Variational Quantum Algorithms for Dimensionality Reduction and Classification

TL;DR

This paper introduces quantum algorithms for dimensionality reduction and classification that leverage variational methods, achieving exponential speedups over classical algorithms and providing versatile outputs for quantum or classical use.

Contribution

The work presents novel quantum algorithms for neighborhood preserving embedding and local discriminant embedding, along with a variational quantum eigenvalue solver for generalized eigenproblems.

Findings

Exponential speedup over classical methods

Successful implementation on 32x32 eigenvalue problems

Provides quantum and classical output options

Abstract

In this work, we present a quantum neighborhood preserving embedding and a quantum local discriminant embedding for dimensionality reduction and classification. We demonstrate that these two algorithms have an exponential speedup over their respectively classical counterparts. Along the way, we propose a variational quantum generalized eigenvalue solver that finds the generalized eigenvalues and eigenstates of a matrix pencil $(\mathcal{G},\mathcal{S})$. As a proof-of-principle, we implement our algorithm to solve $2^5\times2^5$ generalized eigenvalue problems. Finally, our results offer two optional outputs with quantum or classical form, which can be directly applied in another quantum or classical machine learning process.