📊 Quantum Principal Component Analysis
Exponential speedup for dimensionality reduction
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0 / 5 completed🌟 Quantum Dimensionality Reduction
Principal Component Analysis (PCA) identifies the most important directions in high-dimensional data—reducing 10,000 features to 10 while preserving 95% of variance. Quantum PCA (qPCA) achieves exponential speedup via quantum phase estimation and matrix exponentiation, solving covariance eigenvalue problems in polylogarithmic time.
💡 Why Quantum PCA?
Classical PCA requires O(nd²) time for n samples and d dimensions—expensive for genomics (millions × thousands), images (pixels × channels), or financial data. Quantum PCA runs in O(log(nd) × k/ε²) where k = principal components, ε = precision—exponential advantage for large datasets.
🎯 What You'll Master
📈 Classical vs Quantum PCA
Compute covariance X^T X → eigendecomposition → sort eigenvalues
Encode density matrix → phase estimation → extract top k eigenvalues
🔬 Key Insight: Quantum Spectral Analysis
Quantum phase estimation extracts eigenvalues λᵢ from e^(iλᵢt) phases. For density matrix ρ = X^T X / n, qPCA finds principal components via quantum eigenvalue sampling—no need to compute full covariance matrix classically. Exponential speedup when k ≪ d.