Linear regression is one of the most fundamental linear algebra problems. Given a dense matrix $A \in \mathbb{R}^{n \times d}$ and a vector $b$, the goal is to find $x'$ such that $ \| Ax' - b \|_2^2 \leq (1+\epsilon) \min_{x} \| A x - b \|_2^2 $. The best classical algorithm takes $O(nd) + \mathrm{poly}(d/\epsilon)$ time [Clarkson and Woodruff STOC 2013, Nelson and Nguyen FOCS 2013]. On the other hand, quantum linear regression algorithms can achieve exponential quantum speedups, as shown in [Wang Phys. Rev. A 96, 012335, Kerenidis and Prakash ITCS 2017, Chakraborty, Gily{\'e}n and Jeffery ICALP 2019]. However, the running times of these algorithms depend on some quantum linear algebra-related parameters, such as $\kappa(A)$, the condition number of $A$. In this work, we develop a quantum algorithm that runs in $\widetilde{O}(\epsilon^{-1}\sqrt{n}d^{1.5}) + \mathrm{poly}(d/\epsilon)$ time. It provides a quadratic quantum speedup in $n$ over the classical lower bound without any dependence on data-dependent parameters. In addition, we also show our result can be generalized to multiple regression and ridge linear regression.

翻译：线性回归是最基础的线性代数问题之一。给定稠密矩阵 $A \in \mathbb{R}^{n \times d}$ 和向量 $b$，目标是找到 $x'$，使得 $ \| Ax' - b \|_2^2 \leq (1+\epsilon) \min_{x} \| A x - b \|_2^2 $。最优经典算法的时间复杂度为 $O(nd) + \mathrm{poly}(d/\epsilon)$ [Clarkson and Woodruff STOC 2013, Nelson and Nguyen FOCS 2013]。另一方面，量子线性回归算法可实现指数级量子加速，如[Wang Phys. Rev. A 96, 012335, Kerenidis and Prakash ITCS 2017, Chakraborty, Gily{\'e}n and Jeffery ICALP 2019]所示。然而，这些算法的运行时间依赖于某些量子线性代数相关参数，例如 $A$ 的条件数 $\kappa(A)$。在本工作中，我们开发了一种量子算法，其运行时间为 $\widetilde{O}(\epsilon^{-1}\sqrt{n}d^{1.5}) + \mathrm{poly}(d/\epsilon)$。该算法在 $n$ 上相对于经典下界实现了二次量子加速，且无需依赖任何数据相关参数。此外，我们还证明该结果可推广至多元回归和岭线性回归。