A rigorous and robust quantum speed-up in supervised machine learning

TL;DR

This paper demonstrates a provable quantum speed-up for supervised classification using a quantum support vector machine that only requires classical data access, establishing advantages over classical methods under certain hardness assumptions.

Contribution

The paper introduces a quantum classifier based on a support vector machine that achieves a rigorous speed-up without relying on quantum data access, unlike previous algorithms.

Findings

Quantum classifier achieves high accuracy on constructed datasets.

Classical learners cannot classify the data better than random guessing under hardness assumptions.

Quantum kernel estimation is robust against sampling errors.

Abstract

Over the past few years several quantum machine learning algorithms were proposed that promise quantum speed-ups over their classical counterparts. Most of these learning algorithms either assume quantum access to data -- making it unclear if quantum speed-ups still exist without making these strong assumptions, or are heuristic in nature with no provable advantage over classical algorithms. In this paper, we establish a rigorous quantum speed-up for supervised classification using a general-purpose quantum learning algorithm that only requires classical access to data. Our quantum classifier is a conventional support vector machine that uses a fault-tolerant quantum computer to estimate a kernel function. Data samples are mapped to a quantum feature space and the kernel entries can be estimated as the transition amplitude of a quantum circuit. We construct a family of datasets and show that no classical learner can classify the data inverse-polynomially better than random guessing, assuming the widely-believed hardness of the discrete logarithm problem. Meanwhile, the quantum classifier achieves high accuracy and is robust against additive errors in the kernel entries that arise from finite sampling statistics.