Quantum Hash Function Based on Spectral Properties of Graphs and Discrete Walker Dynamics

Abstract We present Quantum Graph Hash (QGH-256), a novel spectral hashing algorithm based on quantum techniques for generating fingerprints from message-induced graphs. Each input message is mapped to a weighted graph using a discrete walker on an n × n toroidal grid, where edge weights are determined by the message structure. Quantum Phase Estimation (QPE) is then employed to extract spectral information from the graph Laplacian by estimating the phase value. These features capture both structural and dynamical properties of the graph, enabling the method to distinguish between graphs that may share the same eigenvalue spectrum but differ in structure. The resulting spectral fingerprint is compressed into a fixed 256-bit hash, providing a compact representation of the input message. The proposed approach exhibits strong sensitivity to small input changes, a key requirement for cryptographic hash functions. To demonstrate feasibility, we implement QGH-256 on a 4 × 4 toroidal grid using Qiskit, and perform simulations with a seeded statevector backend to ensure stable and noise-free results. We evaluate the cryptographic properties of QGH-256 and compare its behavior with current standards. We also discuss its relevance in the context of post-quantum cryptography and outline potential applications of the proposed hashing scheme.