QGB-W$k$NN: Quantum Granular-Ball Learning for Robust Classification

Quantum clustering aims to exploit quantum feature representations to uncover complex data structures beyond conventional Euclidean geometry. Yet this sample-level kernel construction requires O(n^2) quantum circuit executions for n data points, creating a major bottleneck under near-term quantum resource constraints. Prior solutions fail to resolve this efficiency-accuracy dilemma: classical granular-ball clustering reduces sample complexity but relies on Euclidean metrics that cannot capture quantum correlations, while existing quantum compression schemes prioritize efficiency over structural preservation, degrading performance on non-convex or noisy data. Here we propose Granular-Ball Quantum Clustering (GBQC), a framework that tightly couples granular-ball structural abstraction with quantum feature learning. GBQC first compresses raw data into compact, representative granular balls via a PCA-guided splitting strategy, reducing kernel evaluations by 80% compared to full-sample methods. A quantum cohesion mechanism then filters noisy granules in Hilbert space to improve clustering robustness. Extensive experiments on synthetic, noisy, overlapping, and real-world datasets demonstrate that GBQC consistently achieves superior clustering accuracy and robustness compared with representative classical and quantum clustering methods. Meanwhile, the proposed granular-ball compression significantly reduces quantum kernel evaluations and computational overhead, enabling quantum clustering experiments on larger datasets within parameterized quantum learning frameworks. These results suggest that granular-ball representations serve not only as a compression mechanism to reduce quantum computational costs but also as an effective structural abstraction mechanism that improves clustering quality by eliminating redundant and structurally ambiguous learning units.

The NISQ era places stringent constraints on quantum computation, where noise and decoherence fundamentally limit performance. In classical deep learning, model robustness and resilience to perturbations are well studied: deep neural networks (DNNs) maintain high performance despite pruning, noise injection, and structural perturbations due to inherent redundancy in their representations. A central challenge in quantum machine learning is to transfer this notion of robustness to quantum neural networks (QNNs) under realistic NISQ noise. While classical deep learning exhibits robustness through structural redundancy, analogous principles for QNNs remain underdeveloped. We propose JGRA: a framework for assessing robustness in noise-aware QNNs via Jacobian geometry, capturing model sensitivity to parameter perturbations induced by noise. Our method includes entropy-matched noise calibration, noise-aware training, and noise-conditioned Jacobian extraction, yielding geometric descriptors that link clean-regime structure to noisy inference behaviour. We also empirically demonstrate that these descriptors encode predictive information about robustness under unseen noise.

The $k$-Nearest Neighbor~(KNN) algorithm is widely used across various tasks. The selection of the $k$ value is a key issue because it significantly impacts performance. In this paper, an adaptive and efficient KNN approach via granular-ball computing is proposed. The method consists of two stages. \textcolor{black}{In the training stage, the dataset is first coarsely partitioned to reduce the complexity of data distributions within a granular ball, and then the Fisher criterion is introduced to control ball splitting and stopping, yielding a multi-granularity granular ball representation. In the prediction stage, the nearest granular ball is first located through a weighted distance mechanism, and an adaptive neighborhood is then constructed around the test sample. The effective $k$ value is dynamically determined by the actual number of samples contained in this neighborhood. The neighborhood induced by the nearest granular ball provides more stable local group information, thereby improving robustness against noise and local perturbations.} Experimental results demonstrate that the proposed method outperforms existing KNN variants across multiple datasets in terms of both accuracy and efficiency. The code has been open-sourced for reproducibility: https://github.com/lianxiaoyu724/Adaptive-GBKNN.