Sampling-based sublinear low-rank matrix arithmetic framework for dequantizing quantum machine learning

TL;DR

This paper introduces a classical algorithmic framework for low-rank matrix operations inspired by quantum algorithms, demonstrating that quantum speedups in certain linear algebra tasks may not be exponential under specific data models.

Contribution

The authors develop a classical SVT framework that matches quantum algorithms' performance, generalizing and improving dequantization results for various quantum machine learning techniques.

Findings

Classical algorithms run in dimension-independent time under sampling assumptions.

Quantum SVT does not provide exponential speedups in the QRAM data structure model.

The framework recovers and enhances dequantization results for multiple quantum machine learning applications.

Abstract

We present an algorithmic framework for quantum-inspired classical algorithms on close-to-low-rank matrices, generalizing the series of results started by Tang's breakthrough quantum-inspired algorithm for recommendation systems [STOC'19]. Motivated by quantum linear algebra algorithms and the quantum singular value transformation (SVT) framework of Gily\'en, Su, Low, and Wiebe [STOC'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: $\ell^2$-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.