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

TL;DR

The paper tackles the prohibitive time complexity of $k$-nearest neighbors on large datasets by marrying granular-ball computing with a quantum-accelerated HNSW framework. It introduces GB-Q$k$NN, which first reduces data via granular-balls and then performs a quantum-enhanced, layer-wise $k$NN search using QRAM, angle encoding, Swap test, and quantum comparators. The authors provide a comprehensive time-complexity analysis showing a construction cost of $O(cdN)$ and a search cost of $O(\log M)$, outperforming both classical graph-based approaches and prior quantum $k$NN methods. This approach offers a scalable path toward fast approximate quantum nearest-neighbor classification on large-scale data, with practicality improving as quantum hardware matures.

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.

Figures (7)

• Figure 1: Quantum circuit for swap test.
• Figure 2: Quantum circuit for comparison.
• Figure 3: Overall flow of the GB-Q$k$NN.
• Figure 4: HNSW method.
• Figure 5: Quantum state encoding circuit.