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

TL;DR

QGH-256 introduces a quantum spectral hash that maps a message to a weighted graph via a discrete walker on a $4\times4$ toroidal grid, then uses Quantum Phase Estimation on the graph Laplacian with a superposition input to generate a multiscale spectral fingerprint. The fingerprint, formed from heat trace values across multiple times, yields a 256-bit digest that is sensitive to input perturbations and differentiates even cospectral graphs by exploiting eigenvector overlaps. The approach combines classical walker dynamics with quantum spectral analysis to deliver a post-quantum, collision-resistant hash suitable for PoW and data integrity applications, while addressing PQC standards and potential Grover threats. Results are demonstrated on a seeded statevector simulator in Qiskit, with discussion of hardware-noise implications and future work toward real-device implementation.

Abstract

We present Quantum Graph Hash (QGH-256), a novel quantum spectral hashing algorithm that generates high-entropy fingerprints from message-induced graphs. Each input message is mapped to a weighted graph via a discrete random walk on an n X n toroidal grid, where the walk dynamics determine the edge weights. Quantum Phase Estimation (QPE) is then used to extract the phase spectrum of the graph Laplacian. Unlike standard QPE settings, the phase estimation is performed with respect to a superposition state (a uniform superposition over all node basis states) rather than an eigenvector, ensuring that all eigencomponents contribute to the resulting spectrum. This yields spectral features that distinguish even co-spectral but non-isomorphic message-induced graphs. The final spectral fingerprint is converted into a 256-bit digest, producing a compact representation of the input. As the fingerprint encodes both spectral and dynamical properties of the message-induced graph, the resulting hash exhibits strong sensitivity to input perturbations and provides a structurally rich foundation for post-quantum hashing. To demonstrate the feasibility of the approach, we implement QGH-256 on a 4 X 4 toroidal grid, chosen empirically: smaller grids exhibit collisions, whereas larger grids significantly increase execution time. The entire pipeline is implemented in Qiskit, and we use a seeded statevector simulator to obtain stable, noise-free results.

Figures (10)

• Figure 1: One-dimensional classical random walk along the toroidal path
• Figure 2: Two-dimensional classical random walk along the toroidal path
• Figure 3: Message induced discrete walker path traced in 4X4 toroidal grid for the message "Hi"
• Figure 4: Graph Laplacian of the weighted graph produced by the message "Hi"
• Figure 5: A simple 3-node weighted graph. Node $v_1$ is connected to $v_2$ with weight 1, and $v_2$ is connected to $v_3$ with weight 2. There is no direct edge between $v_1$ and $v_3$.