Quantum spectral clustering algorithm for unsupervised learning

TL;DR

This paper presents a quantum spectral clustering algorithm that leverages quantum phase estimation and Grover's search to achieve significant speedups in unsupervised learning tasks, addressing limitations of quantum k-means.

Contribution

It introduces a novel quantum circuit design for spectral clustering that does not require quantum RAM or adiabatic processes, expanding quantum clustering capabilities.

Findings

Effective in solving clustering problems in simulations

Provides substantial quantum speedup over classical methods

Complements quantum k-means algorithms in unsupervised learning

Abstract

Clustering is one of the most crucial problems in unsupervised learning, and the well-known $k$-means clustering algorithm has been shown to be implementable on a quantum computer with a significant speedup. However, many clustering problems cannot be solved by $k$-means, and a powerful method called spectral clustering is introduced to solve these problems. In this work, we propose a circuit design to implement spectral clustering on a quantum processor with a substantial speedup, by initializing the processor into a maximally entangled state and encoding the data information into an efficiently-simulatable Hamiltonian. Compared with the established quantum $k$-means algorithms, our method does not require a quantum random access memory or a quantum adiabatic process. It relies on an appropriate embedding of quantum phase estimation into Grover's search to gain the quantum speedup. Simulations demonstrate that our method is effective in solving clustering problems and will serve as an important supplement to quantum $k$-means for unsupervised learning.