Kernel Target Alignment Optimization
Overview
Optimizes quantum kernel parameters to maximize centered kernel-target alignment (KTA), providing task-specific quantum feature maps for improved classification performance.
| Property | Value |
|---|---|
| Category | Machine Learning |
| Difficulty | Advanced |
| Framework | PennyLane |
| Qubits | 4 |
| Depth | Variable |
| Gates | RY, RZ, CNOT |
Centered Alignment
More robust than standard KTA:
CODECKA(K, Y) = ⟨K̃, Ỹ⟩_F / (||K̃||_F · ||Ỹ||_F)
Where K̃, Ỹ are centered versions:
CODEK̃ = HKH, H = I - (1/n)11^T
Parameterized Encoding
Data-dependent + trainable parameters:
CODE┌───────────┐┌────────┐ ┌───────────────┐ q_0: ┤ RY(x₀·θ₀) ├┤ RZ(θ₄) ├──■──┤ RZ(x₀x₁·θ₈) ├ ├───────────┤├────────┤┌─┴─┐└───────────────┘ q_1: ┤ RY(x₁·θ₁) ├┤ RZ(θ₅) ├┤ X ├──────────────── ├───────────┤├────────┤└───┘ q_2: ┤ RY(x₀·θ₂) ├┤ RZ(θ₆) ├──■── ├───────────┤├────────┤┌─┴─┐ q_3: ┤ RY(x₁·θ₃) ├┤ RZ(θ₇) ├┤ X ├ └───────────┘└────────┘└───┘
Running the Circuit
PYTHONfrom circuit import run_circuit result = run_circuit(n_samples=12, n_epochs=20) print(f"Initial alignment: {result['initial_alignment']:.4f}") print(f"Final alignment: {result['final_alignment']:.4f}") print(f"Improvement: {result['improvement']:.4f}")
Expected Output
| Phase | Alignment |
|---|---|
| Random init | ~0.0-0.2 |
| After training | ~0.5-0.8 |
Optimization Details
Loss Function
CODEL(θ) = -CKA(K_θ, Y)
Gradient Computation
Uses PennyLane's automatic differentiation:
- Parameter-shift rule for quantum gradients
- Adam optimizer for stable training
Hyperparameters
- Learning rate: 0.1
- Epochs: 20-50
- Layers: 2-3
Test Case: XOR Pattern
The circuit is tested on XOR-like data:
CODEClass 1: (x > 0) == (y > 0) ⊕⊕ Class -1: (x > 0) != (y > 0) ⊖⊕
This is non-linearly separable, making it a good test for quantum advantage.
Applications
- Dataset-specific kernels: Learn optimal feature space
- Automated feature engineering: No manual feature design
- Transfer learning: Adapt kernels to new domains