Quantum Kernel Estimation

Overview

Quantum kernels measure similarity between data points in a high-dimensional quantum feature space. They enable quantum-enhanced support vector machines (SVMs) and other kernel methods.

The Concept

Classical kernels: K(x, y) = ⟨φ(x), φ(y)⟩

Quantum kernels: K(x, y) = |⟨φ(x)|φ(y)⟩|²

where |φ(x)⟩ is a quantum state encoding x.

The Circuit

To compute K(x₁, x₂):

CODE
|0⟩ ─┤ Feature Map(x₂) ├─┤ Feature Map(x₁)† ├─┤M├
|0⟩ ─┤                   ├─┤                   ├─┤M├

Kernel value = P(00) = probability of measuring |00⟩

Feature Map (ZZFeatureMap)

CODE
┌───┐┌────────┐     ┌───────────────┐     ┌───┐┌────────┐
q_0: ┤ H ├┤ RZ(2x₀)├──■──┤               ├──■──┤ H ├┤ RZ(2x₀)├
     ├───┤├────────┤┌─┴─┐│RZ(2(π-x₀)(π-x₁))│┌─┴─┐├───┤├────────┤
q_1: ┤ H ├┤ RZ(2x₁)├┤ X ├┤               ├┤ X ├┤ H ├┤ RZ(2x₁)├
     └───┘└────────┘└───┘└───────────────┘└───┘└───┘└────────┘

Running the Circuit

PYTHON
from circuit import run_circuit, compute_kernel_matrix

# Single kernel computation
result = run_circuit(x1=[0.5, 0.3], x2=[0.6, 0.4])
print(f"K(x1, x2) = {result['kernel']:.4f}")

# Full kernel matrix
X = [[0.0, 0.0], [0.5, 0.5], [1.0, 0.0]]
K = compute_kernel_matrix(X)
print(K)

Expected Output

Points	Kernel Value	Interpretation
Same point	~1.0	Identical
Similar	0.5-0.9	High similarity
Different	0.1-0.5	Low similarity
Orthogonal	~0.0	No similarity

Properties

Symmetric: K(x, y) = K(y, x)
Bounded: 0 ≤ K(x, y) ≤ 1
Self-similar: K(x, x) = 1
Positive semi-definite: Valid kernel matrix

Use with SVM

PYTHON
from sklearn.svm import SVC

# Compute quantum kernel matrix
K_train = compute_kernel_matrix(X_train)

# Train SVM with precomputed kernel
clf = SVC(kernel='precomputed')
clf.fit(K_train, y_train)

Quantum Advantage

Quantum kernels can:

Access exponentially large feature spaces
Compute classically intractable kernels
Potentially achieve quantum speedup for certain problems