We propose fast and practical quantum-inspired classical algorithms for solving linear systems. Specifically, given sampling and query access to a matrix $A\in\mathbb{R}^{m\times n}$ and a vector $b\in\mathbb{R}^m$, we propose classical algorithms that produce a data structure for the solution $x\in\mathbb{R}^{n}$ of the linear system $Ax=b$ with the ability to sample and query its entries. The resulting $x$ satisfies $\|x-A^{+}b\|\leq\epsilon\|A^{+}b\|$, where $\|\cdot\|$ is the spectral norm and $A^+$ is the Moore-Penrose inverse of $A$. Our algorithm has time complexity $\widetilde{O}(\kappa_F^4/\kappa\epsilon^2)$ in the general case, where $\kappa_{F} =\|A\|_F\|A^+\|$ and $\kappa=\|A\|\|A^+\|$ are condition numbers. Compared to the prior state-of-the-art result [Shao and Montanaro, arXiv:2103.10309v2], our algorithm achieves a polynomial speedup in condition numbers. When $A$ is $s$-sparse, our algorithm has complexity $\widetilde{O}(s \kappa\log(1/\epsilon))$, matching the quantum lower bound for solving linear systems in $\kappa$ and $1/\epsilon$ up to poly-logarithmic factors [Harrow and Kothari]. When $A$ is $s$-sparse and symmetric positive-definite, our algorithm has complexity $\widetilde{O}(s\sqrt{\kappa}\log(1/\epsilon))$. Technically, our main contribution is the application of the heavy ball momentum method to quantum-inspired classical algorithms for solving linear systems, where we propose two new methods with speedups: quantum-inspired Kaczmarz method with momentum and quantum-inspired coordinate descent method with momentum. Their analysis exploits careful decomposition of the momentum transition matrix and the application of novel spectral norm concentration bounds for independent random matrices. Finally, we also conduct numerical experiments for our algorithms on both synthetic and real-world datasets, and the experimental results support our theoretical claims.

翻译：我们提出了快速实用的量子启发式经典算法用于求解线性系统。具体而言，给定对矩阵 $A\in\mathbb{R}^{m\times n}$ 和向量 $b\in\mathbb{R}^m$ 的采样与查询访问，我们提出了经典算法，该算法可为线性系统 $Ax=b$ 的解 $x\in\mathbb{R}^n$ 构建一个支持对其元素进行采样与查询的数据结构。所得解 $x$ 满足 $\|x-A^{+}b\|\leq\epsilon\|A^{+}b\|$，其中 $\|\cdot\|$ 为谱范数，$A^+$ 为 $A$ 的摩尔-彭若斯逆。在一般情况下，我们的算法时间复杂度为 $\widetilde{O}(\kappa_F^4/\kappa\epsilon^2)$，其中 $\kappa_{F} =\|A\|_F\|A^+\|$ 与 $\kappa=\|A\|\|A^+\|$ 为条件数。相较于此前最优结果 [Shao and Montanaro, arXiv:2103.10309v2]，我们的算法在条件数上实现了多项式级加速。当 $A$ 为 $s$-稀疏矩阵时，算法复杂度为 $\widetilde{O}(s \kappa\log(1/\epsilon))，在 $\kappa$ 与 $1/\epsilon$ 上达到求解线性系统的量子下界（至多相差多对数因子）[Harrow and Kothari]。当 $A$ 为 $s$-稀疏且对称正定矩阵时，算法复杂度为 $\widetilde{O}(s\sqrt{\kappa}\log(1/\epsilon))$。从技术层面看，我们的主要贡献在于将重球动量法应用于求解线性系统的量子启发式经典算法，并提出了两种加速新方法：带动量的量子启发式卡奇玛兹法与带动量的量子启发式坐标下降法。其分析利用了动量转移矩阵的精细分解以及针对独立随机矩阵的新型谱范数集中界。最后，我们在合成数据集与真实数据集上对算法进行了数值实验，实验结果支持了我们的理论结论。