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arXiv:2501.10431 · IEEE Transactions on Quantum Engineering

Quantum Annealing for Robust Principal Component Analysis

Ian Tomeo, Panos P. Markopoulos, Andreas E. Savakis

Published Jan 1, 2026• 2 citations
arXivHTMLPDF

Abstract

Principal component analysis is commonly used for dimensionality reduction, feature extraction, denoising, and visualization. The most commonly used principal component analysis method is based upon optimization of the L2-norm, however, the L2-norm is known to exaggerate the contribution of errors and outliers. When optimizing over the L1-norm, the components generated are known to exhibit robustness or resistance to outliers in the data. The L1-norm components can be solved for with a binary optimization problem. Previously, L1-BF has been used to solve the binary optimization for multiple components simultaneously. In this paper we propose QAPCA, a new method for finding principal components using quantum annealing hardware which will optimize over the robust L1-norm. The conditions required for convergence of the annealing problem are discussed. The potential speedup when using quantum annealing is demonstrated through complexity analysis and experimental results. To showcase performance against classical principal component analysis techniques experiments upon synthetic Gaussian data, a fault detection scenario and breast cancer diagnostic data are studied. We find that the reconstruction error when using QAPCA is comparable to that when using L1-BF.

DOI
10.1109/tqe.2026.3661822

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Principal component analysis is commonly used for dimensionality reduction, feature extraction, denoising, and visualization. The most commonly used principal component analysis method is based upon optimization of the L2-norm, however, the L2-norm is known to exaggerate the contribution of errors and outliers. When optimizing over the L1-norm, the components generated are known to exhibit robustness or resistance to outliers in the data. The L1-norm components can be solved for with a binary optimization problem. Previously, L1-BF has been used to solve the binary optimization for multiple components simultaneously. In this paper we propose QAPCA, a new method for finding principal components using quantum annealing hardware which will optimize over the robust L1-norm. The conditions required for convergence of the annealing problem are discussed. The potential speedup when using quantum annealing is demonstrated through complexity analysis and experimental results. To showcase performance against classical principal component analysis techniques experiments upon synthetic Gaussian data, a fault detection scenario and breast cancer diagnostic data are studied. We find that the reconstruction error when using QAPCA is comparable to that when using L1-BF.
Principal Component Analysis (PCA) is commonly used for dimensionality reduction, feature extraction, data denoising, and visualization. The L1-PCA is known to confer robustness or a resistance to outliers in the data. In this paper, a new method for L1-PCA is explored using quantum annealing hardware. To showcase performance increases as compared to other PCA types, results for a fault detection scenario are presented and the speedup of L1-PCA using quantum annealing is demonstrated. Additionally, L1-PCA has better fault detection rates than L2-PCA when in the presence of outliers.
Abstract Kernel principal component analysis (kernel PCA) is a nonlinear dimensionality reduction technique that employs kernel functions to map data into a high-dimensional feature space, thereby extending the applicability of linear PCA to nonlinear data and facilitating the extraction of informative principal components. However, kernel PCA necessitates the manipulation of large-scale matrices, leading to high computational complexity and posing challenges for efficient implementation in big data environments. Quantum computing has recently been integrated with kernel methods in machine learning, enabling effective analysis of input data within intractable feature spaces. Although existing quantum kernel PCA proposals promise exponential speedups, they impose stringent requirements on quantum hardware that are challenging to fulfill. In this work, we propose a quantum algorithm for kernel PCA by establishing a connection between quantum kernel methods and block encoding, thereby diagonalizing the centralized kernel matrix on a quantum computer. The query complexity is logarithmic with respect to the size of the data vector, D , and linear with respect to the size of the dataset. An exponential speedup could be achieved when the dataset consists of a few high-dimensional vectors, wherein the dataset size is polynomial in log ⁡ ( D ) , with D being significantly large. In contrast to existing work, our algorithm enhances the efficiency of quantum kernel PCA and reduces the requirements for quantum hardware. Furthermore, we have also demonstrated that the algorithm based on block encoding matches the lower bound of query complexity, indicating that our algorithm is nearly optimal. Our research has laid down new pathways for developing quantum machine learning algorithms aimed at addressing tangible real-world problems and demonstrating quantum advantages within machine learning.
