修正量子感知机算法复杂度,提出两种量子增强学习方法。
On Quantum Perceptron Learning via Quantum Search
- 用格罗弗搜索和量子行走改进感知机学习的查询效率。
- 发现原算法复杂度受维度影响,在高维下性能下降明显。
- 适合研究量子机器学习理论复杂度的学者参考。
随着量子机器学习兴起,感知机作为传统机器学习的基础单元,成为探索量子算法潜力的重要模型。本文主要贡献有二:首先,重新审视Kapoor等人(2016)提出的量子版本空间感知机算法,指出其复杂度假设存在缺陷,证明该算法的查询复杂度与维度相关,在最坏情况下高维情形下表现显著下降;其次,提出并分析了两种基于量子增强的切平面算法,利用格罗弗搜索和量子行走等成熟量子子程序,给出具体算法构造及查询与算术复杂度分析。在理想无噪声量子计算模型下,结果建立了更优的复杂度边界,揭示了边缘依赖性、维度依赖性与量子资源之间的权衡关系。这些发现为量子感知机模型及其理论计算复杂性提供了更精确的理解。
原文摘要 · Abstract (English)
With the growing interest in quantum machine learning, the perceptron, a fundamental building block in traditional machine learning, has emerged as a valuable model for exploring the potential of quantum algorithms. In this work, we make two principal contributions. First, we revisit the \emph{quantum version space perceptron} algorithm proposed by Kapoor et al. (2016), by identifying and correcting a flawed complexity assumption. We show that the query complexity of the algorithm is dimension-dependent, which has significant implications for its behaviour in high-dimensional regimes under worst-case scenarios. Second, we propose and analyse two \emph{quantum-enhanced} cutting-plane algorithms for perceptron learning. Specifically, we leverage established quantum subroutines such as \emph{Grover's search} and \emph{quantum walk search}, and provide detailed algorithmic constructions together with query and arithmetic complexity analyses. Our results establish improved complexity bounds under an idealised implementation framework and noise-free quantum computational models, offering insights into the trade-offs between margin dependence, dimensional dependence, and quantum resources. These findings provide a refined understanding of quantum perceptron models and their theoretical computational complexity properties.
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