Accelerating Spectral Clustering on Quantum and Analog Platforms

2024-08-16

arXiv.org

AI Summary

To address the high computational complexity—O(N³)—of classical spectral clustering in graph clustering, this paper proposes a quantum–analog hybrid algorithm: Schrödinger dynamics is implemented on quantum hardware over the graph Laplacian manifold, while Dynamic Mode Decomposition (DMD) is performed on analog hardware to extract spectral features. We provide the first rigorous proof that this synergistic approach exactly recovers the eigenvalues and scaled eigenvectors of the normalized graph Laplacian—thereby yielding solutions equivalent to those of standard spectral clustering. This establishes the first quantum–analog cooperative paradigm for spectral clustering, reducing time complexity to O(N). Theoretical analysis and empirical validation jointly confirm both the spectral accuracy of the extracted features and the algorithm’s asymptotic speedup.

Technology Category

• Machine Learning: Quantum Machine Learning
• Search and Optimization: Mixed Discrete/Continuous Search
• Data Mining & Knowledge Management: Graph Mining, Social Network Analysis & Community

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Web Mining and Content Analysis: Normalization, clustering, classification, and summarization of Web text
• Economics, Online Markets and Human Computation: Data quality aspects of human-annotated datasets

Abstract

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 the data generated by Schr""{o}dinger dynamics that evolves on the manifold induced by the graph Laplacian. In particular, we prove and demonstrate that one can extract the eigenvalues and scaled eigenvectors of the normalized graph Laplacian by evolving Schr""{o}dinger dynamics on quantum computers followed by DMD computations on analog devices.

Problem

• Quantum Computing
• Classification Techniques
• Graph Partitioning

Innovation

• Quantum Computing
• Graph Partitioning
• Hybrid Algorithm