# Quantum algorithm and quantum circuit for A-Optimal Projection: dimensionality reduction

## TL;DR

This paper introduces a quantum algorithm and circuit for A-Optimal Projection, achieving exponential speedup over classical methods in dimensionality reduction tasks, with efficient complexity and broad applicability.

## Contribution

The paper presents a novel quantum algorithm and circuit for A-Optimal Projection, improving efficiency and removing quantum-classical interactions compared to prior classical approaches.

## Key findings

- Exponential speedup in feature space dimensions
- Quantum circuit complexity scales logarithmically
- High fidelity with reduced quantum-classical interaction

## Abstract

Learning low dimensional representation is a crucial issue for many machine learning tasks such as pattern recognition and image retrieval. In this article, we present a quantum algorithm and a quantum circuit to efficiently perform A-Optimal Projection for dimensionality reduction. Compared with the best-know classical algorithms, the quantum A-Optimal Projection (QAOP) algorithm shows an exponential speedup in both the original feature space dimension $n$ and the reduced feature space dimension $k$. We show that the space and time complexity of the QAOP circuit are $O\left[ {{{\log }_2}\left( {nk} /{\epsilon} \right)} \right]$ and $O[ {\log_2(nk)} {poly}\left({{\log }_2}\epsilon^{-1} \right)]$ respectively, with fidelity at least $1-\epsilon$. Firstly, a reformation of the original QAOP algorithm is proposed to help omit the quantum-classical interactions during the QAOP algorithm. Then the quantum algorithm and quantum circuit with performance guarantees are proposed. Specifically, the quantum circuit modules for preparing the initial quantum state and implementing the controlled rotation can be also used for other quantum machine learning algorithms.