Matrix multiplication is a fundamental classical computing operation whose efficiency becomes a major challenge at scale, especially for machine learning applications. Quantum computing, with its inherent parallelism and exponential storage capacity, offers a potential solution to these limitations. This work presents a quantum kernel-based matrix multiplication algorithm (QKMM) that achieves an asymptotically optimal computational complexity of $ O(N^2 \log_2 N) $, outperforming the classical optimal complexity of $ O(N^{2.371552}) $, where $N$ denotes the matrix dimension. Through noiseless and noisy quantum simulation experiments, we demonstrate that the proposed algorithm not only exhibits superior theoretical efficiency but also shows practical advantages in runtime performance and stability.

翻译：矩阵乘法是经典计算中的基本运算，其效率在大规模应用（尤其是机器学习领域）中成为一个主要挑战。量子计算凭借其固有的并行性和指数级存储能力，为突破这些限制提供了潜在解决方案。本文提出一种基于量子核的矩阵乘法算法（QKMM），实现了渐近最优计算复杂度$ O(N^2 \log_2 N) $，优于经典最优复杂度$ O(N^{2.371552}) $，其中$N$表示矩阵维度。通过无噪声及含噪声的量子模拟实验，我们证明所提算法不仅具有优越的理论效率，在运行时间性能和稳定性方面也展现出实际优势。