Quantum machine learning (QML) has shown great potential to produce killer applications of quantum computers by introducing the possibility of large quantum speedups for computationally intensive linear algebra tasks. The quantum singular value transformation (QSVT), introduced by Gily\'en, Su, Low and Wiebe, is a unifying framework to obtain QML algorithms. We provide a classical algorithm that matches the performance of QSVT, up to a small polynomial overhead. In particular, given a bounded matrix $A\in\mathbb{C}^{m\times n}$, a vector $b\in\mathbb{C}^{n}$, and a bounded degree-$d$ polynomial $p$, QSVT can output a measurement from the state $|p(A)b\rangle$ in $O(d\|A\|_F )$ time. We show that for any $\epsilon >0$, we can output a vector $v$ such that $\|v-p(A)b\|\le\epsilon$ in $O(d^9\|A\|_F^4/\epsilon^2)$ time after linear-time pre-processing. This improves upon the best known classical algorithm [CGL+'20], which requires $O(d^{22}\|A\|_F^6/\epsilon^6)$ time. Instantiating our algorithm with different polynomials, we obtain fast quantum-inspired algorithms for regression/matrix inversion, recommendation systems and Hamiltonian simulation. We improve upon several recent papers specialized to specific problems, including [CGL+'20,SM'21,GST'22,CCH+'22} for regression, and [Tan'19,CGL+'20,CCH+'22] for recommendation systems. 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 include (a) a non-oblivious matrix sketch for approximately preserving bi-linear forms, (b) a non-oblivious 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 expansion of bounded functions.

翻译：量子机器学习（QML）通过引入计算密集型线性代数任务中的大规模量子加速可能性，展现出产生量子计算机“杀手级应用”的巨大潜力。Gilyén、Su、Low和Wiebe提出的量子奇异值变换（QSVT）是一个统一框架，用于获取QML算法。我们提出了一种经典算法，其性能与QSVT相当，仅存在小幅多项式开销。具体而言，给定有界矩阵$A\in\mathbb{C}^{m\times n}$、向量$b\in\mathbb{C}^{n}$以及有界$d$次多项式$p$，QSVT可在$O(d\|A\|_F )$时间内从状态$|p(A)b\rangle$输出测量结果。我们证明，对任意$\epsilon >0$，在线性时间预处理后，可在$O(d^9\|A\|_F^4/\epsilon^2)$时间内输出向量$v$，使得$\|v-p(A)b\|\le\epsilon$。这改进了已知最优经典算法[CGL+'20]（需$O(d^{22}\|A\|_F^6/\epsilon^6)$时间）。通过将我们的算法与不同多项式实例化，可获得用于回归/矩阵求逆、推荐系统和哈密顿模拟的快速量子启发式算法。我们改进了近期针对特定问题的若干论文，包括用于回归的[CGL+'20,SM'21,GST'22,CCH+'22]和用于推荐系统的[Tan'19,CGL+'20,CCH+'22]。我们的关键洞察在于将计算矩阵多项式的迭代方法——Clenshaw递推与草图技术相结合，以经典方式模拟QSVT。我们引入的工具包括：(a) 用于近似保持双线性形式的非 oblivious 矩阵草图，(b) 基于$\ell_2^2$采样的非 oblivious 非对称近似矩阵乘积草图，(c) Clenshaw递推的新稳定性分析，以及(d) 对有界函数切比雪夫展开系数算术级数进行界定的新技术。