🗺️ Quantum Feature Maps
Transform data into high-dimensional quantum spaces
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0 / 5 completed🌟 The Quantum Kernel Trick
Quantum feature maps embed classical data into exponentially large quantum Hilbert spaces—enabling kernel methods with quantum advantage. Map x → |Φ(x)⟩ creates 2^n dimensional feature space with n qubits, while classical kernels struggle beyond thousands of dimensions.
💡 Why Feature Maps Matter
Classical SVMs map data to high dimensions via kernels K(x,y) = φ(x)·φ(y). Computing this directly is exponential. Quantum feature maps create |Φ(x)⟩ states where the kernel K(x,y) = |⟨Φ(x)|Φ(y)⟩|² is measured via swap test or destructive interference—exponential feature space in polynomial time!
🎯 What You'll Master
📐 Feature Map Mathematics
U_Φ(x) = ∏ᵢ U_entangle · U_rotation(xᵢ)
Alternate between data-encoding rotations and entangling gates
K(x,y) = |⟨0|U†_Φ(y)U_Φ(x)|0⟩|²
Measured by applying inverse map and checking overlap
🔬 Key Insight: Expressivity vs Trainability
More repetitions and entanglement increase expressivity (ability to represent complex functions) but can create barren plateaus where gradients vanish. Balance depth with trainability—typically 2-4 repetitions optimal for NISQ devices.