Spectral methods: crucial for machine learning, natural for quantum computers?

TL;DR

This paper argues that spectral methods are inherently suited for quantum computers and could revolutionize machine learning by enabling more efficient manipulation of models' Fourier spectra.

Contribution

It highlights the potential for quantum computing to directly and efficiently manipulate spectral properties of machine learning models, a task difficult for classical methods.

Findings

Quantum Fourier Transform can manipulate spectral properties of quantum states.

Spectral methods are fundamental to many machine learning techniques.

Quantum computing could enable resource-efficient spectral model design.

Abstract

This article presents an argument for why quantum computers could unlock new methods for machine learning. We argue that spectral methods, in particular those that learn, regularise, or otherwise manipulate the Fourier spectrum of a machine learning model, are often natural for quantum computers. For example, if a generative machine learning model is represented by a quantum state, the Quantum Fourier Transform allows us to manipulate the Fourier spectrum of the state using the entire toolbox of quantum routines, an operation that is usually prohibitive for classical models. At the same time, spectral methods are surprisingly fundamental to machine learning: A spectral bias has recently been hypothesised to be the core principle behind the success of deep learning; support vector machines have been known for decades to regularise in Fourier space, and convolutional neural nets build filters in the Fourier space of images. Could, then, quantum computing open fundamentally different, much more direct and resource-efficient ways to design the spectral properties of a model? We discuss this potential in detail here, hoping to stimulate a direction in quantum machine learning research that puts the question of ``why quantum?'' first.