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:

CODE
CKA(K, Y) = ⟨K̃, Ỹ⟩_F / (||K̃||_F · ||Ỹ||_F)

Where K̃, Ỹ are centered versions:

CODE
K̃ = 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

PYTHON
from 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

CODE
L(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:

CODE
Class 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

Kernel-Target Alignment