Given its widespread application in machine learning and optimization, the Kronecker product emerges as a pivotal linear algebra operator. However, its computational demands render it an expensive operation, leading to heightened costs in spectral approximation of it through traditional computation algorithms. Existing classical methods for spectral approximation exhibit a linear dependency on the matrix dimension denoted by $n$, considering matrices of size $A_1 \in \mathbb{R}^{n \times d}$ and $A_2 \in \mathbb{R}^{n \times d}$. Our work introduces an innovative approach to efficiently address the spectral approximation of the Kronecker product $A_1 \otimes A_2$ using quantum methods. By treating matrices as quantum states, our proposed method significantly reduces the time complexity of spectral approximation to $O_{d,\epsilon}(\sqrt{n})$.

翻译：鉴于克龙尼克积在机器学习和优化领域的广泛应用，它已成为一项关键的线性代数算子。然而，其计算需求使其成为一种昂贵的运算，导致通过传统计算算法对其进行谱近似时成本增加。现有的经典谱近似方法在考虑矩阵尺寸分别为$A_1 \in \mathbb{R}^{n \times d}$和$A_2 \in \mathbb{R}^{n \times d}$时，其计算复杂度与矩阵维度$n$呈线性关系。本研究提出了一种创新方法，利用量子手段高效处理克龙尼克积$A_1 \otimes A_2$的谱近似问题。通过将矩阵视为量子态，我们提出的方法将谱近似的时间复杂度显著降低至$O_{d,\epsilon}(\sqrt{n})$。