A hybrid quantum walk model unifying discrete and continuous quantum walks

npj | quantum information Article Published in partnership with The University of New South Wales https://doi.org/10.1038/s41534-025-01165-z A hybrid quantum walk model unifying discrete and continuous quantum walks Check for updates 1234567890():,; 1234567890():,; Tianen Chen1,2 & Yun Shang1,3 Quantum walks, both discrete and continuous, serve as fundamental tools in quantum information processing with diverse applications. This work introduces a hybrid quantum walk model that integrates the coin mechanism of discrete walks with the Hamiltonian-driven time evolution of continuous walks. Through systematic analysis of probability distributions, standard deviations, and entanglement entropy on fundamental graph structures, we reveal distinctive dynamical characteristics that differentiate our model from conventional quantum walk paradigms. The proposed framework demonstrates unifying capabilities by naturally encompassing existing quantum walk models as special cases. Two significant applications emerge from this hybrid architecture: (1) We develop a novel protocol for perfect state transfer(PST) in general connected graphs, overcoming the limitations of previous graph-specific approaches. A PST on a tree graph has been implemented on a quantum superconducting processor. (2) We devise a quantum algorithm for multiplying K adjacency matrices of n-vertex regular graphs with time complexity O(n2d1 ⋯ dK), outperforming classical matrix multiplication (O(n2.371552)) when vertex degrees di are bounded. The algorithm’s efficacy for triangle counting is experimentally validated through the quantum simulation on PennyLane. These results establish the hybrid quantum walk as a versatile framework bridging discrete and continuous paradigms while enabling practical quantum advantage in graph computation tasks. Quantum walks, as a universal quantum computation model1–5, have demonstrated remarkable computational advantages. They achieve exponential speedups in tasks such as the glued trees problem6 and offer significant improvements in search algorithms7–11, combinatorial optimization problems12, Hamiltonian simulation13, graph isomorphism14–17, the triangle finding18, the element distinction19, the subset finding20, constraint satisfaction problems21 and so on. Quantum walks are primarily classified into discrete and continuous models, corresponding to their classical counterparts22. In the realm of classical and quantum control theory, hybrid systems that integrate both continuous dynamic evolution and discrete controls have garnered increasing attention in academia and industry23–29, particularly for fault-tolerant quantum computing28,30. Combining discrete quantum walks based on coin controls31–34 with the continuous quantum walks6 into hybrid quantum walks may bring new discoveries to quantum algorithms and quantum control theory. However, due to the operational differences between coin-driven models and Hamiltonian-based propagation, creating hybrid frameworks remains challenging. Underwood et al. introduced a discontinuous quantum walk model that integrates discrete and continuous dynamics via edge coloring of graphs and color-switching mechanism1. In this framework, a chromatic subgraph of a specific color is initially selected, followed by the execution of continuous quantum walk evolution on the retained subgraph. Although this model achieves universal quantum computation through engineered perfect state transfers in continuous quantum walks, its discrete dynamics remain limited to graph-switching mechanisms, omitting the essential incorporation of coin operators, which are both a defining characteristic and functional cornerstone of discrete quantum walks33,35–38. The absence of coinbased state control intrinsically restricts the model’s ability to emulate authentic discrete walk features, effectively reducing it to concatenated continuous walks interspersed with chromatic switching. Recent phasespace quantum walk implementations offer alternative hybrid strategies by encoding continuous variables through geometric phase manipulations on circles, lines, or Poincaré disks39,40, suggesting new paradigms for synthesizing discrete and continuous walk properties while preserving their intrinsic operational features. However, existing hybrid models either neglect critical discrete elements such as coin operators or constrain 1 Institute of Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, China. 2School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing, China. 3State Key Laboratory of Mathematical Sciences, Academy of Mathematics and Systems Science, e-mail: Chinese Academy of Sciences, Beijing, China. npj Quantum Information | (2026)12:21 1 Article https://doi.org/10.1038/s41534-025-01165-z continuous evolution to specific geometries, failing to generalize across arbitrary graphs. The challenge to simultaneously maintaining discrete control (via coins) and continuous dynamics on general graphs motivates our work. We propose a hybrid quantum walk framework that unifies discrete and continuous dynamics by integrating discrete coin operators for controlling coin states while introducing Hamiltonian dynamics on arbitrary graphs. This approach embeds coin operations into continuous quantum walks, enabling simultaneous discrete coin controls and continuous propagation via graph Hamiltonians. Systematic analysis on 2-vertex circles, star graphs, and lines reveals unique dynamical signatures in probability distributions and entanglement entropy, distinguishing our model from conventional quantum walks. This hybrid architecture, combining discrete and continuous quantum walks, achieves highly efficient state transfer and superior computational performance on general graphs. Through the joint engineering of coin operators and continuous quantum walks, it enables perfect state transfer on arbitrary connected graphs, overcoming previous limitations imposed by specific topological structures37. For K regular graphs with n vertices and bounded degrees {d1, ⋯ , dK}, the architecture implements a quantum adjacency matrix multiplication algorithm with complexity O(n2d1 ⋯ dK), demonstrating a clear advantage over the classical state-of-the-art method (O(n2.371552))41. Experimental verification on the PennyLane platform confirms its practical effectiveness in solving triangle counting problems42, a fundamental task in graph analysis. By establishing coordinated control between discrete coin operators and continuous-time evolution, our hybrid framework successfully integrates discrete and continuous quantum walk paradigms. We begin by establishing the background on discrete coin-based and continuous quantum walks. Building on this foundation, we introduce our hybrid quantum walk model and describe its dynamic evolutions and properties across 2-vertex circles, star graphs, and lines. We then demonstrate how discontinuous qu (...truncated)