Accelerating Spectral Clustering on Quantum and Analog Platforms

We introduce a novel hybrid quantum-analog algorithm to perform graph clustering that exploits connections between the evolution of dynamical systems on graphs and the underlying graph spectra. This approach constitutes a new class of algorithms that combine emerging quantum and analog platforms to accelerate computations. Our hybrid algorithm is equivalent to spectral clustering and has a computational complexity of $O(N)$, where $N$ is the number of nodes in the graph, compared to $O(N^3)$ scaling on classical computing platforms. The proposed method employs the dynamic mode decomposition (DMD) framework on data generated by Schr\"{o}dinger dynamics embedded into the manifold generated by the graph Laplacian. We prove and demonstrate that one can extract the eigenvalues and scaled eigenvectors of the normalized graph Laplacian from quantum evolution on the graph by using DMD computations.