We study quantum speedups in quantum machine learning (QML) by analyzing the quantum singular value transformation (QSVT) framework. QSVT, introduced by [GSLW, STOC'19, arXiv:1806.01838], unifies all major types of quantum speedup; in particular, a wide variety of QML proposals are applications of QSVT on low-rank classical data. We challenge these proposals by providing a classical algorithm that matches the performance of QSVT in this regime up to a small polynomial overhead. We show that, given a matrix $A \in \mathbb{C}^{m\times n}$, a vector $b \in \mathbb{C}^{n}$, a bounded degree-$d$ polynomial $p$, and linear-time pre-processing, we can output a description of a vector $v$ such that $\|v - p(A) b\| \leq \varepsilon\|b\|$ in $\widetilde{\mathcal{O}}(d^{11} \|A\|_{\mathrm{F}}^4 / (\varepsilon^2 \|A\|^4 ))$ time. This improves upon the best known classical algorithm [CGLLTW, STOC'20, arXiv:1910.06151], which requires $\widetilde{\mathcal{O}}(d^{22} \|A\|_{\mathrm{F}}^6 /(\varepsilon^6 \|A\|^6 ) )$ time, and narrows the gap with QSVT, which, after linear-time pre-processing to load input into a quantum-accessible memory, can estimate the magnitude of an entry $p(A)b$ to $\varepsilon\|b\|$ error in $\widetilde{\mathcal{O}}(d\|A\|_{\mathrm{F}}/(\varepsilon \|A\|))$ time. Our key insight is to combine the Clenshaw recurrence, an iterative method for computing matrix polynomials, with sketching techniques to simulate QSVT classically. We introduce several new classical techniques in this work, including (a) a non-oblivious matrix sketch for approximately preserving bi-linear forms, (b) a new stability analysis for the Clenshaw recurrence, and (c) 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：我们通过分析量子奇异值变换（QSVT）框架研究量子机器学习中的量子加速机制。由[GSLW, STOC'19, arXiv:1806.01838]提出的QSVT统一了所有主要类型的量子加速方法；特别是，大量量子机器学习方案本质上是将QSVT应用于低秩经典数据。我们通过提出一种经典算法来挑战这些方案，该算法在此场景下与QSVT的性能匹配，仅存在较小的多项式开销。我们证明：给定矩阵$A \in \mathbb{C}^{m\times n}$、向量$b \in \mathbb{C}^{n}$、有界次数$d$的多项式$p$以及线性时间预处理，可以在$\widetilde{\mathcal{O}}(d^{11} \|A\|_{\mathrm{F}}^4 / (\varepsilon^2 \|A\|^4 ))$时间内输出向量$v$的描述，使得$\|v - p(A) b\| \leq \varepsilon\|b\|$。这改进了已知最优经典算法[CGLLTW, STOC'20, arXiv:1910.06151]（需$\widetilde{\mathcal{O}}(d^{22} \|A\|_{\mathrm{F}}^6 /(\varepsilon^6 \|A\|^6 ) )$时间），并缩小了与QSVT的差距——后者在通过线性时间预处理将输入加载到量子可访问内存后，可在$\widetilde{\mathcal{O}}(d\|A\|_{\mathrm{F}}/(\varepsilon \|A\|))$时间内估算$p(A)b$的某个分量幅值至$\varepsilon\|b\|$误差。我们的核心洞察是将计算矩阵多项式的迭代方法Clenshaw递推与素描技术结合，以经典方式模拟QSVT。本文引入多项新型经典技术，包括：(a) 用于近似保持双线性形式的非 oblivious 矩阵素描，(b) Clenshaw递推的新稳定性分析，(c) 有界函数切比雪夫级数展开系数算术级数的新界定理，这些技术本身可能具有独立研究价值。