Quantum Support Vector Machines: Leveraging Quantum Kernels for Pattern Classification

Table of Contents

1. Introduction
2. Classical Support Vector Machines (SVMs)
3. Motivation for Quantum SVMs
4. Quantum Kernels in SVMs
5. Quantum Feature Mapping
6. Quantum Kernel Matrix Estimation
7. SVM Decision Function with Quantum Kernels
8. Training Quantum SVM Models
9. Fidelity-Based Kernels and Overlap Circuits
10. Implementation with Qiskit
11. Implementation with PennyLane
12. Optimization and Regularization
13. Noise and Circuit Depth Considerations
14. Performance Benchmarks on Simulators
15. Running Quantum SVMs on Real Hardware
16. Visualization of Quantum Decision Boundaries
17. Multiclass Extensions of Quantum SVMs
18. Applications and Use Cases
19. Limitations and Future Directions
20. Conclusion

1. Introduction

Quantum Support Vector Machines (QSVMs) apply the principles of kernel-based learning using quantum computers. By embedding data into quantum Hilbert space, QSVMs can compute kernel matrices that capture complex, nonlinear relationships.

2. Classical Support Vector Machines (SVMs)

• SVMs find the hyperplane that maximizes margin between data classes.
• Kernel trick allows separation in high-dimensional space using:
• Linear, polynomial, Gaussian (RBF) kernels

3. Motivation for Quantum SVMs

• Quantum computers represent data in exponentially large Hilbert spaces.
• Inner products in these spaces can represent rich similarity metrics.
• QSVMs offer potential advantages in expressiveness and speed.

4. Quantum Kernels in SVMs

• A quantum kernel \( k(x, x’) \) is defined as the squared fidelity:
  \[
  k(x, x’) = |\langle \phi(x) | \phi(x’)
  angle|^2
  \]
• \( \phi(x) \) is a quantum embedding of classical input \( x \)

5. Quantum Feature Mapping

• Feature map \( U(x) \) is a parameterized quantum circuit that encodes data:

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

6. Quantum Kernel Matrix Estimation

• Construct quantum circuits that compute pairwise fidelity between states.
• Kernel matrix \( K_{ij} = k(x_i, x_j) \) is used by classical SVM solvers.

7. SVM Decision Function with Quantum Kernels

• Trained using a classical optimizer (e.g., scikit-learn SVC)
• Prediction:
  \[
  f(x) = \sum_i lpha_i y_i k(x_i, x) + b
  \]

8. Training Quantum SVM Models

• Compute kernel matrix (quantum)
• Solve dual SVM optimization (classical)
• Predict test labels using quantum kernel evaluations

9. Fidelity-Based Kernels and Overlap Circuits

• Overlap test estimates:
  \[
  |\langle \psi(x) | \psi(x’)
  angle|^2
  \]
• Can be implemented using inverse feature maps:

qc.compose(feature_map.inverse()).compose(feature_map)

10. Implementation with Qiskit

from qiskit_machine_learning.kernels import QuantumKernel
qkernel = QuantumKernel(feature_map=feature_map, quantum_instance=backend)

11. Implementation with PennyLane

import pennylane as qml
qml.kernels.square_fidelity(x1, x2, feature_map)

12. Optimization and Regularization

• Kernel SVM regularized with hyperparameter \( C \)
• Prevents overfitting in high-dimensional space

13. Noise and Circuit Depth Considerations

• Shorter feature maps preferred on NISQ devices
• Simulators allow deeper circuits for benchmarking

14. Performance Benchmarks on Simulators

• Use Iris, Wine, or Breast Cancer datasets
• Compare accuracy with classical SVMs
• Analyze kernel matrix separability

15. Running Quantum SVMs on Real Hardware

• Use IBM QPU or AWS Braket
• Shot noise and queue delays must be accounted for
• Use error mitigation where available

16. Visualization of Quantum Decision Boundaries

• Project 2D dataset using PCA
• Visualize kernel-induced boundary with contour plots

17. Multiclass Extensions of Quantum SVMs

• One-vs-rest strategy using binary QSVMs
• Ensemble quantum classifiers

18. Applications and Use Cases

• Fraud detection
• Biometric classification
• Quantum-enhanced recommendation systems

19. Limitations and Future Directions

• Kernel estimation scales quadratically with data size
• Limited by qubit count and fidelity
• Promising for small- to mid-sized structured datasets

20. Conclusion

Quantum Support Vector Machines offer a practical framework for exploring quantum advantage in classification tasks. By leveraging quantum kernels, QSVMs extend the power of classical SVMs into the quantum domain, paving the way for future breakthroughs in quantum-enhanced learning.