As quantum machine learning continues to develop at a rapid pace, the importance of ensuring the robustness and efficiency of quantum algorithms cannot be overstated. Our research presents an analysis of quantum randomized smoothing, how data encoding and perturbation modeling approaches can be matched to achieve meaningful robustness certificates. By utilizing an innovative approach integrating Grover's algorithm, a quadratic sampling advantage over classical randomized smoothing is achieved. This strategy necessitates a basis state encoding, thus restricting the space of meaningful perturbations. We show how constrained $k$-distant Hamming weight perturbations are a suitable noise distribution here, and elucidate how they can be constructed on a quantum computer. The efficacy of the proposed framework is demonstrated on a time series classification task employing a Bag-of-Words pre-processing solution. The advantage of quadratic sample reduction is recovered especially in the regime with large number of samples. This may allow quantum computers to efficiently scale randomized smoothing to more complex tasks beyond the reach of classical methods.

翻译：随着量子机器学习的快速发展，确保量子算法的鲁棒性与效率的重要性不言而喻。本研究分析了量子随机平滑技术，探讨了如何匹配数据编码与扰动建模方法以获得有意义的鲁棒性证明。通过整合Grover算法的创新方法，实现了相对于经典随机平滑的二次采样优势。该策略需要采用基态编码，从而限制了有意义扰动的空间。我们证明了约束$k$-距离汉明权重扰动在此处是合适的噪声分布，并阐明了如何在量子计算机上构建它们。所提出框架的有效性在一个采用词袋预处理解决方案的时间序列分类任务上得到了验证。二次样本减少的优势在样本数量较大的情况下尤为显著。这可能使量子计算机能够将随机平滑高效地扩展到经典方法无法处理的更复杂任务中。