Efficient Quantum Approximate $k$NN Algorithm via Granular-Ball Computing

TL;DR

This paper introduces GB-QkNN, a quantum $k$NN algorithm that combines granular-ball data reduction, hierarchical search acceleration, and quantization to significantly improve efficiency for large datasets.

Contribution

It presents a novel quantum $k$NN algorithm integrating granular-ball data reduction, HNSW acceleration, and quantization, reducing time complexity compared to existing methods.

Findings

Achieves higher efficiency in quantum $k$NN with large data.

Reduces time complexity through granular-ball and quantization techniques.

Provides comprehensive complexity analysis demonstrating improvements.

Abstract

High time complexity is one of the biggest challenges faced by $k$-Nearest Neighbors ($k$NN). Although current classical and quantum $k$NN algorithms have made some improvements, they still have a speed bottleneck when facing large amounts of data. To address this issue, we propose an innovative algorithm called Granular-Ball based Quantum $k$NN(GB-Q$k$NN). This approach achieves higher efficiency by first employing granular-balls, which reduces the data size needed to processed. The search process is then accelerated by adopting a Hierarchical Navigable Small World (HNSW) method. Moreover, we optimize the time-consuming steps, such as distance calculation, of the HNSW via quantization, further reducing the time complexity of the construct and search process. By combining the use of granular-balls and quantization of the HNSW method, our approach manages to take advantage of these treatments and significantly reduces the time complexity of the $k$NN-like algorithms, as revealed by a comprehensive complexity analysis.