Quantum machine learning (QML) has shown great potential to produce large quantum speedups for linear algebra tasks. The quantum singular value transformation (QSVT), introduced by [GSLW, STOC'19, arXiv:1806.01838], is a unifying framework to obtain QML algorithms. We provide a classical algorithm that matches the performance of QSVT on low-rank inputs, up to small polynomial overhead. Under quantum memory assumptions, given a bounded matrix $A\in\mathbb{C}^{m\times n}$, vector $b\in\mathbb{C}^{n}$, and bounded degree-$d$ polynomial $p$, QSVT can output a measurement from the state $|p(A)b\rangle$ in $O(d\|A\|_F)$ time after linear-time pre-processing. We show that, in the same setting, for any $\varepsilon>0$, we can output a vector $v$ such that $\|v - p(A) b\|\leq\varepsilon\|b\|$ in $O(d^9\|A\|_F^4/\varepsilon^2)$ time after linear-time pre-processing. This improves upon the best known classical algorithm [CGLLTW, STOC'20, arXiv:1910.06151], which requires $O(d^{22}\|A\|_F^6/\varepsilon^6)$ time. Instantiating the aforementioned algorithm with different polynomials, we obtain fast quantum-inspired algorithms for regression, recommendation systems, and Hamiltonian simulation. We improve in numerous parameter settings on prior work, including those that use problem-specialized approaches. Our key insight is to combine the Clenshaw recurrence, an iterative method for computing matrix polynomials, with sketching techniques to simulate QSVT classically. The tools we introduce in this work include (a) a matrix sketch for approximately preserving bi-linear forms, (b) an asymmetric approximate matrix product sketch based on $\ell_2^2$ sampling, (c) a new stability analysis for the Clenshaw recurrence, and (d) a new technique to bound arithmetic progressions of the coefficients appearing in the Chebyshev series expansion of bounded functions, each of which may be of independent interest.

翻译：量子机器学习（QML）在线性代数任务中展现出实现巨大量子加速的巨大潜力。由[GSLW, STOC'19, arXiv:1806.01838]引入的量子奇异值变换（QSVT）是获取QML算法的统一框架。我们提出了一种经典算法，在低秩输入上匹配QSVT的性能，仅增加较小的多项式开销。在量子内存假设下，给定有界矩阵 $A\in\mathbb{C}^{m\times n}$、向量 $b\in\mathbb{C}^{n}$ 和有界 $d$ 次多项式 $p$，QSVT 可在线性时间预处理后，于 $O(d\|A\|_F)$ 时间内从状态 $|p(A)b\rangle$ 输出测量结果。我们证明，在相同设定下，对于任意 $\varepsilon>0$，我们可在线性时间预处理后，于 $O(d^9\|A\|_F^4/\varepsilon^2)$ 时间内输出向量 $v$，使得 $\|v - p(A) b\|\leq\varepsilon\|b\|$。这改进了已知最佳的经典算法[CGLLTW, STOC'20, arXiv:1910.06151]，该算法需要 $O(d^{22}\|A\|_F^6/\varepsilon^6)$ 时间。将上述算法与不同多项式实例化，我们获得了用于回归、推荐系统和哈密顿模拟的快速量子启发式算法。我们在许多参数设置上改进了先前工作，包括那些使用问题专用方法的工作。我们的关键见解是将Clenshaw递推（一种计算矩阵多项式的迭代方法）与草图技术相结合，以经典方式模拟QSVT。本文引入的工具包括：(a) 用于近似保持双线性形式的矩阵草图，(b) 基于 $\ell_2^2$ 采样的非对称近似矩阵乘积草图，(c) Clenshaw递推的新的稳定性分析，以及(d) 用于界定有界函数切比雪夫级数展开中系数算术级数的新技术，这些工具各自可能具有独立的研究意义。