# Quantum advantage for learning shallow neural networks with natural data distributions

## TL;DR

This paper shows a quantum algorithm can learn certain neural networks more efficiently than classical methods, especially with non-uniform data distributions.

## Contribution

The paper introduces a quantum algorithm for learning periodic neurons with non-uniform data and proves exponential quantum advantage over classical algorithms.

## Key findings

- Quantum algorithm efficiently learns periodic neurons over non-uniform distributions.
- Problem is hard for classical gradient-based and SQ algorithms.
- Establishes exponential quantum advantage in this learning regime.

## Abstract

Without large quantum computers to empirically evaluate performance, theoretical frameworks such as the quantum statistical query (QSQ) are a primary tool to study quantum algorithms for learning classical functions and search for quantum advantage in machine learning tasks. However, we only understand quantum advantage in this model at two extremes: either exponential advantages for uniform input distributions or no advantage for arbitrary distributions. Our work helps close the gap between these two regimes by designing an efficient quantum algorithm for learning periodic neurons in the QSQ model over a variety of non-uniform distributions and the first explicit treatment of real-valued functions. We prove that this problem is hard not only for classical gradient-based algorithms, which are the workhorses of machine learning, but also for a more general class of SQ algorithms, establishing an exponential quantum advantage.

There is a current void in understanding quantum learning advantages between two extreme cases (exponential advantage for uniform distributions, no advantage for adversarial distributions). Lewis et al. design an efficient quantum algorithm for learning periodic neurons over non-uniform distributions of classical data and show that this is hard for a broad class of classical algorithms, resulting in an exponential quantum advantage.