Trainable Quantum Kernel

Overview

Extends quantum kernels with learnable parameters that can be optimized for specific tasks. Uses kernel-target alignment (KTA) as the training objective.

The Method

Standard Quantum Kernel

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K(x, y) = |⟨φ(x)|φ(y)⟩|²

Trainable Quantum Kernel

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K_θ(x, y) = |⟨φ_θ(x)|φ_θ(y)⟩|²

Where θ are learnable parameters in the feature map.

Kernel-Target Alignment

Measures how well kernel matches the classification task:

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KTA(K, y) = ⟨K, yy^T⟩_F / (||K||_F · ||yy^T||_F)

• KTA = 1: Perfect alignment
• KTA = 0: No correlation
• KTA = -1: Anti-alignment

Circuit Structure

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Layer 1 (Data + Trainable):
     ┌─────────┐┌────────┐┌────────┐
q_0: ┤ RY(x₀π) ├┤ RZ(θ₀) ├┤ RY(θ₁) ├──■──
     ├─────────┤├────────┤├────────┤┌─┴─┐
q_1: ┤ RY(x₁π) ├┤ RZ(θ₂) ├┤ RY(θ₃) ├┤ X ├
     └─────────┘└────────┘└────────┘└───┘

Repeat for n_reps layers...

Running the Circuit

PYTHONCopy
from circuit import run_circuit

result = run_circuit(n_samples=15, n_iterations=20)

print(f"Random KTA: {result['random_kta']:.4f}")
print(f"Trained KTA: {result['trained_kta']:.4f}")
print(f"Improvement: {result['improvement']:.4f}")

Expected Output

Training Process

1. Initialize random parameters θ₀
2. Compute kernel matrix K_θ
3. Evaluate KTA(K_θ, y)
4. Update θ via gradient descent
5. Repeat until convergence

Comparison

Applications

• Task-specific kernels: Optimize for particular datasets
• Transfer learning: Pre-train on related tasks
• Ensemble methods: Multiple trained kernels

Learn More

• Trainable Quantum Kernels
• Kernel-Target Alignment