Quantum principal component analysis

TL;DR

This paper introduces a quantum principal component analysis method that uses the unknown quantum state itself to perform a unitary transformation, enabling faster eigenvector identification compared to classical algorithms.

Contribution

It presents a novel quantum algorithm that leverages the state to perform PCA exponentially faster than existing methods.

Findings

Enables eigenvector extraction of quantum states more efficiently

Uses the state itself to perform unitary transformations

Achieves exponential speedup over classical PCA algorithms

Abstract

The usual way to reveal properties of an unknown quantum state, given many copies of a system in that state, is to perform measurements of different observables and to analyze the measurement results statistically. Here we show that the unknown quantum state can play an active role in its own analysis. In particular, given multiple copies of a quantum system with density matrix \rho, then it is possible to perform the unitary transformation e^{-i\rho t}. As a result, one can create quantum coherence among different copies of the system to perform quantum principal component analysis, revealing the eigenvectors corresponding to the large eigenvalues of the unknown state in time exponentially faster than any existing algorithm.