We construct an efficient classical analogue of the quantum matrix inversion
algorithm (HHL) for low-rank matrices. Inspired by recent work of Tang,
assuming length-square sampling access to input data, we implement the
pseudoinverse of a low-rank matrix and sample from the solution to the problem
$Ax=b$ using fast sampling techniques. We implement the pseudo-inverse by
finding an approximate singular value decomposition of $A$ via subsampling,
then inverting the singular values. In principle, the approach can also be used
to apply any desired "smooth" function to the singular values. Since many
quantum algorithms can be expressed as a singular value transformation problem,
our result suggests that more low-rank quantum algorithms can be effectively
"dequantised" into classical length-square sampling algorithms.

本论文构建了一个高效的量子矩阵反演算法（HHL）的经典模拟器，基于 Tang 的最近研究成果，通过对输入数据进行长度平方采样，实现了低秩矩阵的伪逆并使用快速抽样技术从问题 $Ax=b$ 的解中采样。本文通过子采样来找到 $A$ 的近似奇异值分解，从而实现伪逆。该方法也可用于把任何期望的 “平滑” 函数应用到奇异值上。既然许多量子算法可以表示为奇异值变换问题，我们的结果表明更多的低秩量子算法可以有效地 “去量子化” 成为使用长度平方采样的经典算法。

基于量子理论的低秩随机回归，其维度对数依赖性

Quantum-inspired low-rank stochastic regression with logarithmic  dependence on the dimension

We give a classical analogue to Kerenidis and Prakash's quantum
recommendation system, previously believed to be one of the strongest
candidates for provably exponential speedups in quantum machine learning. Our
main result is an algorithm that, given an $m \times n$ matrix in a data
structure supporting certain $\ell^2$-norm sampling operations, outputs an
$\ell^2$-norm sample from a rank-$k$ approximation of that matrix in time
$O(\text{poly}(k)\log(mn))$, only polynomially slower than the quantum
algorithm. As a consequence, Kerenidis and Prakash's algorithm does not in fact
give an exponential speedup over classical algorithms. Further, under strong
input assumptions, the classical recommendation system resulting from our
algorithm produces recommendations exponentially faster than previous classical
systems, which run in time linear in $m$ and $n$.
The main insight of this work is the use of simple routines to manipulate
$\ell^2$-norm sampling distributions, which play the role of quantum
superpositions in the classical setting. This correspondence indicates a
potentially fruitful framework for formally comparing quantum machine learning
algorithms to classical machine learning algorithms.

通过使用简单例程来操纵 $\ell^2$ -norm 抽样分布，并提供一个给定数据结构的算法来生成从矩阵秩 - $k$ 近似值的 $\ell^2$ -norm 样本，本研究为 Kerenidis 和 Prakash 的量子推荐系统提供了古典类比，进一步证明此算法并未超越经典算法；此外，尽管假定输入数据重程度较大，但我们的经典推荐系统与先前线性 m 和 n 时间运行的经典系统相比能够在指数上更快地产生推荐。