Feature Maps and Quantum Kernels: Enhancing Machine Learning with Quantum Embeddings

Table of Contents

1. Introduction
2. Classical Feature Maps and Kernels
3. Why Quantum Feature Maps?
4. Basics of Quantum Kernel Methods
5. Embedding Data into Hilbert Space
6. Types of Quantum Feature Maps
7. ZZFeatureMap
8. PauliFeatureMap
9. Custom Feature Maps with Entanglement
10. Fidelity-Based Kernel Functions
11. Quantum Kernel Estimation
12. Expressivity and Complexity of Feature Maps
13. Measuring Quantum Kernels
14. Kernel Matrix Computation on QPUs
15. QML with Support Vector Machines
16. Regularization and Overfitting in Quantum Kernels
17. Visualization of Quantum Feature Space
18. Hybrid Classical-Quantum Kernel Methods
19. Software Tools and Frameworks
20. Conclusion

1. Introduction

Feature maps are a core component of many machine learning algorithms. In quantum machine learning, feature maps serve as the foundation for defining quantum kernels — similarity measures between data encoded in quantum states.

2. Classical Feature Maps and Kernels

• In classical ML, feature maps transform input data into high-dimensional spaces where patterns are more separable.
• Kernel trick: compute inner products in high-dimensional feature space without explicitly mapping data.

3. Why Quantum Feature Maps?

Quantum circuits naturally represent exponentially large Hilbert spaces, allowing compact, expressive embeddings of classical data:

• \( x \mapsto |\phi(x)
  angle \)
• Kernel = \( |\langle \phi(x) | \phi(x’)
  angle|^2 \)

4. Basics of Quantum Kernel Methods

• Input data is encoded into quantum states
• Similarity is computed via fidelity or overlap between these states
• Output is a kernel matrix used in downstream tasks (e.g., SVM)

5. Embedding Data into Hilbert Space

Data embedding is performed using parameterized unitary transformations:

• \( U(x) |0
  angle = |\phi(x)
  angle \)
• Goal: maximize class separability in quantum feature space

6. Types of Quantum Feature Maps

Feature maps vary in expressivity, entanglement, and depth. Common choices include:

• ZZFeatureMap
• PauliFeatureMap
• Custom parameterized unitaries

7. ZZFeatureMap

Uses Z-Z entanglement between qubits:

from qiskit.circuit.library import ZZFeatureMap
feature_map = ZZFeatureMap(feature_dimension=3, reps=2)

Good for datasets with feature interactions.

8. PauliFeatureMap

Constructed from exponentiated Pauli operators:

from qiskit.circuit.library import PauliFeatureMap
feature_map = PauliFeatureMap(feature_dimension=3, reps=2, paulis=['X', 'Y', 'Z'])

Provides richer encodings via Pauli strings.

9. Custom Feature Maps with Entanglement

Design unitary encodings with:

• Local rotations \( RY(x_i) \)
• Controlled gates for interactions
• Depth-optimized ansatz

10. Fidelity-Based Kernel Functions

Kernel value:

• \( k(x, x’) = |\langle \phi(x) | \phi(x’)
  angle|^2 \)
• Can be estimated via swap test or overlap circuits

11. Quantum Kernel Estimation

Qiskit supports built-in estimators:

from qiskit_machine_learning.kernels import QuantumKernel

• Kernel matrix computed over pairwise overlaps
• Backend: simulator or real QPU

12. Expressivity and Complexity of Feature Maps

• Deeper feature maps increase expressive power
• Risk: more noise and overfitting
• Balance circuit depth and generalization

13. Measuring Quantum Kernels

• Swap test: measures overlap of \( |\phi(x)
  angle \) and \( |\phi(x’)
  angle \)
• Measurement-based approximation using inverse circuits

14. Kernel Matrix Computation on QPUs

• Requires multiple pairwise evaluations
• Compute \( K_{ij} = |\langle \phi(x_i) | \phi(x_j)
  angle|^2 \)
• Use parallel or batched execution to improve efficiency

15. QML with Support Vector Machines

• Use quantum kernel matrix with classical SVM solver
• Implemented in Qiskit and PennyLane

16. Regularization and Overfitting in Quantum Kernels

• Regularize with SVM margin or dropout
• Visualize kernel matrix spectrum
• Limit feature map depth for stability

17. Visualization of Quantum Feature Space

• Use PCA or t-SNE on kernel matrix
• Plot kernel heatmap or class separation

18. Hybrid Classical-Quantum Kernel Methods

• Combine quantum kernel with classical preprocessor
• Use ensemble methods with multiple quantum feature maps
• Stack kernel SVM with classical layers

19. Software Tools and Frameworks

• Qiskit Machine Learning
• PennyLane kernels module
• sklearn + quantum kernel adapters

20. Conclusion

Quantum feature maps and kernels provide a compelling approach to harnessing quantum power in machine learning. By embedding classical data into quantum Hilbert spaces and exploiting fidelity-based similarity, QML gains access to powerful classification and pattern recognition capabilities.