle is initially localized at a vertex or uniformly distributed at multiple vertices. Our results not only deepen the understanding of the equivalence between the two quantum walks, but also extend the application of the continuous-time quantum walk.
The continuous-time quantum walk is a particle evolving by Schr\"odinger's equation in discrete space. Encoding the space as a graph of vertices and edges, the Hamiltonian is proportional to the discrete Laplacian. In some physical systems, however, the Hamiltonian is proportional to the adjacency matrix instead. It is well known that these quantum walks are equivalent when the graph is regular, i.e., when each vertex has the same number of neighbors. If the graph is irregular, however, the quantum walks evolve differently. In this paper, we show that for some irregular graphs, if the particle is initially localized at a certain vertex, the probability distributions of the two quantum walks are identical, even though the amplitudes differ. We analytically prove this for a graph with five vertices and a graph with six vertices. By simulating the walks on all 1 018 689 568 simple, connected, irregular graphs with 11 vertices or less, we found 64 graphs with this notion of equivalence. We also give eight infinite families of graphs supporting these equivalent walks.
Continuous-time quantum walks (CTQWs) on static graphs provide efficient methods for search and sampling as well as a model for universal quantum computation. We consider an extension of CTQWs to the case of dynamic graphs, in which an ordered sequence of graphs governs free evolution of the quantum walk. We then consider how perfect state transfer during the quantum walk can be used to design dynamic graphs that implement a universal set of quantum logic gates. We give explicit examples for a complete logical basis, and we validate implementations using numerical simulations for quantum teleportation and addition circuits. Finally, we discuss the potential for realizing CTQWs on dynamic graphs using actively controlled quantum optical waveguides.
The connection between coined and continuous-time quantum walk models has been addressed in a number of papers. In most of those studies, the continuous-time model is derived from coined quantum walks by employing dimensional reduction and taking appropriate limits. In this work, we produce the evolution of a coined quantum walk on a generic graph using a continuous-time quantum walk on a larger graph. In addition to expanding the underlying structure, we also have to switch on and off edges during the continuous-time evolution to accommodate the alternation between the shift and coin operators from the coined model. In one particular case, the connection is very natural, and the continuous-time quantum walk that simulates the coined quantum walk is driven by the graph Laplacian on the dynamically changing expanded graph.