Quantum data compression by principal component analysis

TL;DR

This paper introduces a quantum PCA algorithm that exponentially speeds up data compression for high-dimensional, low-rank datasets, enabling more efficient quantum machine learning tasks.

Contribution

It presents a quantum algorithm for PCA-based data compression that achieves exponential speedup over classical methods for low-dimensional projections.

Findings

Achieves exponential speedup over classical PCA

Reduces data dimensionality for quantum machine learning algorithms

Enables practical quantum data processing with fewer resources

Abstract

Data compression can be achieved by reducing the dimensionality of high-dimensional but approximately low-rank datasets, which may in fact be described by the variation of a much smaller number of parameters. It often serves as a preprocessing step to surmount the curse of dimensionality and to gain efficiency, and thus it plays an important role in machine learning and data mining. In this paper, we present a quantum algorithm that compresses an exponentially large high-dimensional but approximately low-rank dataset in quantum parallel, by dimensionality reduction (DR) based on principal component analysis (PCA), the most popular classical DR algorithm. We show that the proposed algorithm achieves exponential speedup over the classical PCA algorithm when the original dataset are projected onto a polylogarithmically low-dimensional space. The compressed dataset can then be further processed to implement other tasks of interest, with significantly less quantum resources. As examples, we apply this algorithm to reduce data dimensionality for two important quantum machine learning algorithms, quantum support vector machine and quantum linear regression for prediction. This work demonstrates that quantum machine learning can be released from the curse of dimensionality to solve problems of practical importance.