Filtered Spectral Projection for Quantum Principal Component Analysis

• URL: http://arxiv.org/abs/2603.13441v1
• Date: Fri, 13 Mar 2026 11:55:39 GMT
• Title: Filtered Spectral Projection for Quantum Principal Component Analysis
• Abstract summary: Filtered Spectral Projection Algorithm (FSPA)<n>For amplitude-encoded centered data, the ensemble density matrix $=sum_i p_i|_iranglelangle_i|$ coincides with the covariance matrix.<n>For uncentered data, $$ corresponds to PCA without centering, and we derive eigenvalue interlacing bounds quantifying the deviation from standard PCA.
• Score: 1.2318267573115809
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Quantum principal component analysis (qPCA) is commonly formulated as the extraction of eigenvalues and eigenvectors of a covariance-encoded density operator. Yet in many qPCA settings, the practical objective is simpler: projecting data onto the dominant spectral subspace. In this work, we introduce a projection-first framework, the Filtered Spectral Projection Algorithm (FSPA), which bypasses explicit eigenvalue estimation while preserving the essential spectral structure. FSPA amplifies any nonzero warm-start overlap with the leading principal subspace and remains robust in small-gap and near-degenerate regimes without inducing artificial symmetry breaking in the absence of bias. To connect this approach to classical datasets, we show that for amplitude-encoded centered data, the ensemble density matrix $ρ=\sum_i p_i|ψ_i\rangle\langleψ_i|$ coincides with the covariance matrix. For uncentered data, $ρ$ corresponds to PCA without centering, and we derive eigenvalue interlacing bounds quantifying the deviation from standard PCA. We further show that ensembles of quantum states admit an equivalent centered covariance interpretation. Numerical demonstrations on benchmark datasets, including Breast Cancer Wisconsin and handwritten Digits, show that downstream performance remains stable whenever projection quality is preserved. These results suggest that, in a broad class of qPCA settings, spectral projection is the essential primitive, and explicit eigenvalue estimation is often unnecessary.