The analysis of data and the development of Quantum Support Vector Machine (QSVM) models for disease classification have rapidly become crucial, commanding significant attention from the research community. Principal Component Analysis (PCA) is a powerful dimensionality reduction technique that effectively eliminates noise and optimizes processing in machine learning applications. When integrated with QSVM, PCA plays an essential role in reducing the number of qubits required and significantly enhancing model performance. In this study, we present a robust breast cancer diagnosis model that seamlessly incorporates PCA into QSVM, clearly demonstrating its effectiveness in improving classification performance on biomedical data compared to traditional methods. Our experimental results decisively show that PCA not only optimizes computational resources but also substantially increases model accuracy, affirming its transformative potential for quantum applications in large-scale data analysis. Our proposed model achieves an exceptional classification accuracy of 98%, unequivocally highlighting the impact of PCA in strengthening QSVM performance for biomedical applications.
Abstract Principal component analysis (PCA) is an important dimensionality reduction method in machine learning and data analysis. Recently, the quantum version of PCA has been established to diagonalize quantum states. Although these quantum algorithms promise quantum advantages, they require substantial resources beyond the reach of state-of-the-art quantum technologies. This work aims to reduce resource requirements and improve the efficiency of quantum PCA. Assuming that the quantum state is accessed through a purified quantum query model and a sampling model, we propose quantum algorithms that use minimal resource requirements for ancillary qubits to reveal properties of eigenvectors and eigenvalues of a state. In particular, we show that estimating eigenvalue λ with error ε and success probability larger than λ ( 1 − η ) requests a query complexity O ~ ( ϵ − 1 ) and a sample complexity O ~ ( ϵ − 2 η − 1 ) , respectively. To our knowledge, our result is the first quantum speedup that achieves asymptotic linear scaling in 1 / ϵ for quantum PCA. As applications, we discuss estimating the minimum relative entropy of entanglement of bipartite pure-states and performing quantum state discrimination tasks. We show that quantum speedups are maintained when the pure state has a low Schmidt number and states of discrimination have a low rank. This study opens up a new quantum PCA method for high-dimensional quantum data analysis and discusses its application in quantum information processing tasks.
Quantum principal component analysis (QPCA) is analogous to the classical method with additive speedup. This is based on the unsupervised dimensionality reduction mechanism that transforms exponentially high dimensional data points into low dimensional space by retaining significant features with the largest variance. To our knowledge, quantum computations are highly exponential and supports implicit parallelism. Therefore, quantum method for finding principal components are proved to achieve exponential speedup over existing classical solution. An advantage of quantum superposition is utilized for the Eigen decomposition of density matrix which is identical to covariance, however, a way through which its computation happen, is truly an interesting factor. This paper presents basic methodological framework to design QPCA algorithm. Our framework includes the methods for quantum data encoding, quantum memory processing, and the requisite quantum circuit implementations to realize QPCA algorithms. In addition, we categorized existing QPCA methods in view of recent advances. We then proposed the possible advancements that can benefit to design quantum based efficient algorithms for principal component analysis. At the last, suggested concepts are compared with existing methods based on complexities and other factors to justify our proposal as significant in perspective to future directions.
Quantum Principal Component Analysis (QPCA) offers a theoretical exponential speedup over classical PCA, but its practical performance under realistic conditions remains underexplored. This work-in-progress research investigates QPCA's viability by implementing a Quantum Phase Estimation (QPE)based QPCA circuit using Qiskit and benchmarking it on ten well known classical datasets in the IBM Quantum simulator environment. We evaluated both the fidelity of eigenvalue and eigenvector reconstruction as well as the downstream impact of QPCA versus classical PCA on regression and classification tasks. Our results confirm that QPCA can reproduce classical eigenvalues with high precision (error<10−4<10^{-4}<10−4), but its eigenvector fidelity degrades in cases of near-degenerate spectra. We further observe that PCA benefits classification more consistently than regression, regardless of the implementation method. These findings establish a preliminary empirical foundation for understanding QPCA's strengths and limitations and serve as a baseline for future work incorporating hardware noise, scalable circuit design, and hybrid approaches.