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Physics Letters A

An improved quantum principal component analysis algorithm based on the quantum singular threshold method

Jie Lin, Wan‐Su Bao, Shuo Zhang, Tan Li, Xiang Wang

Published Jun 20, 2019• 31 citations
Publisher
DOI
10.1016/j.physleta.2019.06.026

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Quantum principal component analysis (qPCA) is the quantum version of principal component analysis (PCA). In this paper, based on the quantum singular value threshold (qSVT), we propose an exact quantum principal component analysis algorithm, which screens the data components through the threshold, rather than output all components of data. Compared with other improved qPCA algorithms, our proposed algorithm does not require to adjust the parameters to obtain estimated results. Instead, it yields exact results directly, and the quantum circuit designed is simpler because almost half of the quantum gates are reduced. We implemented our qPCA algorithm on the IBM quantum computing platform: IBM Quantum Experience, and the experimental results verified correctness of our algorithm.
We present classical and quantum algorithms based on spectral methods for a problem in tensor principal component analysis. The quantum algorithm achieves a q u a r t i c speedup while using exponentially smaller space than the fastest classical spectral algorithm, and a super-polynomial speedup over classical algorithms that use only polynomial space. The classical algorithms that we present are related to, but slightly different from those presented recently in Ref. \cite{wein2019kikuchi}. In particular, we have an improved threshold for recovery and the algorithms we present work for both even and odd order tensors. These results suggest that large-scale inference problems are a promising future application for quantum computers.
Recurrence Plot (RP) is a method employed to analyze the periodicity, chaoticity, and nonlinear characteristics of complex systems. Quantum Principal Component Analysis (QPCA), on the other hand, achieves dimensionality reduction of sample data using density matrices based on quantum circuits. We improve the distance threshold function of the recurrence plot algorithm using a density operator conceptually equivalent to the covariance matrix, integrate it with quantum circuits, and thereby develop a Quantum Recurrence Plot (QRP) algorithm. This algorithm achieves ultra-high efficiency in parallel computing, reduces computational costs, and simultaneously upgrades the traditional grayscale recurrence plot to colored heatmaps, enabling a better revelation of the system's dynamical characteristics.