Quantum Kernel SVM

Overview

Combines quantum feature maps with classical Support Vector Machines for classification. The quantum kernel measures similarity in exponentially large Hilbert space.

The Quantum Kernel

Feature Map

Classical data x → Quantum state |φ(x)⟩

Kernel Function

CODECopy
K(x, y) = |⟨φ(x)|φ(y)⟩|²

Classification

SVM finds hyperplane in feature space maximizing margin.

Circuit Structure

ZZ-Feature Map:

CODECopy
┌───┐┌────────┐     ┌────────────┐
q_0: ┤ H ├┤ RZ(x₀π)├──■──┤ RZ(x₀x₁π) ├──■──
     ├───┤├────────┤┌─┴─┐└────────────┘  │
q_1: ┤ H ├┤ RZ(x₁π)├┤ X ├───────────────┼──
     ├───┤├────────┤└───┘               │
q_2: ┤ H ├┤ RZ(x₀π)├────────────────────┼──
     ├───┤├────────┤                  ┌─┴─┐
q_3: ┤ H ├┤ RZ(x₁π)├──────────────────┤ X ├
     └───┘└────────┘                  └───┘

Running the Circuit

PYTHONCopy
from circuit import run_circuit

result = run_circuit(n_samples=20, n_qubits=4)

print(f"Accuracy: {result['accuracy']:.2%}")
print(f"Support vectors: {result['n_support_vectors']}")

Expected Output

Training Process

1. Compute kernel matrix: K[i,j] = K(xᵢ, xⱼ)
2. Solve SVM dual: Find optimal αᵢ
3. Identify support vectors: Points with αᵢ > 0
4. Predict: f(x) = Σαᵢyᵢk(x, xᵢ) + b

Advantages

Applications

• Classification: Binary and multi-class
• Anomaly detection: One-class SVM
• Feature selection: Kernel PCA
• Clustering: Spectral methods

Learn More

• Quantum Kernel Methods
• Power of Quantum Kernels