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

2025-05-29

AI Summary

To address the high time complexity of k-nearest neighbors (kNN) computation on large-scale datasets and the difficulty of balancing efficiency and accuracy in both classical and quantum kNN algorithms, this paper proposes the Granular-Ball Quantum Approximate kNN (GB-QkNN) algorithm. GB-QkNN is the first method to integrate granular-ball computing with quantized Hierarchical Navigable Small World (HNSW) indexing into a quantum kNN framework: it first compresses data and reduces dimensionality via granular-ball representation; then constructs an efficient approximate index by jointly leveraging hierarchical navigability and vector quantization, while co-optimizing distance computations. Theoretical analysis shows that GB-QkNN achieves sublinear time complexity. Experimental results demonstrate orders-of-magnitude speedup in search latency while preserving approximation accuracy. This work overcomes a key practicality bottleneck in quantum kNN and establishes a novel paradigm for scalable nearest-neighbor retrieval.

Technology Category

• Machine Learning: Quantum Machine Learning
• Search and Optimization: Distributed Search
• Data Mining & Knowledge Management: Data Compression

Application Category

• Graph Algorithms and Modeling for the Web: Querying, indexing, and retrieval in Web-related graphs
• Search and Retrieval-Augmented AI: Efficiency and scalability of Web search engines
• Social Networks and Social Media: Social mining and social search on the Web

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.

Problem

• Reducing high time complexity in kNN algorithms
• Accelerating search via granular-ball data reduction
• Optimizing distance calculation with quantization in HNSW

Innovation

• Granular-ball computing reduces data size
• HNSW method accelerates search process
• Quantization optimizes distance calculation steps