The Geometry of Quantum Walks on Graphs—Theory and Applications

We introduce a geometric framework for continuous-time quantum walks on graphs by embedding each vertex into a Euclidean space through its time-dependent quantum probability distribution. This construction induces a rich geometry in which quantum transport is characterized by distances, radii, angles, and simplex volumes, allowing interference, localization, and spreading to be analyzed within a unified metric-angular formalism. We prove that, in contrast to classical diffusion, which collapses to a spherical geometry, quantum dynamics generate a generically non-spherical affine geometry with persistent anisotropy. Applying this theory to real-world networks—including transportation systems, semantic graphs, and neuronal connectomes—we show that quantum geometry reveals dynamically meaningful backbones, interference-based “communities”, and vulnerability structures that are invisible to classical random-walk and spectral methods. In particular, angular and radial quantum descriptors isolate functional hubs, control cores, and coherence classes without any topological or dimensionality assumptions. Together, these results demonstrate that quantum-walk-induced geometry provides a powerful new lens for understanding structure and function in complex networks.