Quantum Dimensionality Reduction by Linear Discriminant Analysis

TL;DR

This paper introduces a quantum algorithm for linear discriminant analysis that significantly accelerates dimensionality reduction, enabling efficient processing of high-dimensional data for quantum machine learning applications.

Contribution

The paper improves existing quantum LDA algorithms by addressing irreversibility issues and proposes new quantum circuits for low-dimensional data extraction, achieving exponential and quadratic speedups.

Findings

Exponential speedup on the number of vectors M

Quadratic speedup on data dimensionality D

Practical quantum submodule for machine learning tasks

Abstract

Dimensionality reduction (DR) of data is a crucial issue for many machine learning tasks, such as pattern recognition and data classification. In this paper, we present a quantum algorithm and a quantum circuit to efficiently perform linear discriminant analysis (LDA) for dimensionality reduction. Firstly, the presented algorithm improves the existing quantum LDA algorithm to avoid the error caused by the irreversibility of the between-class scatter matrix $S_B$ in the original algorithm. Secondly, a quantum algorithm and quantum circuits are proposed to obtain the target state corresponding to the low-dimensional data. Compared with the best-known classical algorithm, the quantum linear discriminant analysis dimensionality reduction (QLDADR) algorithm has exponential acceleration on the number $M$ of vectors and a quadratic speedup on the dimensionality $D$ of the original data space, when the original dataset is projected onto a polylogarithmic low-dimensional space. Moreover, the target state obtained by our algorithm can be used as a submodule of other quantum machine learning tasks. It has practical application value of make that free from the disaster of dimensionality.