



02222nas a2200169   4500008004100000245010800041210006900149260001400218520167700232100001801909700001901927700001701946700001901963700001501982700001801997856003702015       2020                        eng d00aSampling-based sublinear low-rank matrix arithmetic framework for dequantizing quantum machine learning0 aSamplingbased sublinear lowrank matrix arithmetic framework for   c6/18/20203 a<p>We present an algorithmic framework for quantum-inspired classical algorithms on close-to-low-rank matrices, generalizing the series of results started by Tang&#39;s breakthrough quantum-inspired algorithm for recommendation systems [STOC&#39;19]. Motivated by quantum linear algebra algorithms and the quantum singular value transformation (SVT) framework of Gilyén et al. [STOC&#39;19], we develop classical algorithms for SVT that run in time independent of input dimension, under suitable quantum-inspired sampling assumptions. Our results give compelling evidence that in the corresponding QRAM data structure input model, quantum SVT does not yield exponential quantum speedups. Since the quantum SVT framework generalizes essentially all known techniques for quantum linear algebra, our results, combined with sampling lemmas from previous work, suffice to generalize all recent results about dequantizing quantum machine learning algorithms. In particular, our classical SVT framework recovers and often improves the dequantization results on recommendation systems, principal component analysis, supervised clustering, support vector machines, low-rank regression, and semidefinite program solving. We also give additional dequantization results on low-rank Hamiltonian simulation and discriminant analysis. Our improvements come from identifying the key feature of the quantum-inspired input model that is at the core of all prior quantum-inspired results: ℓ2-norm sampling can approximate matrix products in time independent of their dimension. We reduce all our main results to this fact, making our exposition concise, self-contained, and intuitive.</p>1 aChia, Nai-Hui1 aGilyen, Andras1 aLi, Tongyang1 aLin, Han-Hsuan1 aTang, Ewin1 aWang, Chunhao  uhttps://arxiv.org/abs/1910.06151