Related papers

• Shape Operator PCA: Curvature-Aware Projections for Geometric Machine Learning [52.452902154360565]
  SHOPCA (Shape Operator-based Principal Component Analysis) is a novel method for unsupervised metric learning and dimensionality reduction.<n>We evaluate SHOPCA on more than 50 real-world benchmark datasets.
  arXiv  Detail & Related papers  (2026-08-15T16:37:47Z)
• Eigenstate Preparation Through Near-Optimal Eigenprobability Filtering [3.2651468573861524]
  We introduce the Dominant Eigenstate Filtering via Eigenprobability Amplification and Thresholding (DEFEAT) algorithm.<n>DEFEAT identifies and filters the eigenstate with the largest overlap with the supplied initial state.<n>Our results provide a general eigenvalue-agnostic primitive for dominant eigenstate filtering and preparation, and for estimating properties of dominant eigenstates.
  arXiv  Detail & Related papers  (2026-08-12T17:39:15Z)
• Quantum principal component analysis without eigenvector recovery [9.811303336285961]
  We introduce a measurement-based soft PCA framework replacing the hard top-$k$ projector with an entropy-regularized Fermi--Dirac filter.<n>This filter has a direct interpretation as a quantum measurement, which naturally suggests a quantum approach.<n>For new inputs, the same quantum circuit yields soft principal subspace scores, spectral energy profiles, and postselected filtered states.
  arXiv  Detail & Related papers  (2026-05-27T04:27:47Z)
• Depth, Not Data: An Analysis of Hessian Spectral Bifurcation [29.426396222985563]
  The eigenvalue distribution of the Hessian matrix plays a crucial role in understanding the landscape of deep neural networks.<n>We demonstrate that such spectral Bifurcation can arise purely from the network architecture, independent of data imbalance.<n>Our results suggest that both model architecture and data characteristics should be considered when designing optimization algorithms for deep networks.
  arXiv  Detail & Related papers  (2026-01-31T06:17:00Z)
• Amortized Spectral Kernel Discovery via Prior-Data Fitted Network [0.0]
  We introduce an interpretability-driven framework for amortized spectral discovery from pre-trained PFNs with decoupled attention.<n>We propose decoder architectures that map PFN latents to explicit spectral density estimates and corresponding stationary kernels.<n>This yields orders-of-magnitude reductions in inference time compared to optimization-based baselines.
  arXiv  Detail & Related papers  (2026-01-29T13:51:26Z)
• Quantization of Large Language Models with an Overdetermined Basis [73.79368761182998]
  We introduce an algorithm for data quantization based on the principles of Kashin representation. Our findings demonstrate that Kashin Quantization achieves competitive or superior quality in model performance.
  arXiv  Detail & Related papers  (2024-04-15T12:38:46Z)
• Datacube segmentation via Deep Spectral Clustering [76.48544221010424]
  Extended Vision techniques often pose a challenge in their interpretation. The huge dimensionality of data cube spectra poses a complex task in its statistical interpretation. In this paper, we explore the possibility of applying unsupervised clustering methods in encoded space. A statistical dimensional reduction is performed by an ad hoc trained (Variational) AutoEncoder, while the clustering process is performed by a (learnable) iterative K-Means clustering algorithm.
  arXiv  Detail & Related papers  (2024-01-31T09:31:28Z)
• Improving Expressive Power of Spectral Graph Neural Networks with Eigenvalue Correction [55.57072563835959]
  We propose an eigenvalue correction strategy that can free filters from the constraints of repeated eigenvalue inputs.<n>Concretely, the proposed eigenvalue correction strategy enhances the uniform distribution of eigenvalues, and improves the fitting capacity and expressive power of filters.
  arXiv  Detail & Related papers  (2024-01-28T08:12:00Z)
• Sparse PCA with Oracle Property [115.72363972222622]
  We propose a family of estimators based on the semidefinite relaxation of sparse PCA with novel regularizations. We prove that, another estimator within the family achieves a sharper statistical rate of convergence than the standard semidefinite relaxation of sparse PCA.
  arXiv  Detail & Related papers  (2023-12-28T02:52:54Z)
• Quantum tomography of helicity states for general scattering processes [55.2480439325792]
  Quantum tomography has become an indispensable tool in order to compute the density matrix $rho$ of quantum systems in Physics. We present the theoretical framework for reconstructing the helicity quantum initial state of a general scattering process.
  arXiv  Detail & Related papers  (2023-10-16T21:23:42Z)
• Spectral Estimators for Structured Generalized Linear Models via Approximate Message Passing [28.91482208876914]
  We consider the problem of parameter estimation in a high-dimensional generalized linear model.<n>Despite their wide use, a rigorous performance characterization, as well as a principled way to preprocess the data, are available only for unstructured designs.
  arXiv  Detail & Related papers  (2023-08-28T11:49:23Z)
• Bayes-optimal limits in structured PCA, and how to reach them [21.3083877172595]
  We study the paradigmatic matrix model of principal components analysis (PCA), where a rank-one matrix is corrupted by additive noise. We provide the first characterization of the Bayes-optimal limits of inference in this model. We propose a novel approximate message passing algorithm (AMP), inspired by the theory of Adaptive Thouless-Anderson-Palmer equations.
  arXiv  Detail & Related papers  (2022-10-03T21:31:41Z)
• Covariance matrix preparation for quantum principal component analysis [0.8258451067861933]
  Principal component analysis (PCA) is a dimensionality reduction method in data analysis. Quantum algorithms have been formulated for PCA based on diagonalizing a density matrix. We numerically implement our method for molecular ground-state datasets.
  arXiv  Detail & Related papers  (2022-04-07T15:11:42Z)
• An $\ell_p$ theory of PCA and spectral clustering [23.90245234027558]
  Principal Component Analysis is a powerful tool in statistics and machine learning. In this paper, we develop an $ell_p$ theory for a hollowed version of PCA in Hilbert spaces. For contextual community detection, the $ell_p$ theory leads to a simple spectral algorithm that achieves the information threshold for exact recovery.
  arXiv  Detail & Related papers  (2020-06-24T21:30:28Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.